Physics II: Electricity & Magnetism

Rub a balloon on your hair and it sticks to a wall. Pull a sweater off in dry winter air and you hear a crackle. Both are electric charge at work - the same fundamental property that, scaled up, runs every motor and lights every screen. Charge is a basic property of matter, carried by the particles inside atoms. The proton carries one unit of positive charge, the electron an equal unit of negative charge, and the neutron none. The rule of interaction is simple and famous: like charges repel, opposite charges attract.

This lesson lays the foundation for the whole course. Every field, current, and wave you will meet later is built from charge and the force between charges. Getting clear on what charge is, how it is measured, and how it moves through matter makes everything that follows easier to picture.

Key idea: Electric charge is a fundamental property of matter that comes in two signs, and like signs repel while opposite signs attract.

What charge is, at the level of atoms

An atom has a dense central nucleus of protons and neutrons, surrounded by a cloud of far lighter electrons. The proton's charge is exactly equal in size to the electron's but opposite in sign, so an atom with equal numbers of each is electrically neutral. Ordinary matter is neutral to extraordinary precision, which is why you feel no electric force from a nearby wall most of the time.

Charge appears when this balance is disturbed. Strip an electron from a neutral atom and it becomes a positive ion; add an extra electron and it becomes a negative ion. In solids it is almost always the light, loosely held electrons that move, while the heavy positive nuclei stay locked in place. So a positively charged object is usually one that has lost electrons, not one that has gained protons.

The exact equality of the proton and electron charge magnitudes is one of the most precisely tested facts in physics. Experiments confirm the two match to better than one part in 10^20. If they differed even slightly, ordinary matter would carry a huge net charge and bulk objects would fly apart. Their perfect cancellation is what lets neutral matter exist at all.

Key idea: Neutral atoms hold equal positive and negative charge, and objects become charged when electrons are added or removed, leaving the nuclei fixed.

Two deep rules: conservation and quantization

Charge obeys two laws that never fail. First, charge is conserved: it is never created or destroyed, only transferred from one object to another. When the balloon becomes negative, your hair becomes equally positive - the electrons simply moved. The books balance exactly, so the total charge of an isolated system stays fixed no matter what happens inside it.

Second, charge is quantized: it comes in whole-number multiples of the elementary charge e = 1.60 x 10^-19 coulombs. Any charged object carries a charge of q = n e for some integer n. You never find half an electron's worth of charge on an isolated object. Robert Millikan measured this graininess directly in his oil-drop experiment around 1910, finding that every droplet's charge was a multiple of one basic unit.

The unit of charge is the coulomb (C), and one coulomb is an enormous amount - about 6.24 x 10^18 elementary charges. Everyday static charges are far smaller, measured in nanocoulombs (10^-9 C) or microcoulombs (10^-6 C). That mismatch is worth remembering: a dramatic spark off a doorknob may involve only a few billionths of a coulomb.

Key idea: Total charge is conserved in every process, and charge is quantized in integer multiples of the elementary charge e.

Why negative and positive? A note on convention

The labels are a historical accident. Benjamin Franklin, working in the 1750s long before anyone knew about electrons, chose to call the charge left on glass rubbed with silk positive and the charge on amber or rubber negative. When the electron was finally identified around 1897, it turned out to carry what Franklin had named negative charge.

The naming is arbitrary, but it is now universal, so we keep it. What matters physically is that there are exactly two kinds of charge and that they are opposites. Nothing in the physics would change if the names were swapped; only the signs in our bookkeeping would flip.

Conductors and insulators

Materials differ in how freely charge moves through them. In a conductor such as copper or any metal, some electrons are loosely held and drift easily; charge spreads across the whole object almost instantly. In an insulator such as glass, rubber, or plastic, electrons stay bound to their atoms, so charge placed on one spot stays put. A third class, semiconductors like silicon, sit in between and are the basis of every computer chip.

A useful picture of a metal is a rigid lattice of positive ions bathed in a shared sea of mobile electrons. Those free electrons are what carry current in a wire, and they are also why any excess charge on a metal races to the surface and distributes itself in moments. An insulator has no such sea, so charge you deposit stays trapped where you put it.

This distinction also explains grounding. Connecting a charged conductor to the Earth, itself an enormous conductor, gives excess electrons somewhere to go, and the object returns to neutral. It also explains why static shocks are a winter phenomenon: dry air is a good insulator, so charge that builds up cannot leak away through moisture on surfaces, and it accumulates until it discharges as a spark.

Key idea: Conductors let charge move freely and spread to their surfaces, insulators hold charge in place, and grounding drains excess charge to the Earth.

Three ways to charge an object

• Friction: rubbing two different materials transfers electrons from one to the other, as with the balloon and hair.
• Conduction: touching a charged object to a neutral conductor lets charge flow until both share it.
• Induction: bringing a charged rod near (without touching) a conductor pushes its electrons to one side, separating charge. If you then ground the far side, you can leave the conductor with a net charge of the opposite sign - all without contact.

Charging by friction follows a rough ordering called the triboelectric series, which ranks materials by how strongly they grab electrons. Rub two materials together and the one higher on the grabbing scale ends up negative; the other, having lost electrons, ends up positive. Human hair gives up electrons readily, which is why the balloon rubbed on hair comes away negative and the hair is left positive.

Induction is the subtlest of the three because the object never touches the charged rod. The rod's field merely rearranges the conductor's own electrons. Removing the ground first and the rod second locks in a net charge whose sign is opposite to the rod - a fact the lesson quiz asks you to trace through step by step.

Key idea: Objects gain charge by friction, conduction, or induction, and only induction leaves a net charge without ever touching the source.

Polarization: why a charged object attracts neutral things

Here is a puzzle worth solving. A charged balloon attracts small scraps of paper, yet paper is neutral, with no net charge to pull on. The resolution is polarization. The balloon's field tugs the paper's positive and negative charges in opposite directions, ever so slightly, so the near side of each scrap becomes a little richer in the opposite charge.

Because the force falls off with distance, the attraction on the closer, opposite charge slightly exceeds the repulsion on the farther, like charge. The scrap feels a small net pull toward the balloon. In a conductor the electrons shift freely to make this happen; in an insulator like paper the molecules themselves stretch into tiny dipoles. Either way, a charged object attracts a neutral one.

Key idea: A charged object polarizes nearby neutral matter, drawing opposite charge closer, so it attracts even objects that carry no net charge.

Detecting charge: the electroscope

The classic instrument for detecting charge is the electroscope: a metal knob connected by a rod to two thin metal leaves. Touch a charged object to the knob and charge spreads down to the leaves. Because both leaves receive the same sign of charge, they repel each other and visibly spring apart. The wider they splay, the more charge is present.

An electroscope also demonstrates induction beautifully. Bring a charged rod near the knob without touching, and the leaves diverge even though the device stays neutral overall, because charge has merely separated inside it. Take the rod away and the leaves fall back together. This simple tool made the invisible visible and let early experimenters compare charges long before meters existed.

Key idea: An electroscope reveals charge because like-charged leaves repel, and it shows induction when a nearby charge separates the device's own charge.

Worked example: counting electrons

Given: an object carries a charge of -3.2 x 10^-9 C (that is, -3.2 nC). Find: how many excess electrons it holds.

Solution: The number of elementary charges is the total charge divided by the charge per electron:
n = q / e = (3.2 x 10^-9) / (1.60 x 10^-19) = 2.0 x 10^10 electrons.
So the object has 20 billion excess electrons. The charge is negative, so these are extra electrons, not a shortage.

Worked example: charge from lost electrons

Given: a plastic rod is rubbed and ends up with a charge of +4.8 x 10^-9 C. Find: how many electrons it lost.

Solution: A positive charge means missing electrons, and the count is again the charge divided by e:
n = q / e = (4.8 x 10^-9) / (1.60 x 10^-19) = 3.0 x 10^10 electrons.
The rod gave up 30 billion electrons to whatever it was rubbed against. Notice how small a charge this is in coulombs, yet how staggering the electron count - a reminder of how tiny the elementary charge really is.

Common misconceptions

• A positive object has extra protons. Almost always it has lost electrons; the heavy nuclei do not move in solids.
• Charging creates charge. Charging only transfers existing charge; the total is conserved, and one object gains exactly what another loses.
• Only charged objects feel electric forces. Neutral objects are attracted too, through polarization of their own charges.
• Insulators cannot be charged. They charge readily by friction; what they cannot do is let that charge flow away or spread.
• Charge can take any value. Isolated charge always comes in whole multiples of the elementary charge e.

Recap

• Charge is a property of matter with two signs; like charges repel and opposite charges attract.
• Charge is conserved and quantized in units of e = 1.60 x 10^-19 C, measured in coulombs.
• Conductors let charge move and spread; insulators hold it in place; grounding drains it to the Earth.
• Objects are charged by friction, conduction, or induction, and polarization lets a charged object attract neutral matter.
• To count electrons behind a charge, divide the charge by e.

Sources

1. OpenStax. (2016). 5.1 Electric charge. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 5.2 Conductors, insulators, and charging by induction. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2022). 18.1 Static electricity and charge: Conservation of charge. In College Physics 2e. Rice University. openstax.org
4. Nave, R. (n.d.). Conductors and insulators. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
5. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 1: Electromagnetism. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu
6. National Institute of Standards and Technology. (n.d.). CODATA value: elementary charge. The NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov

How strong is the force between two charges? The answer was measured by Charles-Augustin de Coulomb in the 1780s and is now called Coulomb's law. For two point charges q1 and q2 separated by a distance r, the magnitude of the force each feels is

F = k |q1| |q2| / r^2

where k = 8.99 x 10^9 N m^2 / C^2 is Coulomb's constant. The force is along the line joining the charges - repulsive if the signs match, attractive if they differ. Notice the structure: the force grows with the product of the charges and falls off as the inverse square of the distance. Double the separation and the force drops to one quarter.

Key idea: Coulomb's law gives an inverse-square force proportional to the product of two charges, directed along the line between them.

Reading the law: magnitude and direction

A reliable habit is to split the problem in two. First use the magnitudes of the charges in the formula to get the size of the force. Then look at the signs separately to decide the direction: like signs push apart, opposite signs pull together. Keeping magnitude and direction in separate steps avoids most sign errors.

The inverse-square shape deserves a second look. Because r appears squared in the denominator, the force is fiercely sensitive to distance. Move the charges three times as far apart and the force falls to one ninth; bring them to one third the distance and the force grows nine times. This steep falloff is the same mathematics that governs gravity and the brightness of a lamp.

Key idea: Compute the force magnitude from the charge magnitudes, then set direction from the signs, and remember the force changes with the square of distance.

How Coulomb measured it: the torsion balance

Coulomb did not simply guess the inverse-square form. He built a torsion balance, a delicate apparatus in which a charged ball hangs from a thin fiber near a second charged ball. The electric force twists the fiber, and the twist angle measures the force. By varying the separation and reading the twist, he showed the force weakened as the square of the distance.

The same instrument, in the hands of Henry Cavendish, later measured the far weaker force of gravity. That two such different forces share the identical mathematical form is a hint of deep structure in nature, one that runs through this entire course.

Coulomb's constant and the permittivity of space

The constant k is often written in a second form that will matter when we reach Gauss's law: k = 1 / (4 pi epsilon_0). Here epsilon_0 = 8.85 x 10^-12 C^2 / (N m^2) is the permittivity of free space, a fundamental property of the vacuum itself. The two constants carry the same information; k is just the more convenient package for force calculations.

The units of k, newton meters squared per coulomb squared, are exactly what is needed to turn coulombs and meters into newtons. Checking that the units of an answer come out right is a quick and powerful way to catch mistakes before trusting a number.

The sheer size of k explains why static electricity feels so strong. Two charges of just one coulomb held one meter apart would repel with about nine billion newtons, the weight of a million tonnes. That is why isolated charges of a full coulomb never occur in ordinary life; the forces would be unimaginable. Real charges stay in the micro- and nanocoulomb range.

The same shape as gravity, but far stronger

Coulomb's law has exactly the form of Newton's law of gravitation, with charge in place of mass. But the electric force is vastly stronger. Between a proton and an electron, the electric attraction is about 10^39 times the gravitational attraction. Gravity wins on the scale of planets only because large objects are nearly neutral - their positive and negative charges cancel - while mass always adds up.

The ratio is worth seeing concretely. Taking the electric force k e^2 / r^2 over the gravitational force G m_p m_e / r^2, the distance cancels and the constants give roughly 2 x 10^39. No matter how far apart the two particles sit, electricity outmuscles gravity by that staggering factor. Gravity rules the cosmos only because there is no negative mass to cancel it.

Key idea: The electric and gravitational forces share an inverse-square form, but electricity is about 10^39 times stronger, and gravity dominates large scales only because bulk matter is neutral.

Superposition: forces add as vectors

When more than two charges are present, the force on any one charge is the vector sum of the forces from each of the others taken one at a time. This is the principle of superposition. You compute each pairwise Coulomb force, then add them like the vectors you know from mechanics - by components. Nothing about a third charge changes the force between the first two; each pair acts independently.

A dependable procedure keeps the bookkeeping clean. First, draw the charge and mark the direction of the force from each other charge. Second, compute each force magnitude with Coulomb's law. Third, break each force into x and y components. Fourth, add the components separately, then recombine into a magnitude and angle. On a straight line the method collapses to simple addition with signs.

Key idea: Superposition means the net force is the vector sum of independent pairwise Coulomb forces, best handled by adding components.

The point-charge idealization

Coulomb's law is written for point charges, idealized charges with no size. Real objects have extent, so when does the law apply? It works exactly when the objects are small compared with their separation, so their shape does not matter. It also works for any uniformly charged sphere: from outside, such a sphere pulls and pushes exactly as if all its charge sat at its center.

This is why we can treat charged spheres, ions, and even planets of charge as points in most problems. When two charged objects are large and close, though, their charge rearranges and the simple formula becomes only an approximation. Knowing the idealization's limits is part of using it well.

Finding where the force vanishes

A classic question asks where a test charge feels no net force. Between two equal positive charges, symmetry answers it at once: at the midpoint the two repulsions are equal and opposite, so they cancel and the net force is zero. Nudge the test charge off center and the nearer charge wins, pushing it back, so the midpoint is a balance point.

When the two source charges are unequal, the null point shifts toward the smaller charge, where its stronger nearby push can match the larger charge's weaker distant push. For two charges of opposite sign, no balance point lies between them at all; it sits outside the pair, beyond the smaller charge. Reasoning through these cases builds real fluency with superposition.

Key idea: A net force can vanish where opposing Coulomb forces cancel, at the midpoint for equal charges and shifted toward the smaller charge otherwise.

Worked example: two charges on a line

Given: a charge q1 = +3.0 microC at the origin and q2 = -5.0 microC at x = 0.20 m. (1 microC = 10^-6 C.) Find: the force on q1.

Solution: Use magnitudes first.
F = k |q1| |q2| / r^2 = (8.99 x 10^9)(3.0 x 10^-6)(5.0 x 10^-6) / (0.20)^2
= (8.99 x 10^9)(1.5 x 10^-11) / 0.04 = 0.1349 / 0.04 = 3.4 N.
The signs are opposite, so the force is attractive: q1 is pulled toward q2, in the +x direction.

The force on q2 has the same magnitude, 3.4 N, in the -x direction (Newton's third law). The two charges always feel equal and opposite forces, whatever their sizes, because each acts on the other through the same pair interaction.

Worked example: superposition of three charges

Given: q1 = +2.0 microC at x = 0, q2 = +2.0 microC at x = 0.10 m, and we want the force on q3 = +1.0 microC placed at x = 0.20 m.
Solution: Both q1 and q2 are positive like q3, so both push q3 in the +x direction.

From q2 (r = 0.10 m): F = (8.99 x 10^9)(2.0 x 10^-6)(1.0 x 10^-6)/(0.10)^2 = 1.8 N. From q1 (r = 0.20 m): F = (8.99 x 10^9)(2.0 x 10^-6)(1.0 x 10^-6)/(0.20)^2 = 0.45 N. Both point the same way, so add: 1.8 + 0.45 = 2.25 N in the +x direction.

Worked example: adding forces as vectors

Given: a charge q3 = +2.0 microC at the origin. A charge q1 = +4.0 microC sits on the x-axis at 0.30 m, and q2 = +4.0 microC sits on the y-axis at 0.30 m. Find: the net force on q3.

Solution: Each pushes q3 away. The magnitude from either is
F = (8.99 x 10^9)(4.0 x 10^-6)(2.0 x 10^-6)/(0.30)^2 = 0.80 N.
q1 pushes q3 in the -x direction and q2 pushes it in the -y direction, so the two forces are perpendicular.

Add them as vectors: F_net = sqrt(0.80^2 + 0.80^2) = sqrt(1.28) = 1.13 N, aimed at 45 degrees between the -x and -y axes. Equal perpendicular forces always combine into a resultant that splits the angle between them.

Where this leads: the field idea

Coulomb's law is powerful, but computing forces from many charges quickly grows tedious, and it says nothing about how one charge "knows" another is there. The next lesson reframes the whole picture. Instead of asking about the force between pairs, we say each charge fills the surrounding space with an electric field, and any other charge simply responds to the field at its own location.

That shift, from action at a distance to a field that mediates the force, is one of the most important moves in physics. Everything in this course after Coulomb's law - fields, potential, circuits, magnetism, and light itself - is built on the field concept. Coulomb's law remains the bedrock, but the field is the language.

Common misconceptions

• The bigger charge feels the bigger force. By Newton's third law both charges feel forces of equal magnitude, whatever their sizes.
• A third charge changes the force between the first two. Each pair interacts independently; superposition adds the separate forces.
• Forces from several charges add like plain numbers. They add as vectors; only on a single line does that reduce to signed addition.
• Doubling the distance halves the force. The force falls as 1/r^2, so doubling distance cuts it to one quarter.
• Coulomb's law needs perfect points. It also holds exactly outside any uniformly charged sphere.

Recap

• Coulomb's law: F = k |q1| |q2| / r^2 with k = 8.99 x 10^9 N m^2/C^2, directed along the line of centers.
• Find magnitude from the charge magnitudes, then set direction from the signs.
• The electric force is about 10^39 times stronger than gravity but is hidden at large scale by neutrality.
• Superposition adds pairwise forces as vectors, handled cleanly by components.
• Point charges and uniformly charged spheres both obey the law exactly from outside.

Sources

1. OpenStax. (2016). 5.3 Coulomb's law. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2022). 18.3 Coulomb's law. In College Physics 2e. Rice University. openstax.org
3. Nave, R. (n.d.). Electric forces. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
4. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 4: Electrostatics. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu
5. O'Connor, J. J., & Robertson, E. F. (n.d.). Charles Augustin Coulomb. MacTutor History of Mathematics Archive, University of St Andrews. mathshistory.st-andrews.ac.uk
6. National Institute of Standards and Technology. (n.d.). CODATA value: vacuum electric permittivity. The NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov

Coulomb's law tells you the force between two charges, but it is often more useful to think of a single charge as filling the space around it with an electric field. The field is what a second charge would respond to. Formally, the electric field E at a point is the force per unit charge that a small positive test charge q0 would feel there:

E = F / q0, with units of newtons per coulomb (N/C).

The field is a vector: it has a direction at every point, defined as the direction of the force on a positive charge. Once you know the field at a location, the force on any charge q placed there is simply F = q E. A positive charge feels a force along E; a negative charge feels a force opposite to E.

Key idea: The electric field is the force per unit positive charge at a point, a vector in N/C, and any charge placed there feels F = q E.

Why bother with a field?

Replacing forces with a field may look like extra machinery, but it pays off in two ways. Practically, once you map the field of a complicated charge arrangement, finding the force on any newcomer is a one-step multiplication by its charge. You solve the hard part once.

Conceptually, the field solves a puzzle that troubled physicists for centuries: how can one charge push another across empty space with nothing in between? The field answer is that a charge alters the space around it, and a second charge responds only to the field at its own location. The interaction becomes local, passed along by the field rather than reaching instantly across a gap.

This is more than a bookkeeping trick. Fields carry energy and momentum, and changes in them travel at a finite speed, the speed of light. That last fact, which we reach in the final lesson, means the field is as real as the charges that make it.

Key idea: The field makes force calculations local and reusable, and it is a physical entity that stores energy and propagates at a finite speed.

The test charge must be small

The definition uses a test charge q0, and it must be tiny for a reason. A real probe charge exerts its own force back on the source charges and can push them around, changing the very field you are trying to measure. Imagine the field is set by charges on a soft conductor: a large probe would rearrange them.

To avoid this, we picture q0 shrinking toward zero, so it samples the field without disturbing it. The field is a property of the source charges alone; the test charge only reveals it. That is why the field exists at a point even when no charge sits there to feel it.

Field of a point charge

Dividing Coulomb's law by the test charge gives the field of a single point charge Q at distance r:

E = k |Q| / r^2

It points away from a positive Q and toward a negative Q. Like the force, it obeys superposition: the total field from several charges is the vector sum of their individual fields.

The field is radial, pointing straight out from or in toward the charge, and it weakens as the inverse square of distance, just like the force it came from. At twice the distance the field is one quarter as strong. Notice the field's size does not depend on any test charge; it belongs entirely to the source Q.

Key idea: A point charge produces a radial inverse-square field E = k|Q|/r^2, outward from positive charge and inward toward negative.

Superposition of fields

Because fields add as vectors, mapping the field of several charges means summing their separate contributions at each point. The result can be rich. Two opposite charges form an electric dipole, whose field curls from the positive charge around to the negative one, a pattern that appears throughout chemistry and antenna design.

Two equal charges of the same sign give a different picture. Exactly midway between them, their fields are equal in size but opposite in direction, so they cancel and the field is zero. A test charge placed there feels no push. Such null points are a direct consequence of vector superposition and are worth learning to spot.

Key idea: Fields superpose as vectors, producing dipole patterns from opposite charges and null points where equal like-charge fields cancel.

Beyond points: fields of extended objects

Real charge is often smeared over a line, a surface, or a volume rather than sitting at a single point. In principle you handle these by superposition too: slice the object into tiny pieces, treat each as a point charge, and add all their fields as vectors. In practice the sum becomes an integral, but the idea is unchanged.

Two results are worth previewing because they recur throughout the course. A long straight line of charge produces a field that falls off as 1/r, more slowly than a point charge. A large flat sheet of charge produces a field that is uniform and does not weaken with distance at all. The next lesson derives both cleanly with Gauss's law, but they already follow from adding point-charge fields.

Key idea: Extended charges build their fields by superposing point-charge contributions, giving a 1/r field for a line and a uniform field for a large sheet.

Uniform fields

Not every field spreads out from a point. Between two large, oppositely charged parallel plates, the field is very nearly uniform: the same strength and direction everywhere in the gap, pointing straight from the positive plate to the negative one. The field lines there are evenly spaced parallel arrows.

Uniform fields are the workhorse of the next several lessons. They accelerate charges in a straight line, store energy in capacitors, and make the link between field and voltage especially simple. Whenever you see parallel plates, expect a uniform field in the space between them.

How a charge moves in a field

Put a charge in a field and it feels a force F = qE, so by Newton's second law it accelerates at a = qE/m. In a uniform field that acceleration is constant, and the motion becomes a problem you already know from mechanics. A charge released from rest speeds up in a straight line along the field, gaining energy as it goes.

Fire a charge across a uniform field and something familiar happens: it follows a parabola, exactly like a ball thrown sideways under gravity. The steady sideways force curves the path while the forward motion continues. This is precisely how old television tubes and inkjet printers steer beams of charge, deflecting them with fields to paint an image or a page.

Key idea: A charge in a field accelerates at a = qE/m, moving in a straight line along a uniform field or in a parabola when launched across it.

Field lines

We picture fields with field lines. They start on positive charges and end on negative charges, never cross, and are drawn denser where the field is stronger. The arrow on a line shows the field direction; the tangent to a curved line gives the direction at that point.

Two more rules make the pictures quantitative. The number of lines leaving a charge is proportional to the size of the charge, so a charge of 2Q sprouts twice as many lines as Q. And where lines crowd together the field is strong, while where they spread apart it is weak, so line density is a visual readout of field magnitude.

The rule that lines never cross has a simple reason. If two lines crossed, the field would point in two directions at once at the crossing point, which is impossible. At every point the field has exactly one direction, so exactly one line passes through.

Key idea: Field lines run from positive to negative charge, never cross, and pack together where the field is strong, so their density pictures the field's magnitude.

Worked example: field and force

Given: a point charge Q = +4.0 microC. Find: (a) the field at 0.30 m, and (b) the force on a -2.0 nC charge placed there.

Solution (a): E = kQ/r^2 = (8.99 x 10^9)(4.0 x 10^-6)/(0.30)^2 = 35960/0.09 = 4.0 x 10^5 N/C, pointing away from Q.
Solution (b): F = qE = (2.0 x 10^-9)(4.0 x 10^5) = 8.0 x 10^-4 N. Because the charge is negative, the force points toward Q, opposite to E.

Worked example: adding two fields

Given: a point P with q1 = +4.0 nC at 0.20 m to its left and q2 = -4.0 nC at 0.20 m to its right. Find: the net field at P.

Solution: Each charge gives E = k|Q|/r^2 = (8.99 x 10^9)(4.0 x 10^-9)/(0.20)^2 = 899 N/C. The positive charge on the left points its field to the right; the negative charge on the right also pulls the field to the right. They add:
E_net = 899 + 899 = 1.8 x 10^3 N/C, pointing from the positive toward the negative charge.

This midpoint result captures the dipole in miniature: the field between an opposite pair points from the positive charge to the negative one, and the two contributions reinforce rather than cancel. Contrast it with two like charges, where the midpoint field would be zero. Reading which case you are in before calculating saves time and prevents sign mistakes.

Common misconceptions

• The field needs a charge sitting in it to exist. The field is set by the source charges alone and is there whether or not anything feels it.
• Field and force are the same thing. The field is force per unit charge; multiply by a charge to get a force.
• The field points the way a negative charge is pushed. By definition it points the way a positive charge is pushed; a negative charge feels the opposite.
• Field lines are real paths a charge travels. They are a map of direction, not trajectories.
• Two field lines can cross. They never do, since the field has a single direction at each point.

Recap

• The electric field is force per unit positive charge, E = F/q0, measured in N/C.
• A point charge gives a radial inverse-square field E = k|Q|/r^2, and fields superpose as vectors.
• Field lines run from positive to negative charge, never cross, and crowd where the field is strong.
• Between parallel plates the field is uniform, the setting for the lessons ahead.
• The force on a charge in a field is F = qE, along E for positive charge and opposite for negative.

Sources

1. OpenStax. (2016). 5.4 Electric field. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 5.5 Calculating electric fields of charge distributions. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 5.6 Electric field lines. In University Physics Volume 2. Rice University. openstax.org
4. Nave, R. (n.d.). Electric field. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
5. Nave, R. (n.d.). Electric field, spherical geometry. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
6. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 1: Electromagnetism. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu
7. Massachusetts Institute of Technology. (2007). 8.02 Physics II: Electricity and magnetism [Course materials]. MIT OpenCourseWare. ocw.mit.edu

Adding up point-charge fields with superposition works, but it is tedious. For charge distributions with enough symmetry, there is a far more elegant tool: Gauss's law. It relates the field on a closed surface to the charge inside, and it is one of the four Maxwell equations that summarize all of electromagnetism.

Gauss's law does not contain any physics beyond Coulomb's law; the two are mathematically equivalent. What it offers is a change of viewpoint that turns certain hard field calculations into one-line answers. Learning to recognize when that shortcut applies is the real skill of this lesson.

Key idea: Gauss's law restates Coulomb's law in terms of a closed surface, giving an elegant shortcut whenever the charge arrangement is symmetric.

Electric flux

Electric flux measures how much field passes through a surface - picture field lines poking through a net. For a flat area A in a uniform field E, the flux is

Phi = E A cos(theta)

where theta is the angle between the field and the normal (the direction perpendicular to the surface). Flux is maximum when the field is perpendicular to the surface (theta = 0, so cos = 1) and zero when the field skims along the surface (theta = 90 degrees). Its units are N m^2 / C.

It helps to treat area as a vector pointing along the normal. Then flux is the field's component along that normal times the area. A tilted surface catches fewer lines, just as a window lets in less rain when it is turned edge-on to a slanting shower. Only the field passing straight through counts.

For a closed surface we agree that the normal points outward. Field lines leaving the surface then count as positive flux and lines entering count as negative. A surface enclosing nothing has every line that enters also leaving, so its positive and negative contributions cancel to zero.

Key idea: Flux is the field passing perpendicularly through a surface, Phi = E A cos(theta), counted positive outward through a closed surface.

Gauss's law

Gauss's law states that the total electric flux through any closed surface (a "Gaussian surface") equals the charge enclosed divided by a constant:

Phi_total = Q_enclosed / epsilon_0

where epsilon_0 = 8.85 x 10^-12 C^2 / (N m^2) is the permittivity of free space, related to Coulomb's constant by k = 1 / (4 pi epsilon_0). The law says something remarkable: the flux depends only on the enclosed charge, not on how it is arranged inside and not on any charges outside the surface. Charges outside contribute zero net flux, because every line that enters the closed surface also leaves it.

Why should this be true? It traces directly to the inverse-square law. The field of a point charge weakens as 1/r^2, but the area of a surrounding sphere grows as r^2. The two effects cancel exactly, so the same total number of field lines crosses any sphere around the charge, near or far. Gauss's law is the inverse-square law in disguise.

Key idea: The net flux through any closed surface equals the enclosed charge over epsilon_0, independent of arrangement, a direct consequence of the inverse-square law.

Gauss's law among Maxwell's equations

This is the first of the four Maxwell equations that the final lesson assembles. It is the electric Gauss law, and it says that electric field lines begin and end on charges. Positive charge is a source from which lines spring; negative charge is a sink into which they dive.

There is a magnetic partner to this law, and its content is strikingly different: the net magnetic flux through any closed surface is always zero. That difference encodes a deep fact about nature, that isolated magnetic poles do not exist, which we meet in the magnetism modules. For now, notice how much a single statement about flux can say about the structure of a field.

Key idea: The electric Gauss law is one of Maxwell's four equations and states that field lines start and end only on charges.

Using symmetry

Gauss's law becomes a calculation tool when you choose a surface on which E is constant and either parallel or perpendicular to the surface everywhere. Three classic results follow:

• Sphere / point charge: a spherical surface of radius r around charge Q gives E (4 pi r^2) = Q/epsilon_0, so E = kQ/r^2 - recovering Coulomb's law.
• Infinite line of charge with charge per length lambda: E = lambda / (2 pi epsilon_0 r), falling off as 1/r.
• Infinite sheet of charge with charge per area sigma: E = sigma / (2 epsilon_0) - a uniform field that does not depend on distance.

Each case follows the same recipe. Pick a Gaussian surface matched to the symmetry - a sphere for a point, a cylinder for a line, a pillbox for a sheet. On it the field has constant magnitude and a fixed angle to the surface, so the flux integral collapses to E times an area. Setting that equal to Q_enclosed / epsilon_0 and solving for E finishes the job.

The power here is dramatic. Finding the field of an infinite sheet by adding point-charge contributions would take a demanding integral; Gauss's law delivers sigma / (2 epsilon_0) in a few lines. The catch is that the trick only works when symmetry makes the field constant on a well-chosen surface.

Key idea: Choose a Gaussian surface matching the symmetry so the flux becomes E times an area, then solve E times area = Q_enclosed/epsilon_0 for the field.

Spelling out the line and the sheet

For an infinite line of charge, wrap a cylinder of radius r and length L coaxial with the line. The field points radially outward, so no flux passes through the flat end caps, and the curved side has area 2 pi r L. Gauss's law reads E (2 pi r L) = (lambda L)/epsilon_0, and the L cancels to give E = lambda/(2 pi epsilon_0 r).

For an infinite sheet, use a pillbox that pokes through the sheet with a face of area A on each side. Flux leaves through both faces, so the enclosed charge sigma A gives E (2A) = (sigma A)/epsilon_0, hence E = sigma/(2 epsilon_0). The factor of two comes from field escaping both faces of the box.

Key idea: A coaxial cylinder yields the line field and a pillbox yields the sheet field, with the pillbox's factor of two coming from its two faces.

Conductors in equilibrium

Gauss's law explains a key fact about conductors: in electrostatic equilibrium the field inside a conductor is zero, and any excess charge sits entirely on the outer surface. If a field existed inside, the free electrons would move until it did not. This is why a car or a metal cage shields its interior from external fields.

Draw a Gaussian surface just inside the metal, where E is zero, and the enclosed charge must be zero too. So excess charge cannot live in the bulk; it is forced onto the surface. Just outside the surface the field is perpendicular to the metal, because any sideways component would drive the surface charges into motion until it vanished.

This shielding is the principle of the Faraday cage. A hollow conductor keeps external fields out of its cavity, protecting sensitive electronics and the passengers of a car struck by lightning. Remarkably, the shielding works no matter how strong the outside field, so long as the conductor can supply the surface charge to cancel it within.

Key idea: In a conductor at equilibrium the interior field is zero and excess charge lies on the surface, so a hollow conductor shields its interior as a Faraday cage.

The field just outside a conductor

Gauss's law also pins down the field right at a charged conductor's surface. Slip a tiny pillbox half inside the metal and half outside. Inside, the field is zero, so no flux leaves the inner face. Outside, the field is perpendicular to the surface, so all the flux leaves the outer face of area A.

The enclosed surface charge is sigma A, so E A = (sigma A)/epsilon_0, giving E = sigma/epsilon_0 just outside the conductor. Notice this is twice the field of an isolated sheet, because here all the field is pushed out to one side rather than splitting both ways. Small differences like this factor of two reward careful attention to which surface encloses what.

Key idea: Just outside a charged conductor the field is E = sigma/epsilon_0, twice an isolated sheet's field because it emerges on one side only.

Worked example: flux through a box

Given: a closed box encloses a net charge of +8.85 x 10^-9 C. Find: the total electric flux through the box.

Solution: Phi = Q/epsilon_0 = (8.85 x 10^-9) / (8.85 x 10^-12) = 1.0 x 10^3 N m^2/C. The shape of the box does not matter - only the enclosed charge. A crumpled bag holding the same charge would pass exactly the same total flux.

Worked example: field of a charged sphere

Given: a small sphere carries Q = +5.0 nC. Find: the field 0.10 m from its center, outside the sphere.

Solution: Wrap a Gaussian sphere of radius 0.10 m around it. By symmetry E (4 pi r^2) = Q/epsilon_0, which rearranges to E = kQ/r^2:
E = (8.99 x 10^9)(5.0 x 10^-9)/(0.10)^2 = 44.95/0.01 = 4.5 x 10^3 N/C, pointing radially outward. From outside, the sphere behaves exactly like a point charge at its center. Inside a uniformly charged sphere the story differs, since a smaller Gaussian sphere encloses only part of the charge, and the field there actually grows with radius.

Common misconceptions

• Charges outside the surface change the flux. They do not; only enclosed charge sets the net flux, since outside charges send in as many lines as they send out.
• Gauss's law gives the field for any shape. It always holds, but it only solves for E when symmetry makes the field constant on a chosen surface.
• Zero flux means zero field. A surface can have equal inward and outward flux, so net flux is zero while the field is nonzero everywhere on it.
• Excess charge spreads through a conductor's volume. It resides entirely on the outer surface.
• Flux depends on where inside the charge sits. It depends only on how much charge is enclosed, not its position.

Recap

• Electric flux is field through a surface, Phi = E A cos(theta), positive outward through a closed surface.
• Gauss's law: Phi_total = Q_enclosed / epsilon_0, equivalent to the inverse-square law.
• With symmetry it yields the sphere, line, and sheet fields in a few lines each.
• Inside a conductor at equilibrium the field is zero and charge lives on the surface, giving Faraday-cage shielding.
• Only enclosed charge matters; the surface's shape and outside charges do not affect net flux.

Sources

1. OpenStax. (2016). 6.1 Electric flux. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 6.2 Explaining Gauss's law. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 6.3 Applying Gauss's law. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 6.4 Conductors in electrostatic equilibrium. In University Physics Volume 2. Rice University. openstax.org
5. Nave, R. (n.d.). Gauss's law. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
6. Nave, R. (n.d.). Gaussian surfaces. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 5: Application of Gauss' law. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu

Lifting a book stores gravitational potential energy; the same idea works for charge. Moving a positive charge against an electric field takes work and stores electric potential energy. Because tracking energy is often easier than tracking force, this leads to one of the most-used quantities in all of physics: voltage.

Energy has a decisive advantage over force: it is a scalar, a single number with no direction to track. Where fields and forces demand vector bookkeeping, energy and potential add as plain numbers. That simplicity is why engineers speak of volts far more often than of newtons per coulomb, even though the two describe the same underlying physics.

Key idea: Electric potential energy tracks the work to assemble charges, and because energy is a scalar it is often easier to use than force or field.

Potential energy of two charges

For two point charges the electric potential energy is

U = k q1 q2 / r

Note it goes as 1/r, not 1/r^2 like the force. The sign carries meaning: for like charges U is positive (you had to do work to push them together, and they will fly apart if released); for opposite charges U is negative (they are bound, and energy is needed to separate them).

This energy is the work stored when the charges are brought from far apart to a separation r. Release two like charges and that stored energy converts to kinetic energy as they rush apart, exactly as a compressed spring launches a block. Opposite charges, by contrast, sit in an energy well and must be pulled apart, just as a planet is bound to the Sun.

For three or more charges, the total energy is the sum over every pair. Assemble the charges one at a time and add the work each new arrival takes against those already in place. The bookkeeping is again scalar, so you simply add signed k q_i q_j / r_ij terms for all pairs, with no directions to track.

Key idea: Two charges store energy U = k q1 q2 / r, positive and unbound for like charges, negative and bound for opposite charges, and many charges add pairwise.

The zero of potential energy

Where is U equal to zero? By the formula, U vanishes as r grows without limit, so we take the reference point to be infinite separation. Two charges infinitely far apart have zero shared energy, and bringing them closer either stores energy or releases it depending on their signs.

Only differences in energy have physical meaning; the choice of where to call zero is a convention. This mirrors gravity, where we freely pick the ground, a tabletop, or sea level as the zero of height. What matters is the change as a charge moves, never the absolute label.

Electric potential: energy per charge

The electric potential V at a point is the potential energy per unit charge that a test charge would have there:

V = U / q, measured in volts (V), where 1 volt = 1 joule per coulomb.

Potential is a scalar - just a number at each point, with no direction - which makes it much easier to work with than the field vector. For a single point charge, V = kQ/r (this can be positive or negative depending on the sign of Q). The potential from several charges is the plain algebraic sum of each kQ/r, adding signs, with no vectors involved.

This scalar sum is a genuine labor saver. To find the potential of a dozen charges you add a dozen signed numbers; to find their field you would juggle a dozen vectors. Once the potential map is known, the field can be recovered from how steeply the potential changes, as we see shortly.

Key idea: Potential is energy per unit charge in volts, a scalar equal to the signed sum of kQ/r from each source charge.

Equipotential surfaces

Points that share the same potential form an equipotential surface. Around a point charge these are spheres; between parallel plates they are flat sheets. Moving a charge along an equipotential takes no work, because the potential, and so the energy, does not change.

Equipotentials are always perpendicular to field lines. If they were not, the field would have a component along the surface, which would do work as a charge slid along it, contradicting the definition. A contour map makes the analogy: field lines run straight downhill, and equipotentials are the level contours that circle the slope.

A conductor in equilibrium is itself an equipotential. Since no field exists inside and none runs along its surface, every point of the conductor sits at the same voltage. This is why we can speak of "the voltage" of a wire or a metal plate as a single value.

Key idea: Equipotential surfaces cost no work to move along, run perpendicular to field lines, and include the whole surface of any conductor.

Potential difference and the field

What actually drives current and does work is the potential difference (voltage) between two points, delta V = V_b - V_a. Moving a charge q through a potential difference changes its energy by delta U = q delta V. In a uniform field (as between two parallel plates) the relationship is simple: delta V = E d, where d is the distance along the field. This is why field can be quoted in volts per meter, the same as N/C.

The field always points from high potential toward low potential, the downhill direction for a positive charge. Its strength equals how quickly the potential falls with distance: a steep voltage drop over a short gap means a strong field. Rearranging the plate relation gives E = delta V / d, a formula you will use constantly with capacitors and circuits.

Key idea: The field points from high to low potential, and in a uniform field E = delta V / d, linking volts per meter to newtons per coulomb.

From potential back to field

The plate relation is a special case of a general truth: the field is the rate at which potential changes with position. Where the potential drops steeply over a short distance, the field is strong; where the potential is nearly flat, the field is weak. The field always points in the direction of steepest decrease of V.

This is powerful in practice. It is usually far easier to compute the scalar potential everywhere and then read off the field from how V varies than to add field vectors directly. Potential maps and field lines are two views of the same physics: the contours and the downhill arrows of one landscape.

Key idea: The field equals how fast potential falls with distance and points toward steepest decrease, so a potential map fully determines the field.

Why voltage runs the world

A battery is a device that maintains a fixed potential difference between its terminals. Connect a wire and that voltage drives charge through it, delivering energy at a rate P = I V, the power that lights bulbs and spins motors. Every circuit in the coming lessons is organized around voltages.

Voltage is also what instruments actually measure. A voltmeter reads the potential difference between two points, and the "12 volts" on a car battery or the "120 volts" in a wall outlet is precisely this quantity. Because potential is scalar and additive, tracing voltages around a circuit is far simpler than tracking fields inside every wire.

The electron-volt

A handy energy unit at the atomic scale is the electron-volt (eV): the energy an electron gains crossing a 1-volt difference. 1 eV = e x 1 V = 1.60 x 10^-19 J.

The electron-volt is sized to the physics it describes. Visible-light photons carry a few eV, the energy that holds outer electrons in atoms and drives chemistry. X-ray photons carry thousands of eV (keV), and nuclear processes release millions (MeV). Quoting these energies in joules would bury the meaning under strings of tiny powers of ten, so the eV earns its keep.

Worked example: accelerating an electron

Given: an electron starts at rest and is accelerated through a potential difference of 100 V. Find: its kinetic energy and speed.

Solution: The energy gained is KE = q delta V = (1.60 x 10^-19)(100) = 1.6 x 10^-17 J (equivalently 100 eV). Setting this equal to (1/2) m v^2 with the electron mass m = 9.11 x 10^-31 kg:
v = sqrt(2 KE / m) = sqrt(2 x 1.6 x 10^-17 / 9.11 x 10^-31) = sqrt(3.51 x 10^13) = 5.9 x 10^6 m/s.
The electron reaches about 5.9 million m/s.

This is exactly how an electron gun works: a voltage accelerates electrons to precise speeds in old picture tubes, electron microscopes, and X-ray machines. A larger accelerating voltage means a faster, more energetic beam, which is why these instruments quote their electron energies directly in kilovolts.

Worked example: potential where the field is not zero

Given: a point P sits 0.10 m from q1 = +2.0 nC and 0.10 m from q2 = -2.0 nC. Find: the potential at P.

Solution: Potential is a signed scalar sum:
V = kq1/r + kq2/r = (8.99 x 10^9)(2.0 x 10^-9)/0.10 + (8.99 x 10^9)(-2.0 x 10^-9)/0.10 = 180 - 180 = 0 V.
The potential is zero, yet the field there is not: recall the same dipole midpoint had a strong field. Potential and field are different quantities, and one can vanish where the other does not.

Worked example: field between plates

Given: two parallel plates 0.020 m apart with 60 V across them. Find: the field in the gap.

Solution: The field is the voltage drop per meter:
E = delta V / d = 60 / 0.020 = 3.0 x 10^3 V/m, or equivalently 3000 N/C, pointing from the positive plate to the negative one.

Common misconceptions

• Potential and potential energy are the same. Potential is energy per unit charge; multiply by a charge to get energy.
• Zero potential means zero field. The dipole midpoint has zero potential but a strong field; the two are independent.
• Potential is a vector. It is a scalar and adds with signs, no directions involved.
• A positive charge speeds up toward higher voltage. Left free, it moves toward lower potential, downhill in energy.
• The absolute value of potential matters. Only differences are physical; the zero point is a free choice.

Recap

• Two charges store energy U = k q1 q2 / r, signed by whether they are alike or opposite.
• Potential is energy per unit charge, V = U/q in volts, and adds as a signed scalar.
• Equipotentials cost no work to traverse and run perpendicular to the field; conductors are equipotentials.
• In a uniform field E = delta V / d, and the field points from high to low potential.
• The electron-volt is the natural energy unit for atomic and subatomic processes.

Sources

1. OpenStax. (2016). 7.1 Electric potential energy. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 7.2 Electric potential and potential difference. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 7.4 Determining field from potential. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 7.5 Equipotential surfaces and conductors. In University Physics Volume 2. Rice University. openstax.org
5. OpenStax. (2022). 19.1 Electric potential energy: Potential difference. In College Physics 2e. Rice University. openstax.org
6. Nave, R. (n.d.). Electric potential for different charge geometries. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 4: Electrostatics. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu

A capacitor is a device that stores charge and energy in an electric field. In its simplest form it is two conducting plates separated by a small gap. Connect it to a battery and charge piles up - positive on one plate, negative on the other - until the voltage across the plates matches the battery. Capacitors are everywhere: smoothing power supplies, timing circuits, camera flashes, and the memory cells in electronics.

The two plates always carry equal and opposite charge, so the capacitor as a whole stays neutral. What it stores is separated charge and the electric field that separation creates in the gap. This lesson turns that simple picture into precise relations among charge, voltage, energy, and geometry.

The idea is old. The first capacitor, the Leyden jar of 1745, was a glass jar lined inside and out with metal foil, and it startled its inventors by delivering a powerful shock. That jar is the direct ancestor of every capacitor in your phone, though the modern versions are smaller, safer, and far better understood.

Key idea: A capacitor stores energy by holding equal and opposite charge on two conductors, creating an electric field in the gap between them.

Definition of capacitance

The capacitance C measures how much charge a capacitor stores per volt applied:

C = Q / V, measured in farads (F), where 1 farad = 1 coulomb per volt.

A farad is a very large unit, so real capacitors are usually microfarads (10^-6 F) or picofarads (10^-12 F). For a parallel-plate capacitor with plate area A and separation d, geometry sets the capacitance: C = epsilon_0 A / d. Bigger plates or a smaller gap store more charge per volt.

That geometric formula is worth understanding, not just memorizing. The field between the plates is E = Q/(epsilon_0 A), and the voltage across the gap is V = E d = Q d/(epsilon_0 A). Dividing charge by voltage cancels Q and leaves C = epsilon_0 A/d. Larger plates hold more charge at a given field, and a thinner gap needs less voltage for the same charge, so both raise capacitance.

Key idea: Capacitance is charge stored per volt, C = Q/V in farads, and for parallel plates C = epsilon_0 A/d grows with plate area and shrinks with gap.

How big is a capacitor?

Because epsilon_0 is tiny, a one-farad capacitor made of bare plates would need an enormous area, which is why the farad is such a large unit. Typical circuit capacitors range from a few picofarads in radio tuning up to thousands of microfarads in power supplies. A capacitance you can hold in your hand is usually far below a single farad.

Modern supercapacitors reach several farads or more by using vast internal surface areas and microscopically thin charge layers. They bridge the gap between ordinary capacitors and batteries, storing enough energy to back up memory or smooth demand while still charging and discharging far faster than any battery.

Key idea: Real capacitors run from picofarads to thousands of microfarads, and only special supercapacitors reach a full farad or beyond.

Energy stored

Charging a capacitor takes work, which is stored as energy in the field. Three equivalent expressions give it:

U = (1/2) Q V = (1/2) C V^2 = Q^2 / (2 C).

The middle form, (1/2) C V^2, is the one you will use most.

Why the factor of one half? The first bit of charge crosses at almost zero voltage, while the last bit crosses at the full voltage V, so the average cost is V/2 per unit charge. Multiplying the average voltage by the total charge Q gives (1/2) Q V. The energy is not Q V, a mistake worth guarding against.

Key idea: A charged capacitor stores U = (1/2) C V^2, with the factor of one half arising because voltage climbs from zero to V during charging.

Where the energy lives: the field

It is natural to ask where the stored energy actually sits. The answer is: in the electric field itself, filling the gap. The energy per unit volume of any electric field is u = (1/2) epsilon_0 E^2, so a stronger field packs more energy into each cubic meter of space.

Multiply that density by the gap volume of a parallel-plate capacitor and you recover (1/2) C V^2 exactly, a satisfying consistency check. This idea, that fields carry energy, is not a mere accounting choice. It becomes essential in the final lesson, where electromagnetic waves carry field energy across empty space with no charges present at all.

Key idea: The stored energy resides in the field with density u = (1/2) epsilon_0 E^2, the same field energy that later travels as light.

Combining capacitors

Connection	Rule	Behavior
Parallel	C_total = C1 + C2 + ...	Same voltage across each; capacitances add.
Series	1/C_total = 1/C1 + 1/C2 + ...	Same charge on each; total is less than the smallest.

Notice these rules are the opposite of the resistor rules you will meet next - capacitors add in parallel, while resistors add in series.

The reasoning is short. In parallel, both capacitors feel the same voltage, and their charges add, so Q = C1 V + C2 V gives a combined C = C1 + C2. In series, the same charge sits on each, while the voltages add, so V = Q/C1 + Q/C2 gives 1/C = 1/C1 + 1/C2. Tracing which quantity is shared tells you at once which rule applies.

Key idea: Parallel capacitors share voltage and add directly, while series capacitors share charge and add as reciprocals, the reverse of resistors.

Dielectrics

Slipping an insulating material (a dielectric) between the plates increases the capacitance by a factor called the dielectric constant kappa (kappa greater than 1): C = kappa epsilon_0 A / d. The dielectric's molecules polarize and partly cancel the internal field, so more charge can be held at the same voltage. Dielectrics also let the plates sit closer without touching and raise the voltage the capacitor can survive.

Every insulator can withstand only so strong a field before it breaks down and conducts, a limit called its dielectric strength. A good dielectric both multiplies capacitance and tolerates a high field, letting a small device store useful energy. This is why real capacitors are built as thin foils wound around a dielectric film rather than as bare plates in air.

The multiplying factor kappa varies widely by material. Air is close to 1 and barely helps, paper and common plastics sit around 2 to 5, and water is a striking 80. Ceramic dielectrics engineered for the job can reach into the hundreds or thousands, which is how a fingertip-sized ceramic capacitor can rival much larger air-gap designs.

These properties make capacitors workhorses of technology. A camera flash dumps a capacitor's stored energy through a bulb in a millisecond; a defibrillator does the same through a patient's chest. Tiny capacitors hold each bit in a computer's memory, and large ones smooth the ripples in power supplies.

Key idea: A dielectric multiplies capacitance by kappa and raises the breakdown voltage, which is why practical capacitors are wound from thin dielectric films.

Power versus energy: capacitors against batteries

A capacitor and a battery both store energy, but they play different roles. A battery holds far more total energy and releases it slowly at a steady voltage through chemical reactions. A capacitor holds much less energy but can dump or absorb it almost instantly, delivering enormous power for a brief moment.

That contrast decides which device a job wants. You want a battery to run a phone for a day, but a capacitor to fire a flash, launch a defibrillator pulse, or catch a sudden surge. A capacitor's voltage also sags as it discharges, unlike a battery's steady output, so circuits that need a capacitor's energy are built to expect that falling voltage.

Key idea: Batteries store more energy and release it slowly, while capacitors store less but deliver it in an instant, so each suits different tasks.

Inserting a dielectric: two cases

Sliding a dielectric into a capacitor multiplies C by kappa, but what happens to charge, voltage, and energy depends on whether a battery is still attached. The two cases are a favorite exam trap, and reasoning them out cements the relations Q = C V and U = (1/2) C V^2.

If the capacitor stays connected to the battery, the voltage V is held fixed. Then C rises by kappa, so the charge Q = C V rises by kappa and the energy (1/2) C V^2 rises by kappa too; the battery supplies the extra. If instead the capacitor is disconnected first, the charge Q is trapped and fixed. Now C rises by kappa, so V = Q/C falls by kappa and the energy Q^2/(2C) falls by kappa, as the dielectric is pulled in and does work.

Key idea: With the battery connected a dielectric raises charge and energy at fixed voltage, while with the capacitor isolated it lowers voltage and energy at fixed charge.

Worked example: a parallel-plate capacitor

Given: a 5.0 microF capacitor is charged to 12 V. Find: the charge stored and the energy stored.

Solution: Charge is Q = C V = (5.0 x 10^-6)(12) = 6.0 x 10^-5 C = 60 microC.
Energy is U = (1/2) C V^2 = (1/2)(5.0 x 10^-6)(12)^2 = (1/2)(5.0 x 10^-6)(144) = 3.6 x 10^-4 J.
So the capacitor holds 60 microC and stores 0.36 mJ.

Worked example: series and parallel

Given: a 3.0 microF and a 6.0 microF capacitor. Find: the combined capacitance in parallel and in series.

Solution: In parallel they simply add: C = 3.0 + 6.0 = 9.0 microF. In series the reciprocals add: 1/C = 1/3.0 + 1/6.0 = 2/6 + 1/6 = 3/6, so C = 2.0 microF. As always, the series combination is smaller than either capacitor, and the parallel combination is larger than both.

Common misconceptions

• A capacitor stores net charge. Its plates hold equal and opposite charge, so the device stays neutral overall.
• The stored energy is Q V. It is (1/2) Q V, because the voltage rises from zero to V while charging.
• Capacitors combine like resistors. They are reversed: parallel capacitors add, series capacitors add as reciprocals.
• A dielectric just fills space. It polarizes, cutting the internal field, so it raises capacitance and the safe voltage.
• Bigger gap means bigger capacitance. Capacitance goes as 1/d, so a wider gap stores less charge per volt.

Recap

• Capacitance is charge per volt, C = Q/V in farads; for parallel plates C = epsilon_0 A/d.
• A charged capacitor stores U = (1/2) C V^2, held in the field at density (1/2) epsilon_0 E^2.
• Parallel capacitors add; series capacitors add as reciprocals, the opposite of resistors.
• A dielectric multiplies capacitance by kappa and raises the breakdown voltage.
• Capacitors power flashes, defibrillators, memory cells, and smoothing in power supplies.

Sources

1. OpenStax. (2016). 8.1 Capacitors and capacitance. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 8.2 Capacitors in series and in parallel. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 8.3 Energy stored in a capacitor. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 8.4 Capacitor with a dielectric. In University Physics Volume 2. Rice University. openstax.org
5. OpenStax. (2016). 8.5 Molecular model of a dielectric. In University Physics Volume 2. Rice University. openstax.org
6. Nave, R. (n.d.). Capacitance. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 10: Dielectrics. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu

So far charges have been static. Now we let them flow. Electric current is the rate at which charge passes a point in a wire:

I = Q / t, measured in amperes (A), where 1 ampere = 1 coulomb per second.

By convention, current direction is the direction positive charge would move - which is opposite to the actual drift of the negative electrons in a metal. This conventional current convention predates the discovery of the electron; it works out fine as long as you are consistent.

Key idea: Current is the rate of charge flow in amperes, and by convention it points the way positive charge would move, opposite to the electron drift.

A microscopic look at current

Inside a metal wire, the free electrons are always in frenzied random motion at high speed, but with no net direction. Connect a battery and a small drift is added on top of that chaos, a slow collective creep along the wire. It is this net drift, not the frantic random speed, that we count as current.

The drift is astonishingly slow, often less than a millimeter per second. So why does a lamp light the instant you flip the switch? Because the electric field that pushes the electrons is established along the whole wire almost at the speed of light. Every electron starts nudging forward at once, like a train of already-touching train cars that all move the moment the engine pulls.

A single ampere is a torrent of charge at the particle level: about 6.2 x 10^18 electrons drift past each second. They manage this enormous count precisely because a wire holds so many free electrons that even a crawl adds up to a large current. Counting charge, not tracking any one electron, is what the ampere measures.

Key idea: Current is a slow net drift added to fast random motion, yet devices respond instantly because the driving field spreads through the wire near light speed.

Resistance and Ohm's law

Push charge through a material and it resists, converting some electrical energy to heat. Resistance R measures this opposition. For many materials, current is proportional to the voltage across them - a relationship called Ohm's law:

V = I R

Resistance is measured in ohms (the symbol is the Greek capital omega). A material that obeys V = IR over a range of voltages is called ohmic. Resistance depends on the material and the shape: R = rho L / A, where rho is the resistivity of the material, L the length, and A the cross-sectional area. A long thin wire resists more than a short thick one, just as a narrow pipe restricts water flow.

Resistivity rho is the material's own contribution, independent of shape. Copper and silver have very low resistivity, which is why wires are made of them; rubber and glass have enormous resistivity and serve as insulators. Multiply the material property rho by the shape factor L/A and you get the resistance of a particular object.

Key idea: Ohm's law V = IR ties current to voltage, and R = rho L/A splits resistance into a material property and a shape factor.

The water-circuit analogy

A flowing-water picture makes circuits intuitive. Voltage is like water pressure, the push that drives flow. Current is like the flow rate, the amount of water passing each second. Resistance is like a narrow section of pipe that throttles the flow, and a battery is the pump that keeps the pressure up.

In this picture Ohm's law reads naturally: more pressure drives more flow, and a narrower pipe allows less. The analogy even captures power, since a fast flow through a big pressure drop delivers energy quickly. Like all analogies it has limits, but it is a reliable guide for building a first intuition about a circuit.

Key idea: Voltage is like pressure, current like flow rate, resistance like a narrow pipe, and a battery like a pump, so Ohm's law reads as pressure driving flow.

What causes resistance

Resistance arises because drifting electrons collide with the vibrating atoms of the lattice, losing energy to them as heat. The more crowded and jittery the lattice, the harder the electrons' passage. This picture explains why resistance usually rises with temperature in a metal: hotter atoms vibrate more and scatter electrons more often.

A light-bulb filament shows this clearly. Cold, it has a low resistance; blazing at operating temperature, its resistance is much higher. So a filament is non-ohmic: its R changes with conditions, and a plot of current against voltage bends rather than staying straight. Diodes bend even more sharply, conducting well one way and hardly at all the other.

At the opposite extreme, some materials cooled below a critical temperature lose all resistance and become superconductors. A current started in a superconducting loop can circulate for years without a battery, since nothing dissipates its energy. Superconducting coils make the powerful magnets in hospital scanners and research accelerators, though keeping them cold enough remains costly.

Key idea: Resistance comes from electrons scattering off lattice vibrations, so it usually grows with temperature, and it vanishes entirely in a superconductor.

What drives the current: EMF

Current needs a push, and that push comes from a source of electromotive force, or EMF, such as a battery. Despite the name, EMF is not a force but a voltage: the energy per unit charge a source gives to charges as it drives them around a circuit. A 1.5-volt cell raises each coulomb's energy by 1.5 joules.

A battery does this through chemical reactions that pump charge from its low-potential terminal to its high-potential one, maintaining the voltage difference that the external circuit then uses. Generators, solar cells, and thermocouples are other EMF sources, each converting some form of energy into the electrical push that sustains a current.

Electrical power

Current through a resistance dissipates energy as heat at a rate given by the power:

P = I V, and using Ohm's law, P = I^2 R = V^2 / R.

Power is in watts. The energy used over a time t is Energy = P t. Your electric bill is charged in kilowatt-hours, the energy of 1000 watts running for one hour.

The three forms each answer a different question. Use P = IV when you know current and voltage, P = I^2 R when you know the current through a known resistor, and P = V^2/R when you know the voltage across it. All three give the same watts; choose whichever matches the numbers you have.

Key idea: Electrical power is P = IV = I^2 R = V^2/R in watts, and energy delivered is power times time, billed in kilowatt-hours.

Why power lines run at high voltage

The form P = I^2 R explains a decision worth billions. Sending power over a long line wastes some as heat in the wire's own resistance, and that loss grows as the square of the current. Cut the current and the loss plummets.

To deliver a given power P = IV with less current, you raise the voltage. That is exactly what the grid does, stepping up to hundreds of thousands of volts for long-distance transmission and stepping back down near homes. High voltage is dangerous but efficient, and transformers make the trade practical, a story completed in the induction lessons ahead.

Key idea: Because line loss goes as I^2 R, the grid transmits at very high voltage to keep current and wasted heat low.

Direct and alternating current

Current comes in two styles. Direct current (DC) flows steadily in one direction, the kind a battery supplies and the kind these formulas describe most simply. Flashlights, phones, and cars run on DC, and it is the natural starting point for learning circuits.

Alternating current (AC) reverses direction many times a second, sixty times per second in North American outlets and fifty in much of the world. The grid uses AC because transformers can raise and lower its voltage easily, which is exactly what efficient long-distance transmission demands. We return to AC in depth once induction gives us the tools to understand it.

Key idea: DC flows one way from sources like batteries, while AC reverses many times a second and powers the grid because its voltage is easy to transform.

Worked example: a light bulb

Given: a bulb draws 0.50 A when connected to 120 V. Find: its resistance and power.

Solution: Resistance from Ohm's law: R = V/I = 120/0.50 = 240 ohms. Power: P = IV = 0.50 x 120 = 60 W. This is a 60-watt bulb. Check with another form: P = V^2/R = 120^2/240 = 14400/240 = 60 W, which agrees.

Worked example: charge delivered

Given: a current of 2.0 A flows for 30 s. Find: the charge that passes.

Solution: Q = I t = 2.0 x 30 = 60 C. That is 60 coulombs, or about 3.7 x 10^20 electrons.

Worked example: resistance of a wire

Given: a copper wire 10 m long with cross-sectional area 1.0 x 10^-6 m^2, and copper's resistivity rho = 1.7 x 10^-8 ohm m. Find: its resistance.

Solution: R = rho L / A = (1.7 x 10^-8)(10)/(1.0 x 10^-6) = 1.7 x 10^-1 = 0.17 ohms. The tiny resistivity of copper keeps even a long wire well under an ohm, which is why household wiring wastes little power.

Measuring current and voltage

Two instruments recur throughout circuit work. An ammeter measures current, so it is placed in series, in the path the charge follows, and it is built with almost no resistance so it barely disturbs the flow. A voltmeter measures potential difference, so it is placed in parallel across a component, and it is built with very high resistance so it draws almost no current for itself.

Key idea: An ammeter goes in series and must have low resistance, while a voltmeter goes in parallel and must have high resistance, so neither disturbs the circuit it measures.

Common misconceptions

• Electrons zip through wires near light speed. Their drift is under a millimeter per second; the field travels near light speed.
• Current is used up in a device. Charge is conserved; the same current returns to the source, but energy is delivered along the way.
• Everything obeys Ohm's law. Only ohmic materials do; filaments and diodes do not.
• Conventional current is the electron flow. It points the opposite way, since electrons are negative.
• Thicker wires have more resistance. More area means less resistance, since R = rho L/A.

Recap

• Current is charge per time, I = Q/t in amperes, directed as positive charge would move.
• Ohm's law V = IR holds for ohmic materials, with R = rho L/A.
• Resistance comes from electron scattering and usually grows with temperature.
• Power is P = IV = I^2 R = V^2/R in watts; the grid uses high voltage to cut I^2 R loss.
• An EMF source such as a battery supplies the voltage that drives the current.

Sources

1. OpenStax. (2016). 9.1 Electrical current. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 9.2 Model of conduction in metals. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 9.3 Resistivity and resistance. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 9.4 Ohm's law. In University Physics Volume 2. Rice University. openstax.org
5. OpenStax. (2016). 9.5 Electrical energy and power. In University Physics Volume 2. Rice University. openstax.org
6. Nave, R. (n.d.). Ohm's law. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. O'Connor, J. J., & Robertson, E. F. (n.d.). Georg Simon Ohm. MacTutor History of Mathematics Archive, University of St Andrews. mathshistory.st-andrews.ac.uk

Real circuits contain many resistors. To analyze them you reduce combinations to a single equivalent resistance, then work backward for the details. Two building blocks cover most cases: series and parallel.

Mastering these two patterns is the core skill of circuit analysis. Almost any resistor network, however tangled it looks, can be simplified by spotting series and parallel groups, replacing each with its equivalent, and repeating until a single resistor remains. Then you reverse the steps to find the current and voltage everywhere.

Key idea: Circuits are analyzed by reducing series and parallel groups to one equivalent resistance, then working backward for each part.

Series: one path

Resistors in series sit end to end on a single path, so the same current flows through each. Their resistances simply add:

R_series = R1 + R2 + R3 + ...

The battery voltage divides among them in proportion to their resistances (the biggest resistor drops the most voltage). The sum of the voltage drops equals the source voltage.

This proportional sharing is called a voltage divider, and it is one of the most used ideas in electronics. Two resistors in series across a source split the voltage in the ratio of their resistances, so choosing the resistors lets you tap off any fraction of the supply voltage you need. Sensors and volume controls work this way.

Key idea: Series resistors share one current and add directly, dividing the source voltage in proportion to their resistances.

Parallel: multiple paths

Resistors in parallel connect across the same two nodes, so each feels the same voltage. The current splits among them. The reciprocals add:

1 / R_parallel = 1/R1 + 1/R2 + 1/R3 + ...

The equivalent resistance is always less than the smallest resistor in the group - adding another path makes it easier for current to flow. For just two resistors, a handy form is R = R1 R2 / (R1 + R2).

The current splits in the opposite proportion to the voltage: the smaller resistance carries the larger share of the current, since it offers the easier path. This is a current divider. It is why the bulk of the current in a household circuit flows through whichever appliance has the lowest resistance.

Key idea: Parallel resistors share one voltage and combine as reciprocals to less than the smallest, with the smallest resistance carrying the most current.

A handy shortcut for equal resistors

When several equal resistors are in parallel, the reciprocal sum collapses to a simple rule: N equal resistors of value R combine to R/N. Three 6 ohm resistors in parallel, for instance, give 6/3 = 2.0 ohms. The more identical paths you add, the more the current spreads and the lower the equivalent resistance falls.

The same shortcut warns against a common slip. Two equal resistors in parallel give half of one, not double, because parallel paths ease the flow. Keeping the direction of the effect in mind, down for parallel and up for series, catches many arithmetic errors before they happen.

Key idea: N equal resistors in parallel combine to R/N, so more parallel paths always lower the equivalent resistance.

A summary table

	Series	Parallel
Current	Same through each	Splits between branches
Voltage	Divides among them	Same across each
Equivalent R	Adds up (larger)	Reciprocals add (smaller)

Reading down these columns is often enough to start a problem. Ask first which quantity is shared - current in series, voltage in parallel - and the rest of the analysis follows from Ohm's law applied to each resistor in turn.

Why your house is wired in parallel

Home outlets are wired in parallel, and for good reason. Each device then receives the same full supply voltage, about 120 volts, regardless of what else is plugged in. And because each sits on its own branch, switching off a lamp or unplugging a toaster leaves every other device running.

Contrast this with a series string, like old-style holiday lights: the same current runs through every bulb, so a single burned-out bulb breaks the path and the whole string goes dark. Series wiring also means each bulb gets only a fraction of the supply voltage. The parallel choice trades a little more wiring for independence and a steady voltage at every socket.

Key idea: Parallel house wiring gives every outlet the full voltage and keeps devices independent, unlike a series string where one failure stops them all.

A strategy for mixed networks

Most real circuits mix the two patterns. The method is to work from the inside out. Find a group that is purely series or purely parallel, replace it with its single equivalent, and redraw the simpler circuit. Repeat until one resistor remains, which gives the total current from the source.

Then travel back outward. The total current sets the voltage across each stage, and at every parallel junction the current re-splits among the branches. Careful, patient reduction turns a daunting web of resistors into a short sequence of easy steps.

Key idea: Reduce mixed networks inside out into one equivalent resistance, then work back outward to find each current and voltage.

A real battery has internal resistance

An ideal battery holds its voltage no matter the load, but a real one has a small internal resistance of its own. When it drives a current I, some voltage is lost inside it, so the terminal voltage you actually get is V = EMF - I r, a little below the rated EMF.

This is why a car's headlights dim for a moment when the starter motor draws a huge current: the large I makes the I r drop significant, and the terminal voltage sags. Under light loads the effect is tiny, which is why we usually treat batteries as ideal in first calculations.

Key idea: A real battery loses I r inside itself, so its terminal voltage V = EMF - I r falls below the rated value under heavy current.

Which resistor runs hottest?

A useful question is where the power goes. In a series circuit the same current flows through every resistor, so by P = I^2 R the largest resistance dissipates the most power and runs hottest. It also drops the most voltage, which fits, since it does the most work per unit charge.

In a parallel circuit every resistor feels the same voltage, so by P = V^2/R the smallest resistance dissipates the most power. The reversal is easy to remember once you note which quantity is shared. Picking the right power formula for the shared quantity settles these questions immediately.

Key idea: With shared current in series the largest resistor dissipates most, while with shared voltage in parallel the smallest resistor dissipates most.

Short circuits and open circuits

Two extreme cases are worth naming. A short circuit is a near-zero-resistance path, and by Ohm's law it draws a very large current. A short across a battery can overheat wires and start fires, which is exactly what a fuse or circuit breaker guards against by cutting the current when it grows dangerous.

An open circuit is the opposite: a break in the path, an effectively infinite resistance, so no current flows at all. A blown fuse, a flipped switch, or a snapped wire all create open circuits. Recognizing these extremes helps you diagnose real circuits, where a fault is usually one or the other.

Key idea: A short circuit is a near-zero-resistance path drawing huge current, while an open circuit is a break carrying none, and fuses protect against shorts.

When series and parallel are not enough

Some circuits cannot be reduced by these two rules at all, because their resistors are neither purely in series nor purely in parallel. A bridge network, or a circuit with more than one battery in different loops, resists the inside-out method.

For those cases the next lesson introduces Kirchhoff's laws, two conservation statements that apply to any circuit whatsoever. Series and parallel rules remain the fast first tools; Kirchhoff is the general method held in reserve for when they run out.

Worked example: series

Given: a 4.0 ohm and a 6.0 ohm resistor in series across a 20 V battery. Find: the current and the voltage across each.

Solution: Equivalent: R = 4.0 + 6.0 = 10 ohms. Current: I = V/R = 20/10 = 2.0 A (same in both). Voltage drops: V1 = I R1 = 2.0 x 4.0 = 8.0 V and V2 = 2.0 x 6.0 = 12 V. These add to 20 V, as they must.

Worked example: parallel

Given: a 4.0 ohm and a 6.0 ohm resistor in parallel across a 12 V battery. Find: the equivalent resistance and the total current.

Solution: 1/R = 1/4.0 + 1/6.0 = 3/12 + 2/12 = 5/12, so R = 12/5 = 2.4 ohms (less than the smaller resistor). Total current: I = V/R = 12/2.4 = 5.0 A. Checking branch currents: 12/4.0 = 3.0 A and 12/6.0 = 2.0 A, which sum to 5.0 A.

Worked example: a mixed network

Given: a 2.0 ohm resistor in series with a parallel pair of 3.0 ohm and 6.0 ohm resistors, all across a 12 V battery. Find: the total current and the branch currents.

Solution: First the parallel pair: 1/R = 1/3.0 + 1/6.0 = 3/6, so R = 2.0 ohms. Add the series resistor: R_total = 2.0 + 2.0 = 4.0 ohms. Total current: I = 12/4.0 = 3.0 A.

Now work back. The series resistor drops 3.0 x 2.0 = 6.0 V, leaving 6.0 V across the pair. Branch currents: 6.0/3.0 = 2.0 A and 6.0/6.0 = 1.0 A, which add to the 3.0 A total, confirming the analysis.

A final energy check ties it together. The source delivers P = V I = 12 x 3.0 = 36 W. The series resistor burns I^2 R = 3.0^2 x 2.0 = 18 W, and the parallel pair burns the other 18 W across its 6.0 V. The books balance, a reassuring sign that no step went astray.

Common misconceptions

• Adding a parallel resistor raises total resistance. It lowers it, since each new path lets more current flow.
• Series resistors share the voltage equally. They share it in proportion to their resistances; the biggest drops the most.
• Current is used up as it passes through resistors. The same current returns to the source in series; only energy is spent.
• Parallel resistors share the current equally. The smaller resistance takes the larger share.
• Batteries deliver their full rated voltage always. Internal resistance makes the terminal voltage sag under heavy load.

Recap

• Series resistors add and share one current, dividing the voltage by resistance ratio.
• Parallel resistors combine as reciprocals to less than the smallest and share one voltage.
• Reduce mixed networks inside out, then work back outward for currents and voltages.
• Homes are wired in parallel so each outlet gets full voltage and devices stay independent.
• A real battery's terminal voltage is EMF - I r, below its rating under load.

Sources

1. OpenStax. (2016). 10.2 Resistors in series and parallel. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 10.1 Electromotive force. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 10.6 Household wiring and electrical safety. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2022). 21.1 Resistors in series and parallel. In College Physics 2e. Rice University. openstax.org
5. OpenStax. (2022). 21.2 Electromotive force: Terminal voltage. In College Physics 2e. Rice University. openstax.org
6. Nave, R. (n.d.). Resistance and resistivity. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Massachusetts Institute of Technology. (2007). Syllabus. In 8.02 Physics II: Electricity and magnetism. MIT OpenCourseWare. ocw.mit.edu

Some circuits cannot be reduced by series and parallel rules alone - for example, two batteries in different loops. For these, we use two conservation statements called Kirchhoff's laws, which apply to any circuit whatsoever.

The two laws are nothing more than conservation of charge and conservation of energy, dressed for circuits. Because they rest on those bedrock principles, they never fail, no matter how tangled the network. Where series and parallel shortcuts run out, Kirchhoff's laws always finish the job.

Key idea: Kirchhoff's two laws restate conservation of charge and energy for circuits and solve any network, even where series and parallel rules cannot.

The junction (current) rule

Kirchhoff's current law (KCL): at any junction, the total current flowing in equals the total current flowing out. This is conservation of charge - charge does not pile up at a point. If 3 A and 2 A flow into a node, then 5 A must flow out.

The picture is a branching river or a plumbing tee: whatever water arrives must leave, since none is created or stored at the junction. In a circuit the same holds for charge every instant. Writing a KCL equation at each junction gives one relation among the branch currents, and these relations are the first half of any Kirchhoff analysis. The law named after Gustav Kirchhoff, who stated both rules in 1845 while still a student, remains the foundation of all circuit theory.

Key idea: KCL says current in equals current out at every junction, expressing conservation of charge.

The loop (voltage) rule

Kirchhoff's voltage law (KVL): around any closed loop, the sum of all voltage changes is zero. This is conservation of energy - a charge returning to its start has the same potential it began with. Going around a loop, you add the EMF of a battery (from - to +) and subtract IR drops across resistors (in the direction of current).

Think of walking a hilly loop trail. You may climb and descend many times, but on returning to the trailhead your net change in height is exactly zero. Potential is the electrical height, batteries are the climbs, and resistors are the descents. The round trip must break even.

Key idea: KVL says the voltage changes around any closed loop sum to zero, expressing conservation of energy.

Bookkeeping the signs

Most Kirchhoff mistakes are sign errors, so a fixed routine helps. First choose a direction to walk each loop. Crossing a battery from its minus to its plus terminal is a voltage rise, counted positive; crossing the other way is a drop. Crossing a resistor in the direction of the current is a drop of I R, counted negative; against the current it is a rise.

Guess the current directions before you start. If a guess turns out backward, the algebra simply returns a negative value for that current, telling you it really flows the other way. There is no need to guess correctly, only to stay consistent once you have chosen.

Key idea: Fix a loop direction, count battery minus-to-plus as a rise and IR-with-current as a drop, and let negative answers flag a reversed current guess.

A multi-loop strategy

1. Label a current in each branch and pick a direction (a wrong guess just gives a negative answer).
2. Write a junction equation (KCL) at the nodes.
3. Write a loop equation (KVL) for each independent loop.
4. Solve the simultaneous equations for the currents.

The number of independent equations always matches the number of unknown currents, so the system can be solved. With practice the setup becomes routine, and the only real work is the algebra of solving two or three equations together.

These same two laws scale far beyond hand calculation. The circuit simulators that design microchips solve exactly Kirchhoff's equations, but for millions of nodes at once, using a computer to handle the enormous system of equations. The physics a student applies to a three-resistor loop is identical to the physics inside a chip-design tool; only the size of the algebra differs.

Worked example: a two-source loop

Given: a single loop with a 12 V battery and a 6 V battery opposing it, in series with a 3 ohm resistor. Find: the current.

Solution: Going around the loop, the net EMF is 12 - 6 = 6 V. By KVL, 6 - I(3) = 0, so I = 6/3 = 2.0 A in the direction the 12 V battery drives.

Worked example: a genuine two-loop circuit

Given: two branches meet at a node and share a middle resistor. Branch 1 has a 10 V battery and a 2 ohm resistor carrying current I1; branch 2 has a 9 V battery and a 3 ohm resistor carrying I2; the shared middle branch has a 2 ohm resistor carrying I3. Find: the three currents.

Solution: KCL at the node gives I3 = I1 + I2. KVL on the two loops gives 10 = 2 I1 + 2 I3 and 9 = 3 I2 + 2 I3. Substitute I3 = I1 + I2 into both:
10 = 4 I1 + 2 I2 and 9 = 2 I1 + 5 I2.

Halve the first equation to get 2 I1 + I2 = 5, then subtract it from the second: 4 I2 = 4, so I2 = 1.0 A. Back-substituting gives I1 = 2.0 A and I3 = 3.0 A. All came out positive, so every guessed direction was correct.

RC circuits: charging over time

When a resistor and capacitor are connected in series to a battery, the capacitor does not charge instantly - the resistor limits the current. The charge builds up smoothly toward its final value, governed by the time constant:

tau = R C (in seconds).

After one time constant the capacitor reaches about 63% of full charge; after about 5 time constants it is essentially fully charged. Discharging follows the mirror image: the charge falls to about 37% of its start after one tau. This exponential behavior is the basis of timing circuits, camera flashes, and the blinker in a car.

The curve is steep at first and flattens as it goes. Early on, the full battery voltage drives a large charging current, so charge piles on fast. As the capacitor fills, it pushes back, the current dwindles, and the approach to full charge slows. The capacitor never quite reaches the final value in finite time, which is why we measure progress in time constants rather than a fixed finish line.

Written as a formula, the charge grows as Q(t) = Q_max (1 - e^(-t/tau)), where e is the base of natural logarithms. At t = tau the bracket equals 1 - e^(-1) = 0.63, the source of the 63% figure. Discharging follows the mirror form Q(t) = Q_0 e^(-t/tau), which falls to e^(-1) = 0.37 of its start in one time constant.

Key idea: In an RC circuit charge follows Q = Q_max(1 - e^(-t/tau)) when charging and Q = Q_0 e^(-t/tau) when discharging, with tau = RC setting the pace.

Discharging: the mirror image

Disconnect the battery and let the charged capacitor drive current back through the resistor, and the process runs in reverse. The charge decays from its full value toward zero, again governed by tau = RC. After one time constant only about 37% of the charge remains; after five, almost none.

The current is largest at the first instant, when the capacitor's voltage is highest, and it fades as the capacitor empties. This is the discharge that fires a camera flash or a defibrillator: stored energy released quickly through a low resistance gives a brief, powerful surge. A larger resistance stretches the same energy over a longer, gentler pulse.

Key idea: Discharging mirrors charging, with charge falling to about 37% after one time constant and the current largest at the start.

Where the charging energy goes

Charging a capacitor through a resistor hides a surprising result. The battery supplies energy Q V, yet the capacitor keeps only (1/2) Q V. The other half is dissipated as heat in the resistor, and remarkably this holds no matter how large or small the resistance is.

A smaller resistor charges the capacitor faster but with a bigger current, and the two effects cancel to leave the same heat lost. This is why charging is never perfectly efficient in a simple RC circuit, a fact that matters when designing energy-conscious electronics.

Key idea: Charging a capacitor through any resistor stores half the battery's delivered energy and dissipates the other half as heat, regardless of the resistance.

Worked example: time constant

Given: a 2.0 kilo-ohm resistor in series with a 100 microF capacitor. Find: the time constant.

Solution: tau = R C = (2.0 x 10^3)(100 x 10^-6) = (2000)(0.0001) = 0.20 s. So the capacitor reaches 63% of full charge in about 0.20 s and is nearly full after roughly 1 second (5 tau).

Worked example: how fast it discharges

Given: the same circuit, now discharging from full charge. Find: the charge remaining after 0.20 s and after 1.0 s.

Solution: One time constant is 0.20 s, so after 0.20 s the charge is at e^(-1) = 37% of full. After 1.0 s, which is 5 time constants, it is at e^(-5) = 0.7%, essentially fully discharged. The same tau governs both charging and discharging.

RC timing in the real world

The steady, predictable pace of an RC circuit makes it a natural clock. Choosing R and C sets a time constant, and that timing runs windshield-wiper delays, the blink of a turn signal, and the tone-shaping in synthesizers. Change a resistor and the rhythm changes with it.

RC circuits also filter signals by how fast they vary. A slow-changing signal passes while rapid wiggles are smoothed away, or the reverse, depending on where the output is taken. This lets a circuit separate a wanted signal from noise, a role RC networks play in nearly every piece of audio and radio equipment.

Key idea: By setting a time constant, RC circuits time delays and blinks and filter signals by how quickly they change.

Common misconceptions

• KVL means the voltages are all equal. It means their signed changes around a loop sum to zero, not that they match.
• You must guess current directions correctly. A wrong guess just yields a negative current; the magnitude is still right.
• A capacitor charges instantly. The series resistance sets a time constant, and full charge is approached only after several tau.
• After one time constant the capacitor is full. It is at about 63%; it needs roughly five tau to be essentially full.
• Kirchhoff's laws are new physics. They are just conservation of charge and energy applied to circuits.

Recap

• KCL: currents into a junction equal currents out (charge conservation).
• KVL: voltage changes around a loop sum to zero (energy conservation).
• Label currents, write junction and loop equations, and solve them together.
• An RC circuit charges and discharges exponentially with time constant tau = RC.
• One time constant reaches about 63% of full charge; about five reach nearly all.

Sources

1. OpenStax. (2016). 10.3 Kirchhoff's rules. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 10.5 RC circuits. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2022). 21.3 Kirchhoff's rules. In College Physics 2e. Rice University. openstax.org
4. OpenStax. (2022). 21.6 DC circuits containing resistors and capacitors. In College Physics 2e. Rice University. openstax.org
5. Nave, R. (n.d.). Charging a capacitor. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
6. Nave, R. (n.d.). Capacitor discharging. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. O'Connor, J. J., & Robertson, E. F. (n.d.). Gustav Kirchhoff. MacTutor History of Mathematics Archive, University of St Andrews. mathshistory.st-andrews.ac.uk

Magnetism and electricity are two faces of one force, but magnetism has its own character. A magnetic field (symbol B) is produced by moving charges and by magnets, and it acts only on other moving charges. A charge sitting still feels no magnetic force at all - motion is essential.

This single fact, that magnetism couples only to moving charge, sets the whole subject apart from electrostatics. It leads to forces that act sideways to the motion, to circular and spiral paths, and eventually to the deep link between electricity and magnetism that produces light. We begin with the force on one moving charge.

Key idea: A magnetic field is made by moving charges and magnets and exerts a force only on other charges that are moving.

Measuring the field: teslas and gauss

The strength of a magnetic field is measured in teslas (T), and one tesla is a strong field. The Earth's field at the surface is only about 50 microteslas, a refrigerator magnet a few milliteslas, and a hospital MRI scanner a mighty 1.5 to 3 teslas. Because the tesla is so large, older work often quotes fields in gauss, where 1 tesla equals 10,000 gauss.

Keeping a feel for these scales helps you judge answers. A field of several teslas is laboratory-grade and usually needs a superconducting magnet, while the fields around everyday wires and gadgets are thousands of times weaker. An answer of many teslas from a small current should prompt a second look.

The magnetic force law

For a charge q moving with speed v at an angle theta to a magnetic field B, the force magnitude is

F = q v B sin(theta)

The field B is measured in teslas (T); one tesla is a strong field, so everyday fields are often given in gauss (1 T = 10,000 gauss). Three features make this force unusual:

• It depends on the angle: maximum when the velocity is perpendicular to the field (theta = 90 degrees) and zero when the charge moves parallel to the field (theta = 0).
• Its direction is perpendicular to both v and B, given by a right-hand rule.
• Because the force is always perpendicular to the motion, it does no work - it changes the charge's direction but never its speed.

The angle dependence is worth dwelling on. Only the part of the velocity that cuts across the field produces a force; motion straight along the field lines is ignored entirely. This is quite unlike the electric force, which acts on a charge whether it moves or not.

Key idea: The magnetic force is F = qvB sin(theta), perpendicular to both velocity and field, greatest across the field and zero along it.

The right-hand rule

To find the direction of the force on a positive charge: point the fingers of your right hand along the velocity v, curl them toward the field B, and the thumb points along the force. For a negative charge, the force is the opposite direction.

Because the force is perpendicular to both v and B, it points out of the plane those two vectors define. That three-dimensional twist is why magnetic problems reward a careful hand gesture over guesswork. Practicing the rule on a few cases builds a reliable instinct for which way a charge will veer.

Key idea: The right-hand rule gives the force direction for a positive charge, and a negative charge feels the opposite, always perpendicular to the v-B plane.

Why the force does no work

A force that is always perpendicular to the motion can change a body's direction but never its speed, because work requires a force component along the displacement. The magnetic force is exactly such a force, so it cannot speed up or slow down a charge; it only bends the path.

This is why a magnetic field alone cannot add energy to a particle. Accelerators still use electric fields to raise a particle's energy and reserve magnetic fields for steering. The magnetic force is the perfect rudder: it turns without pushing.

Key idea: Being perpendicular to velocity, the magnetic force does no work, changing a charge's direction but never its speed.

Circular motion

A charge moving perpendicular to a uniform field feels a constant sideways push, which bends its path into a circle. The magnetic force provides the centripetal force: q v B = m v^2 / r, which solves to a radius

r = m v / (q B).

Faster or heavier particles curve in larger circles; stronger fields curve them tighter. This principle runs mass spectrometers and particle accelerators, and it steers charged particles in the aurora.

A remarkable consequence hides in this result. The time for one full circle, the period T = 2 pi m / (q B), does not depend on the speed or the radius. A fast particle traces a big circle and a slow one a small circle, but both complete a lap in the same time. This speed-independent rhythm, the cyclotron frequency, is what let early cyclotrons accelerate particles with a steady alternating voltage.

Key idea: A charge circles at radius r = mv/(qB), and its period T = 2 pi m/(qB) is independent of speed, the basis of the cyclotron.

Where the field itself comes from

We have treated B as simply given, but it is worth previewing its source. Every magnetic field, without exception, is produced by moving charge. An electromagnet's field comes from the current in its coils, and even a permanent bar magnet gets its field from the ceaseless motion of electrons within its atoms.

So magnetism is doubly tied to motion: moving charges make magnetic fields, and magnetic fields push only on moving charges. The next module works out how currents create fields, but keep this symmetry in mind. It hints that electricity and magnetism are not two separate forces but two aspects of one.

Key idea: Every magnetic field arises from moving charge, so magnetism both springs from motion and acts only on motion.

Spirals and velocity selectors

If a charge has some velocity along the field as well as across it, the along-field part is untouched while the across-field part circles. The two combine into a helix, a corkscrew path winding along the field lines. Charged particles from the Sun spiral down Earth's field lines this way, lighting the aurora near the poles.

Cross an electric and a magnetic field and you can build a velocity selector. The electric force qE and the magnetic force qvB can be made to oppose each other, and only particles with the exact speed v = E/B pass straight through undeflected. Faster or slower ones are pushed aside. This trick sorts particles by speed at the entrance to a mass spectrometer.

Key idea: A velocity along the field turns circular motion into a helix, and crossed E and B fields select the single speed v = E/B.

Applications: sorting and steering charges

The circular-motion formula is a measuring tool. In a mass spectrometer, ions of a known speed enter a field and curve; since r = mv/(qB), heavier ions swing wider, so the landing position reveals each ion's mass-to-charge ratio. Chemists use this to identify molecules and to separate isotopes atom by atom.

The same physics steers beams elsewhere. A cyclotron spirals particles outward while an alternating voltage speeds them up each half-turn, exploiting the speed-independent period. Old bubble chambers photographed the curved tracks of particles to measure their momenta, and today's accelerators ring their tunnels with magnets to hold beams on course.

Key idea: Because the curvature encodes mass, speed, and charge, magnetic fields sort ions in mass spectrometers and steer beams in cyclotrons and accelerators.

The Hall effect

Send a current through a flat conductor sitting in a magnetic field, and the moving charges are pushed to one edge. Charge piles up there until its own electric field balances the magnetic push, leaving a small measurable voltage across the strip called the Hall voltage.

This effect is doubly useful. The size of the Hall voltage measures the magnetic field, so cheap Hall sensors detect fields in phones, cars, and motors. Its sign reveals whether the moving charges are positive or negative, which is how physicists first confirmed that electrons carry current in metals.

Key idea: The Hall effect produces a voltage across a current-carrying strip in a field, used both to measure fields and to identify the sign of the charge carriers.

Worked example: force on a proton

Given: a proton (q = 1.6 x 10^-19 C) moves at v = 2.0 x 10^6 m/s perpendicular to a 0.50 T field. Find: the magnetic force.

Solution: F = q v B sin 90 = (1.6 x 10^-19)(2.0 x 10^6)(0.50)(1) = 1.6 x 10^-13 N. This force is perpendicular to the motion and curves the proton into a circle.

Worked example: radius of the circle

Given: the same proton (mass m = 1.67 x 10^-27 kg) moving at 2.0 x 10^6 m/s perpendicular to the 0.50 T field. Find: the radius of its circular path.

Solution: r = m v / (q B) = (1.67 x 10^-27)(2.0 x 10^6) / [(1.6 x 10^-19)(0.50)] = (3.34 x 10^-21)/(8.0 x 10^-20) = 0.042 m. The proton circles with a radius of about 4.2 cm, small enough to fit in a tabletop apparatus.

Worked example: the cyclotron period

Given: the same proton in the 0.50 T field. Find: the time for one full circle, and check that it does not depend on the speed.

Solution: T = 2 pi m / (q B) = 2 pi (1.67 x 10^-27) / [(1.6 x 10^-19)(0.50)] = (1.05 x 10^-26)/(8.0 x 10^-20) = 1.3 x 10^-7 s. The speed v never appears, so a faster proton simply traces a bigger circle in the same 0.13 microseconds, exactly the property a cyclotron relies on.

Common misconceptions

• A magnetic field pushes any charge. It acts only on moving charge, and only on the part of the motion across the field.
• The magnetic force speeds charges up. It does no work and cannot change speed, only direction.
• The force points along the field. It points perpendicular to both the velocity and the field.
• Heavier particles always curve more. A larger mass gives a larger radius, so it curves less sharply.
• The circling period depends on speed. It does not; period and cyclotron frequency are independent of speed.

Recap

• The magnetic force is F = qvB sin(theta), perpendicular to both v and B.
• Its direction follows the right-hand rule, reversed for negative charges.
• Being perpendicular to velocity, it does no work and only bends the path.
• Perpendicular motion gives a circle of radius r = mv/(qB) with a speed-independent period.
• Motion along the field adds a helix, and crossed E and B fields select one speed.

Sources

1. OpenStax. (2016). 11.2 Magnetic fields and lines. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 11.3 Motion of a charged particle in a magnetic field. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 11.6 The Hall effect. In University Physics Volume 2. Rice University. openstax.org
4. Nave, R. (n.d.). Magnetic forces. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
5. Nave, R. (n.d.). Hall effect. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
6. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 29: The motion of charges in electric and magnetic fields. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu

A current is just charges in motion, so a wire carrying current in a magnetic field feels a force too. This is the effect that turns every electric motor and moves the needle in analog meters.

This lesson scales the single-charge force up to whole wires and then bends those wires into loops. Out of that comes the torque that spins motors and deflects meter needles, the mechanical output of electromagnetism that does the visible work of the modern world.

Key idea: Because current is moving charge, a magnetic field pushes on current-carrying wires, and this force runs motors, meters, and loudspeakers.

From one charge to a whole wire

The wire force follows directly from the single-charge law. Each drifting charge in the wire feels qvB, and a length L of wire holds a great many such charges. Adding their forces, the charge-per-length and drift speed combine into exactly the current I, and the total force becomes F = B I L for a wire perpendicular to the field.

So no new physics is needed; the wire force is the moving-charge force counted up over all the carriers. That is why the same right-hand rule and the same angle dependence carry straight over from the previous lesson. A current is simply a tidy, controllable stream of the moving charges we already understand.

Key idea: The force on a wire is the single-charge force summed over all its carriers, which is why current gives F = BIL with the same rules as before.

Force on a wire

A straight wire of length L carrying current I in a field B, at angle theta between the wire and the field, feels a force

F = B I L sin(theta).

Like the force on a single charge, it is maximum when the wire is perpendicular to the field and zero when the wire lies along the field. Its direction is perpendicular to both the wire and the field, again from a right-hand rule (point fingers along the current, curl toward B, thumb gives the force).

Key idea: A current-carrying wire feels F = BIL sin(theta), greatest across the field, zero along it, and directed perpendicular to both.

The loudspeaker: force on a wire at work

A loudspeaker is this formula made audible. A coil of wire, the voice coil, sits in the field of a permanent magnet and is glued to a paper cone. Feed the audio signal as a current through the coil, and the force F = BIL pushes the coil back and forth in step with the current.

The cone follows, shoving air to make sound. Louder passages send more current and a bigger force; higher notes reverse the current faster. Nearly every speaker and headphone works this way, a direct everyday use of the force on a current in a field.

Key idea: A loudspeaker drives a wire coil in a magnetic field, turning an audio current into the back-and-forth force that moves a cone and makes sound.

Torque on a current loop

Now bend the wire into a loop. In a uniform field, the forces on opposite sides of the loop are equal and opposite, so there is no net force - but they act on different sides, producing a torque that tries to rotate the loop. For a loop of area A carrying current I with N turns, the torque is

tau = N B I A sin(theta),

where theta is the angle between the field and the loop's normal (the line perpendicular to the loop's plane). The torque is greatest when the loop's plane is parallel to the field and zero when the loop has rotated so its normal lines up with the field. The quantity mu = N I A is called the magnetic moment of the loop.

Key idea: A current loop feels no net force in a uniform field but a torque tau = N B I A sin(theta) that twists it toward alignment.

The magnetic moment and alignment

The magnetic moment mu = N I A captures a loop's response to a field in one quantity, and the torque can be written compactly as tau = mu B sin(theta). The moment points along the loop's normal, in the direction your right thumb gives when your fingers curl the way the current flows.

A current loop therefore behaves just like a tiny compass needle. The torque swings it until its magnetic moment lines up with the field, where the torque vanishes and the energy is lowest. Nudged from alignment, it swings back, and the whole picture explains why bar magnets, compass needles, and current loops all seek to align with a field.

Key idea: The magnetic moment mu = NIA acts like a compass needle, and the field torques it until the moment aligns and the torque drops to zero.

Why so many turns?

Notice that N, the number of turns, multiplies the torque. Each loop of a coil feels the same twist, and stacking N loops in series stacks N torques while the same current flows through them all. This is why motors, meters, and electromagnets are wound with many turns rather than a single loop.

The payoff is leverage. A modest current in a many-turn coil can produce a strong torque even in a weak field, which lets compact devices do real mechanical work. Winding wire into coils is the standard way engineers turn small electrical inputs into large magnetic effects throughout this subject.

Key idea: Because torque scales with the number of turns, coils let a small current in a modest field deliver a strong twist.

How a motor works

This torque is the heart of the electric motor. A current loop in a magnetic field twists toward alignment; just as it gets there, a switching contact called the commutator reverses the current, so the torque keeps pushing the same way and the loop spins continuously. Convert that rotation to a shaft and you can drive a fan, a wheel, or a pump.

The commutator, brushed by sliding contacts, is the clever trick: without it the loop would merely swing to alignment and stop, like a settling compass. By flipping the current every half-turn, it keeps the torque always driving the same way. As the motor spins it also generates a voltage that opposes its supply, a back-EMF we meet in the induction lessons, which is what limits a motor's current and speed.

Motors of this family are everywhere, from the fan cooling a laptop to the traction motors driving an electric car. Real machines use many coils set at different angles so that some are always well placed to feel a strong torque, giving smooth, powerful rotation. Yet every one of them rests on the single idea of this lesson: a current loop twisting in a magnetic field.

Key idea: A motor uses a commutator to reverse the loop current each half-turn, converting the alignment torque into continuous rotation of a shaft.

Galvanometers and analog meters

The same loop torque measures current. In a galvanometer, a coil in a magnetic field is held by a spring; a current produces a torque that twists the coil until the spring balances it. The deflection of the attached needle is proportional to the current, so the scale reads amps directly.

Dress a galvanometer with the right resistors and it becomes an ammeter or a voltmeter, the classic analog meters. Though digital displays have largely replaced them, the swinging-needle movement is a direct mechanical readout of the magnetic torque on a current loop.

Key idea: A galvanometer balances the loop torque against a spring, giving a needle deflection proportional to current, the basis of analog meters.

More machines: rail launchers and maglev

The straight-wire force does more than move speaker cones. A rail launcher sends a large current through a sliding bar between two rails in a magnetic field, and the resulting F = BIL flings the bar forward at high speed. The same principle, on a gentler scale, propels some experimental trains.

Maglev transport uses carefully arranged magnetic forces both to lift a train off its track and to push it along, removing rolling friction entirely. In every case the core idea is unchanged: a current in a field feels a force, and steering that force does useful mechanical work.

Key idea: Rail launchers and maglev trains both harness the F = BIL force on currents to accelerate and levitate large masses.

Worked example: force on a wire

Given: a 0.25 m wire carries 4.0 A perpendicular to a 0.30 T field. Find: the force on it.

Solution: F = B I L sin 90 = (0.30)(4.0)(0.25)(1) = 0.30 N.

Worked example: torque on a coil

Given: a 50-turn coil of area 0.020 m^2 carries 3.0 A in a 0.10 T field, with the field in the plane of the loop (theta = 90 degrees). Find: the torque.

Solution: tau = N B I A sin 90 = (50)(0.10)(3.0)(0.020)(1) = 0.30 N m.

Worked example: torque as the loop turns

Given: the same coil, now rotated so its normal makes a 30 degree angle with the field (theta = 30 degrees). Find: the torque.

Solution: tau = N B I A sin 30 = (50)(0.10)(3.0)(0.020)(0.5) = 0.15 N m, half the peak value. As the loop keeps rotating toward alignment (theta going to 0), the torque keeps shrinking and vanishes at theta = 0, which is exactly why a motor needs its commutator.

Worked example: current from a measured force

Given: a 0.50 m wire perpendicular to a 0.40 T field feels a force of 0.60 N. Find: the current it carries.

Solution: Rearrange F = B I L to I = F/(B L) = 0.60/[(0.40)(0.50)] = 0.60/0.20 = 3.0 A. This reverse calculation is how a current balance can measure a current from a force.

Common misconceptions

• A current loop is thrown across a uniform field. It feels no net force there, only a torque that rotates it.
• The torque keeps growing as the loop turns. It falls to zero once the loop's normal aligns with the field.
• A motor spins because the loop reaches alignment. Alignment would stop it; the commutator flips the current to keep it going.
• The wire force is new physics. It is the single-charge force summed over all the carriers in the wire.
• Force is greatest when the wire lies along the field. It is zero then and greatest when the wire is perpendicular.

Recap

• A current-carrying wire feels F = BIL sin(theta), the single-charge force summed over its carriers.
• A current loop feels a torque tau = NBIA sin(theta) and no net force in a uniform field.
• The magnetic moment mu = NIA makes a loop align with the field like a compass needle.
• A commutator reverses the current each half-turn, turning that torque into continuous rotation in a motor.
• Loudspeakers and galvanometers are the same force and torque put to practical use.

Sources

1. OpenStax. (2016). 11.4 Magnetic force on a current-carrying conductor. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 11.5 Force and torque on a current loop. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 11.7 Applications of magnetic forces and fields. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2022). 22.8 Torque on a current loop: Motors and meters. In College Physics 2e. Rice University. openstax.org
5. Nave, R. (n.d.). Magnetic force on a current-carrying wire. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
6. Nave, R. (n.d.). Magnetic moment. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 13: Magnetostatics. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu

In 1820 Hans Christian Oersted noticed a compass needle deflect near a current-carrying wire. That accident revealed a deep truth: moving charges create magnetic fields. Every magnetic field, even that of a bar magnet, ultimately comes from currents (in a magnet, tiny atomic current loops).

Oersted's chance observation, made during a lecture demonstration, joined two subjects that had seemed unrelated. Until then electricity and magnetism were separate studies; his deflecting needle showed they were linked. Within a decade that link would grow into a full theory, and this lesson takes the first step by finding the field a current makes.

Key idea: Oersted's deflected compass showed that electric currents create magnetic fields, tying electricity and magnetism together.

Field of a long straight wire

A long straight wire carrying current I produces a magnetic field that circles around it. At a distance r from the wire, the field magnitude is

B = mu_0 I / (2 pi r),

where mu_0 = 4 pi x 10^-7 T m / A is the permeability of free space. The field falls off as 1/r and forms concentric circles around the wire. The direction comes from a right-hand rule: point your right thumb along the current, and your fingers curl in the direction of the field.

The permeability mu_0 plays the same role for magnetism that the permittivity epsilon_0 played for electricity, setting the strength of the field a source makes. The two constants are more than analogous: they combine to give the speed of light through c = 1/sqrt(mu_0 epsilon_0), a clue that the final lesson develops into the theory of electromagnetic waves.

Notice that this field falls off as 1/r, more slowly than the 1/r^2 of a point charge. A single straight wire is an extended source, and its gentle falloff means its field reaches noticeably far, which is why a compass responds a good distance from a current.

Key idea: A straight wire's field is B = mu_0 I/(2 pi r), circling the wire by the right-hand grip rule and falling off as 1/r.

Superposition of wire fields

When several wires carry current, their fields add as vectors, exactly as electric fields do. To find the field at a point you compute each wire's contribution from B = mu_0 I/(2 pi r), get its direction from the grip rule, and sum the vectors.

The result can reinforce or cancel. Midway between two wires carrying current in opposite directions, both fields point the same way and add to a strong field. For currents in the same direction, the two fields oppose at the midpoint and partly cancel. Reading which case you have before calculating, just as with charges, prevents sign mistakes.

Key idea: Fields from several wires superpose as vectors, reinforcing or cancelling depending on the current directions.

The general rule: Biot-Savart

Where does the wire formula come from? The underlying law is the Biot-Savart law, which says every tiny segment of current makes its own small contribution to the field, falling off as the inverse square of distance from that segment. Adding up all the segments of a long straight wire yields the 1/r result we used.

Biot-Savart handles any shape of wire, but its sums can be laborious. The next lesson introduces Ampere's law, a shortcut that, like Gauss's law, delivers the field in one step whenever the geometry is symmetric. Biot-Savart is the general engine; Ampere's law is the fast tool for special cases.

Key idea: The Biot-Savart law builds any current's field by adding the inverse-square contributions of every current segment, and it underlies the simpler formulas.

Field of a loop and a solenoid

Bend the wire into a loop and the circular field lines pass through the center all in the same direction, concentrating the field there - the loop acts like a small magnet with a north and south face. Stack many loops into a coil and you get a solenoid. Inside a long solenoid the field is remarkably uniform and parallel to the axis:

B = mu_0 n I,

where n is the number of turns per unit length. A solenoid is the practical way to make a strong, controllable magnetic field, and it is the basis of electromagnets.

A single loop of radius R produces a field B = mu_0 I/(2R) at its center, and the loop's two faces behave exactly like the poles of a tiny bar magnet. Notice the solenoid's field depends on the turns per meter n, not on how wide the coil is, so a tightly wound coil gives a strong field even at modest current.

Slide an iron core inside a solenoid and the field can grow hundreds of times stronger, because the iron's own atomic magnets line up and add to it. This is how a powerful electromagnet is built, and unlike a permanent magnet it can be switched off, which is why scrapyard cranes and door locks use them.

Key idea: A loop concentrates field like a small magnet, a solenoid gives a uniform interior field B = mu_0 n I, and an iron core turns it into a strong, switchable electromagnet.

Two parallel wires

Two parallel wires carrying current interact through their fields: currents in the same direction attract, and currents in opposite directions repel. This force is actually how the ampere was historically defined.

The mechanism ties this lesson to the last one. Each wire sits in the magnetic field created by the other, so each feels the F = BIL force on a current in a field. Working out the directions with the right-hand rules gives attraction for parallel currents and repulsion for antiparallel ones. The force per length is mu_0 I1 I2/(2 pi d) for a separation d.

Because this force depends only on measurable currents and distances, it once served to define the ampere: the current that produces a specified force between two wires one meter apart. It is a satisfying closing of the loop, with the field a current makes now pushing on another current.

Key idea: Parallel currents attract and antiparallel ones repel, each wire feeling the F = BIL force in the other's field, a relation once used to define the ampere.

The right-hand rules, gathered

Magnetism uses the right hand in a few related ways, and keeping them straight is worth a moment. For the field around a wire, point the thumb along the current and the curled fingers show the circling field. For the field of a coil, curl the fingers along the current and the thumb points out the coil's north face.

For the force on a moving charge or a current, point the fingers along the velocity or current, curl toward B, and the thumb gives the force. All three are the same underlying geometry seen from different angles. A little practice makes the correct gesture automatic, and it rarely fails you.

Key idea: One right hand gives the field around a wire, the north face of a coil, and the force on a current, three views of the same geometry.

Magnetic fields around us

These field sources appear throughout technology and nature. Electromagnets built as solenoids lift cars in scrapyards, click the relays that switch heavy machinery, and provide the intense, steady field of an MRI scanner. Every loudspeaker and motor pairs a coil's field with a permanent magnet.

A relay shows the idea in miniature: a small current energizes a solenoid, whose field pulls an iron lever that closes a second, heavier circuit. This lets a weak signal switch a powerful load, the same trick behind the starter in a car and the controls in industrial machinery. Switchability is the electromagnet's great advantage over a permanent magnet.

Nature makes fields the same way. The Earth's magnetic field, which swings your compass, is generated by vast electric currents churning in its molten iron core. From a tabletop wire to a planet, the rule is one and the same: current makes field.

Key idea: From electromagnets and MRI scanners to the Earth's core, magnetic fields everywhere are made by electric currents.

How magnetic field lines differ from electric ones

There is a striking contrast with the electric case. Electric field lines begin on positive charges and end on negative ones. Magnetic field lines, by contrast, never begin or end anywhere; they always close on themselves into complete loops, circling the wire and returning.

This reflects a deep fact: there are no isolated magnetic poles, no lone north without a south. Break a bar magnet in half and each piece grows a new set of poles rather than yielding a separate north. Because field lines close rather than terminate, the net magnetic flux through any closed surface is always zero, a statement that becomes one of Maxwell's four equations.

Key idea: Magnetic field lines form closed loops rather than starting and ending on poles, because isolated magnetic poles do not exist.

Worked example: field near a wire

Given: a long wire carries 10 A. Find: the field 0.050 m away.

Solution: B = mu_0 I / (2 pi r) = (4 pi x 10^-7)(10) / (2 pi x 0.050). Cancel pi: = (2 x 10^-7)(10)/0.050 = (2.0 x 10^-6)/0.050 = 4.0 x 10^-5 T. So the field is 4.0 x 10^-5 T (about the strength of the Earth's field).

Worked example: field inside a solenoid

Given: a solenoid of 800 turns over a length of 0.40 m carries 3.0 A. Find: the interior field.

Solution: First the turns per meter: n = 800/0.40 = 2000 turns/m. Then B = mu_0 n I = (4 pi x 10^-7)(2000)(3.0) = 7.5 x 10^-3 T. This is far stronger than the straight-wire field above, because the solenoid stacks the contributions of many turns into one uniform interior field.

Common misconceptions

• A wire's field points along the wire. It circles the wire in concentric loops, perpendicular to the current.
• The straight-wire field falls off as 1/r^2. It falls off as 1/r, more slowly than a point charge's field.
• A solenoid's field depends on its diameter. It depends on turns per meter and current, not the coil's width.
• Permanent magnets are a separate kind of magnetism. Their field also comes from current, in the form of atomic-scale electron motion.
• Parallel currents repel. Parallel currents attract; only opposite currents repel.

Recap

• Oersted showed currents make magnetic fields, uniting electricity and magnetism.
• A straight wire gives B = mu_0 I/(2 pi r), circling the wire and falling off as 1/r.
• A solenoid gives a uniform interior field B = mu_0 n I, the basis of electromagnets.
• Parallel currents attract and antiparallel currents repel, once used to define the ampere.
• All magnetic fields, from wires to the Earth's core, trace back to moving charge.

Sources

1. OpenStax. (2016). 12.1 The Biot-Savart law. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 12.2 Magnetic field due to a thin straight wire. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 12.3 Magnetic force between two parallel currents. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 12.4 Magnetic field of a current loop. In University Physics Volume 2. Rice University. openstax.org
5. OpenStax. (2016). 11.1 Magnetism and its historical discoveries. In University Physics Volume 2. Rice University. openstax.org
6. Nave, R. (n.d.). Magnetic fields of currents. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 14: The magnetic field in various situations. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu

Just as Gauss's law provides a symmetry shortcut for electric fields, Ampere's law does the same for magnetic fields created by currents. It is another of the four Maxwell equations.

Where the Biot-Savart law of the last lesson is the general but laborious engine, Ampere's law is the elegant shortcut. When a current arrangement has enough symmetry, it hands you the field in a single step, turning a hard integral into a one-line calculation. Recognizing that symmetry is the whole art.

Key idea: Ampere's law is a symmetry shortcut for magnetic fields, the magnetic counterpart of Gauss's law and one of Maxwell's four equations.

Statement of the law

Ampere's law relates the magnetic field summed around a closed loop (an Amperian loop) to the current passing through that loop:

(sum of B parallel to the loop) x (loop length) = mu_0 I_enclosed.

In the common textbook form for a path on which B is constant and along the path, this is B L = mu_0 I_enclosed. The law says the circulation of B around any closed path is set only by the current threading that path - fields from currents outside the loop contribute nothing to the total.

Key idea: The circulation of B around any closed loop equals mu_0 times the current enclosed, regardless of currents outside the loop.

What "circulation" means

The left side of the law, the circulation, deserves a plain description. Walk once around the chosen loop and, step by step, add up the component of B that points along your direction of travel, weighted by the length of each step. That running total, all the way around, is the circulation.

Where the field runs with you, it adds positively; where it runs against you, it subtracts; where it crosses your path at right angles, it adds nothing. On a well-chosen symmetric loop the field has the same size and runs along the whole path, so the sum collapses to simply B times the loop's length. That is why the law becomes so easy in symmetric cases.

Key idea: Circulation is the running sum of B along a closed loop, which collapses to B times the loop length when the field is uniform and follows the path.

The parallel with Gauss's law

The two great shortcut laws are close cousins, and seeing the parallel makes both easier to hold. Gauss's law relates the flux of the electric field through a closed surface to the charge enclosed. Ampere's law relates the circulation of the magnetic field around a closed loop to the current enclosed.

Each law is always true, but each becomes a practical calculating tool only when symmetry makes the field constant along the chosen boundary. And each ignores its sources on the outside: external charge adds no net flux, external current adds no net circulation. Learning one law deepens your grasp of the other.

Key idea: Ampere's law mirrors Gauss's law, trading flux through a surface for circulation around a loop and enclosed charge for enclosed current.

Choosing a good loop

Success with Ampere's law rests on picking the right Amperian loop, just as Gauss's law needs the right surface. A good loop follows the field's symmetry, so that B is either constant and along the path or else perpendicular to it, contributing nothing.

For a straight wire the natural loop is a circle centered on the wire, matching the field's circular symmetry. For a solenoid it is a rectangle with one side down the uniform interior. The current you count is only the current that actually threads the loop, with its sign set by a right-hand rule: curl your fingers along the loop and your thumb gives the positive current direction.

Key idea: Choose an Amperian loop that follows the field's symmetry, and count only the current threading it, signed by the right-hand rule.

Ampere and the birth of electrodynamics

The law bears the name of Andre-Marie Ampere, who in the 1820s raced to build a mathematical theory of currents within weeks of hearing about Oersted's discovery. He measured the forces between wires, worked out the rules of direction, and laid the foundations of what he called electrodynamics.

The compact circulation form used today was polished later, but the physics is his. It is worth appreciating how quickly the subject grew: a chance deflection of a compass in 1820 became, within a few years, a quantitative science of currents and their fields, and within a few decades the theory of light.

Key idea: Ampere built the first quantitative theory of currents and forces in the 1820s, founding the electrodynamics that this law summarizes.

Deriving the wire field

For a long straight wire, choose a circular Amperian loop of radius r centered on the wire. By symmetry B has the same magnitude all around and points along the circle, so the left side is B (2 pi r). The enclosed current is I. Setting them equal:

B (2 pi r) = mu_0 I, giving B = mu_0 I / (2 pi r) - exactly the result from before, now derived in one line.

Compare the effort. Adding up Biot-Savart contributions from every segment of an infinite wire takes a real integral, yet Ampere's law reaches the same answer in a single step. That economy is the whole reason the law earns its place beside Gauss's law.

Key idea: A circular Amperian loop gives B (2 pi r) = mu_0 I and the straight-wire field in one line, where Biot-Savart would need an integral.

Deriving the solenoid field

For a solenoid, a rectangular Amperian loop that runs along the axis inside and returns outside (where B is nearly zero) encloses N turns of current. The result is the uniform interior field B = mu_0 n I. Ampere's law is the cleanest route to both of these standard fields.

The same method handles a toroid, a solenoid bent into a doughnut. A circular loop threading the ring encloses all N turns, giving an interior field B = mu_0 N I/(2 pi r) that is confined almost entirely within the doughnut. Toroids are prized in electronics precisely because they keep their field to themselves, avoiding interference with nearby parts.

Key idea: A rectangular loop yields the solenoid field B = mu_0 n I, and a circular loop yields the toroid field, both confined to the coil's interior.

When it works

Ampere's law is always true, but it only lets you solve for B when the situation has high symmetry - a long straight wire, an ideal solenoid, or a toroid - so that B is constant and simply related to the path. For irregular geometries you fall back on the Biot-Savart law (an integral over the current), which is beyond this quick treatment but rests on the same physics.

There is one more subtlety, and it turned out to be historic. As stated, the law fails for a circuit with a charging capacitor, where current stops at the plates. James Clerk Maxwell fixed this by adding a term for a changing electric field, the so-called displacement current. That repair, which we reach in the final lesson, is what let electromagnetism predict light itself.

Key idea: Ampere's law solves for B only under high symmetry, and Maxwell's added changing-field term completes it for all cases, including charging capacitors.

The field profile of a real wire

Ampere's law reveals the full field of a thick, uniformly conducting wire, inside and out. Inside, an Amperian circle encloses only the current within it, which grows as the area, so the field rises linearly from zero at the axis to a maximum right at the surface.

Outside, the loop encloses the whole current, so the field falls off as the familiar 1/r. The two pieces meet smoothly at the surface, where the field is strongest. This tidy profile, zero at the center, peaking at the edge, then decaying outward, would be painful to obtain by direct summation but drops out of Ampere's law almost for free.

Key idea: Inside a uniform wire the field rises linearly to a peak at the surface, then falls as 1/r outside, a full profile Ampere's law gives easily.

Where the law shows its power

The handful of geometries Ampere's law solves cleanly happen to be the ones engineers use most. The solenoid is the standard electromagnet and the sensing coil in countless devices. The toroid, with its self-contained field, is the shape of choice for transformer cores and inductors that must not leak field into their neighbors.

Even the humble coaxial cable yields to the law: the field between its inner wire and outer shield follows the straight-wire result, while outside the shield the equal and opposite currents enclose zero net current, so the field vanishes. That is exactly why coaxial cable carries signals without radiating them away. Symmetry, once spotted, turns each of these into a one-line calculation.

Key idea: Solenoids, toroids, and coaxial cables all have the symmetry Ampere's law needs, which is why the law is a workhorse of real engineering.

Worked example: field inside a wire loop path

Given: an Amperian circle of radius 0.10 m encloses a wire carrying 8.0 A. Find: the field on the circle.

Solution: B = mu_0 I / (2 pi r) = (4 pi x 10^-7)(8.0)/(2 pi x 0.10) = (2 x 10^-7)(8.0)/0.10 = 1.6 x 10^-5 T.

Worked example: inside a thick wire

Given: a solid wire of radius 2.0 mm carries 8.0 A spread uniformly across its cross-section. Find: the field at 1.0 mm from the axis, inside the wire.

Solution: An Amperian circle of radius 1.0 mm encloses only the fraction of current inside it: I_enc = I (r/R)^2 = 8.0 (1.0/2.0)^2 = 2.0 A. Then B = mu_0 I_enc/(2 pi r) = (4 pi x 10^-7)(2.0)/(2 pi x 0.0010) = (2 x 10^-7)(2.0)/0.0010 = 4.0 x 10^-4 T. Inside the wire the field grows with radius, then falls off as 1/r outside.

Common misconceptions

• Ampere's law only holds for symmetric currents. It always holds; only solving for B needs symmetry.
• Currents outside the loop change the circulation. They contribute nothing to the net circulation, though they do affect the field at points.
• The field inside a wire is zero. It grows linearly from zero at the axis to a maximum at the surface.
• Ampere's law and Gauss's law are unrelated. They are close analogs, one for circulation, one for flux.
• The original law is complete. It needed Maxwell's changing-field term to handle charging capacitors.

Recap

• Ampere's law: the circulation of B around a loop equals mu_0 I_enclosed.
• It is the magnetic analog of Gauss's law and works cleanly only with symmetry.
• A circular loop gives the straight-wire field; a rectangle gives the solenoid field; a toroid confines its field inside.
• Inside a uniform wire the field rises with radius, then falls as 1/r outside.
• Maxwell later added a changing-electric-field term that completed the law.

Sources

1. OpenStax. (2016). 12.5 Ampere's law. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 12.6 Solenoids and toroids. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2022). 22.9 Magnetic fields produced by currents: Ampere's law. In College Physics 2e. Rice University. openstax.org
4. Nave, R. (n.d.). Ampere's law. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
5. Nave, R. (n.d.). Solenoids as magnetic field sources. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
6. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 13: Magnetostatics. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu
7. O'Connor, J. J., & Robertson, E. F. (n.d.). Andre-Marie Ampere. MacTutor History of Mathematics Archive, University of St Andrews. mathshistory.st-andrews.ac.uk

Oersted showed that current makes a magnetic field. The reverse question - can a magnetic field make a current? - was answered by Michael Faraday in 1831. The answer is yes, but with a crucial twist: it takes a changing magnetic field. This discovery, electromagnetic induction, is how nearly all the world's electricity is generated.

The word "changing" is the whole story. A magnet resting inside a coil, however strong, drives no current. Only while the magnetic conditions are shifting does a voltage appear. Faraday's insight turned magnetism from a curiosity into the engine of the electrical age, and it is the principle behind every power plant on the grid.

Key idea: A changing magnetic field induces a voltage, the effect called electromagnetic induction that generates almost all of the world's electricity.

Magnetic flux

First we need magnetic flux, the magnetic analog of electric flux - the amount of field passing through a loop:

Phi_B = B A cos(theta),

where A is the loop area and theta is the angle between the field and the loop's normal. Flux is measured in webers (Wb). You can change the flux three ways: change the field strength B, change the loop area A, or rotate the loop to change theta.

Picture flux as the number of field lines threading the loop. Turn the loop face-on to the field and it catches the most lines; turn it edge-on and it catches none. Because there are three separate ways to change the flux, there are three distinct ways to generate electricity, and real machines exploit each of them.

Key idea: Magnetic flux Phi_B = B A cos(theta) counts the field lines through a loop and can be changed by altering B, the area, or the angle.

Faraday's law

Faraday's law says an EMF (voltage) is induced in a loop equal to the rate of change of flux through it:

EMF = - N (change in Phi_B) / (change in time),

for N turns. The faster the flux changes, the bigger the induced voltage. A magnet sitting still in a coil induces nothing; a magnet moving through the coil induces a voltage that drives current. This is exactly how a generator works - rotating a coil in a magnetic field continuously changes the flux and produces alternating current.

Two factors set the size of the EMF: how fast the flux changes and how many turns the coil has. Doubling the turns doubles the voltage, because each turn contributes its own induced EMF and they add in series. This is why generator and transformer coils carry many hundreds of turns.

Key idea: Faraday's law gives an induced EMF equal to N times the rate of change of flux, so faster change and more turns both raise the voltage.

Motional EMF: a moving conductor

One clean case of induction needs no magnet to move at all. Slide a straight conductor of length L across a magnetic field at speed v, and the free charges inside it feel the magnetic force qvB, which drives them to one end. Charge piles up until its electric field balances the push, leaving a voltage across the rod:

EMF = B L v.

This motional EMF is Faraday's law in disguise: as the rod moves, it sweeps out area, so the flux through the circuit changes at exactly the rate B L v. The rod acts like a little battery, and if the circuit is closed, a current flows. It is the seed of every generator, where moving conductors continuously cut across field lines.

Key idea: A conductor of length L moving at speed v across a field develops a motional EMF B L v, which is Faraday's law seen as swept-out area.

Lenz's law: the minus sign

The minus sign is Lenz's law: the induced current flows in the direction that opposes the change in flux that produced it. If you push a magnet's north pole toward a coil, the coil's near face becomes a north pole to push back. Pull the magnet away, and the coil's face becomes a south pole to pull it back. This is energy conservation in action - you must do work against this opposition to generate electricity, and that work is what becomes the electrical energy. If the induced current aided the change instead, you would get energy for free.

Lenz's law is really a promise that induction cannot cheat. The opposition is nature's way of paying for the electrical energy with mechanical work. A generator is hard to turn precisely because the current it makes pushes back on the coil; switch off the electrical load and it spins more freely.

Key idea: Lenz's law says the induced current opposes the change that made it, enforcing energy conservation so that induction always costs mechanical work.

Eddy currents and magnetic braking

Induction happens in solid metal too, not just in wire loops. A changing flux through a chunk of conductor drives swirling eddy currents within it. By Lenz's law these currents oppose the change, and their most visible effect is a braking force.

Drop a magnet down a copper tube and it falls in slow motion. The moving magnet changes the flux through each ring of the tube, inducing eddy currents whose own field pushes back on the magnet, braking its fall. The same effect gives smooth, contactless brakes on trains and roller coasters, and it heats the metal in an induction cooktop and lights the coils of a metal detector.

Key idea: A changing flux in solid metal drives eddy currents that, by Lenz's law, brake motion and generate heat, used in magnetic brakes and induction cooktops.

Generators and the grid

A generator is Faraday's law turned into a machine. Spin a coil in a magnetic field, or spin a magnet past a coil, and the flux rises and falls smoothly through each turn, inducing a voltage that reverses every half-rotation. The output is alternating current, the sinusoidal voltage that fills the grid.

Almost all electricity is made this way. A turbine driven by steam, falling water, or wind spins the generator, converting mechanical energy into electrical energy through induction. Only the source of the spin differs between a coal plant, a hydroelectric dam, and a wind farm; the induction at the heart of each is identical.

Key idea: A generator spins a coil in a field to induce alternating current, and nearly all grid electricity is made by driving such generators with turbines.

Transformers: sharing flux between coils

Wind two coils on the same iron core and Faraday's law couples them. An alternating current in the first coil makes a changing flux, and that same flux threads the second coil and induces a voltage in it. This is a transformer, and it works only with alternating current, because a steady current gives no change of flux and no output.

The voltage ratio equals the turns ratio: more turns on the output coil step the voltage up, fewer step it down. This is the device that makes the grid practical, raising voltage to hundreds of thousands of volts for efficient transmission and lowering it again near homes. The link between electricity and magnetism, first hinted at by Oersted, here becomes indispensable infrastructure.

Key idea: A transformer couples two coils through shared changing flux, stepping AC voltage up or down by the turns ratio, which is why the grid uses AC.

Faraday's world-changing idea

Michael Faraday came to science with little formal mathematics, yet his physical intuition about lines of force reshaped the subject. His 1831 experiments with coils and moving magnets revealed induction, and within his lifetime the principle grew into the dynamo, the first practical generator of electricity.

It is hard to overstate the consequence. Every power station, from the largest dam to a bicycle dynamo, runs on the effect Faraday found. When a politician asked what use his discovery could possibly have, he is said to have replied that one day it might be taxed, a wry forecast of the industry it would create.

Key idea: Faraday's 1831 discovery of induction grew into the dynamo and the generator, the foundation of the entire electric-power industry.

Worked example: a shrinking loop

Given: a single loop of area 0.040 m^2 sits perpendicular to a field that grows from 0.10 T to 0.50 T in 0.20 s. Find: the induced EMF.

Solution: With theta = 0, flux is Phi = B A. The change is (0.50 - 0.10)(0.040) = 0.016 Wb. The EMF magnitude is |EMF| = (change in Phi)/(time) = 0.016/0.20 = 0.080 V. The direction, by Lenz's law, opposes the increase - the induced current makes a field pointing against the growing external field.

Worked example: a sliding rod

Given: a rod 0.20 m long slides at 3.0 m/s along rails in a 0.50 T field perpendicular to its motion. Find: the motional EMF, and the current if the rails close through a 6.0 ohm resistor.

Solution: EMF = B L v = (0.50)(0.20)(3.0) = 0.30 V. With the resistor, I = EMF/R = 0.30/6.0 = 0.050 A. By Lenz's law the current creates a force opposing the rod's motion, so you must keep pushing to keep it sliding.

Looking ahead: a coil induces on itself

So far the changing flux has come from outside the loop. But a coil's own current makes a flux through itself, so when that current changes, the coil induces a voltage in itself that, by Lenz's law, opposes the change. This self-induction is the subject of the next lesson.

The consequence is a kind of electrical inertia: a coil resists sudden changes in its current, just as mass resists sudden changes in motion. Recognizing that Faraday's law applies to a coil's own field, not only to external magnets, opens the door to inductors and to the behavior of alternating-current circuits.

Key idea: A coil's own changing current induces an opposing voltage in itself, a self-induction that gives coils an electrical inertia explored in the next lesson.

Common misconceptions

• A steady field induces a current. Only a changing flux induces an EMF; a constant flux induces nothing.
• Induction gives energy for free. Lenz's law guarantees the induced current opposes the change, so mechanical work always pays for it.
• Only a moving magnet can induce. Changing the field, the area, or the angle all change the flux and induce an EMF.
• Eddy currents need a wire loop. They swirl freely inside solid metal whenever the flux there changes.
• The minus sign is a bookkeeping quirk. It is Lenz's law, an expression of energy conservation.

Recap

• Magnetic flux is Phi_B = B A cos(theta), measured in webers.
• Faraday's law: the induced EMF equals N times the rate of change of flux.
• A conductor moving across a field gives a motional EMF B L v.
• Lenz's law makes the induced current oppose the change, conserving energy.
• Generators and eddy-current brakes are induction put to work.

Sources

1. OpenStax. (2016). 13.1 Faraday's law. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 13.2 Lenz's law. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 13.3 Motional emf. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 13.5 Eddy currents. In University Physics Volume 2. Rice University. openstax.org
5. OpenStax. (2016). 13.6 Electric generators and back emf. In University Physics Volume 2. Rice University. openstax.org
6. Nave, R. (n.d.). Faraday's law. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 17: The laws of induction. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu
8. Faraday, M. (1832). V. Experimental researches in electricity. Philosophical Transactions of the Royal Society of London, pp. 125-162. doi.org/10.1098/rstl.1832.0006

Faraday's law has a consequence for any coil carrying a changing current: the coil's own changing field induces a voltage in itself that opposes the change. This property is self-inductance, and a component built to have it is an inductor - usually just a coil of wire, sometimes around an iron core.

An inductor is the magnetic mirror of the capacitor. Where a capacitor stores energy in an electric field and resists changes in voltage, an inductor stores energy in a magnetic field and resists changes in current. Together these two components, with the resistor, make up the basic vocabulary of every analog circuit.

Key idea: An inductor is a coil that opposes changes in its own current through self-induction, storing energy in a magnetic field as a capacitor does in an electric one.

Defining inductance

The inductance L relates the flux linkage of a coil to the current through it, and it sets how much voltage a changing current induces:

EMF = - L (change in I) / (change in time).

Inductance is measured in henries (H). The equation says an inductor resists changes in current: try to increase the current quickly and it pushes back with a voltage, like electrical inertia. A steady current, however, passes freely - once the current stops changing, the inductor's voltage is zero.

The minus sign is Lenz's law again: the induced voltage always opposes whatever the current is trying to do. Ramp the current up and the inductor fights the rise; let it fall and the inductor fights the fall, trying to keep the current going. This stubbornness is the defining behavior of an inductor.

Key idea: Inductance in henries sets the self-induced voltage EMF = -L (change in I)/(time), which always opposes the change in current.

What sets a coil's inductance

Like capacitance, inductance is fixed by geometry. For a solenoid it is L = mu_0 n^2 A l, where n is the turns per meter, A the cross-sectional area, and l the length. The turns enter squared, so winding more turns raises inductance sharply, one factor because each turn links more flux and another because there are more turns to induce voltage in.

Slipping an iron core inside multiplies the inductance further, often by hundreds, because the iron concentrates the magnetic flux. This is why practical inductors are compact coils on magnetic cores rather than long air-wound solenoids. The same design choices that make a strong electromagnet also make a large inductance.

Key idea: A coil's inductance grows with the square of its turns and with an iron core, so many turns on a core give a large inductance.

How big is an inductor?

The henry, like the farad, is a large unit. The small coils that tune radios measure in microhenries, general-purpose inductors in millihenries, and the big iron-cored chokes in power supplies reach a henry or more. A one-henry air coil would be bulky, which is why cores and many turns are used to pack inductance into a small package.

Keeping these scales in mind helps you sanity-check answers, just as with capacitors. A circuit inductance of many henries points to a large iron-cored coil, while the traces on a circuit board carry only nanohenries. The number tells you roughly what the component looks like.

Key idea: Practical inductances run from microhenries in radio coils to henries in power chokes, with cores and many turns packing inductance into a small size.

Energy stored in an inductor

Building up current in an inductor stores energy in its magnetic field, just as charging a capacitor stores energy in an electric field. The stored energy is

U = (1/2) L I^2.

Compare this with the capacitor's (1/2) C V^2 - the same shape, with L and current in place of C and voltage. This stored magnetic energy is why breaking an inductive circuit can cause a spark: the collapsing field dumps its energy suddenly.

As with the capacitor, the energy truly resides in the field. The magnetic energy density is u = B^2 / (2 mu_0), so a stronger field packs more energy into each cubic meter. That same field energy, in both electric and magnetic form, is what an electromagnetic wave carries through empty space in the final lesson.

Key idea: An inductor stores U = (1/2) L I^2 in its magnetic field, at density B^2/(2 mu_0), mirroring the capacitor's electric energy.

Opening a switch: the inductive kick

Because an inductor fights any change in current, suddenly opening a switch in an inductive circuit is dramatic. The current tries to drop to zero in an instant, so the rate of change is enormous, and the induced voltage L (change in I)/(time) spikes far above the supply, often arcing across the opening switch as a spark.

Engineers tame this inductive kick with a protective diode across the coil, giving the current a safe path as the field collapses. The same effect is put to use in a car's ignition coil, which deliberately interrupts a current to generate the thousands of volts that jump the spark-plug gap. What is a hazard in one circuit is the whole point in another.

Key idea: Interrupting an inductor's current produces a large voltage spike, a hazard tamed by a diode but harnessed in an ignition coil to make sparks.

RL circuits

A resistor and inductor in series with a battery form an RL circuit. Because the inductor opposes sudden change, the current does not jump to its final value but rises smoothly, with a time constant tau = L / R. After one time constant the current has reached about 63% of its maximum - the same exponential shape as the RC circuit, but for current instead of charge.

The parallel with the RC circuit is exact in form. There, a resistor slowed the buildup of charge on a capacitor; here it slows the buildup of current through an inductor. In both, the approach to the final state is exponential and the timescale is set by a single time constant, though the RL constant is L/R rather than RC.

Key idea: In an RL circuit the current rises exponentially with time constant tau = L/R, reaching about 63% of its final value in one tau.

Inductors and capacitors together

Pair an inductor with a capacitor and something new appears: oscillation. Energy sloshes back and forth between the capacitor's electric field and the inductor's magnetic field, first one full then the other, over and over. The circuit rings at a natural frequency set by L and C, much as a pendulum swings between height and speed.

This LC oscillation is the heart of the tuning circuit in a radio, which selects one station's frequency from the crowd. It also foreshadows the alternating-current behavior of the next module, where inductors and capacitors respond in opposite ways to a changing drive and can be balanced at resonance.

The natural frequency is f = 1 / (2 pi sqrt(L C)). A larger inductance or capacitance slows the oscillation, just as a heavier pendulum or a longer spring swings more slowly. Tuning a radio to a station means adjusting L or C until this frequency matches the station's, so the circuit rings strongly at that one frequency and ignores the rest.

Key idea: An LC pair oscillates at f = 1/(2 pi sqrt(LC)), trading energy between magnetic and electric fields, and tuning a radio matches this frequency to a station.

Inductors at work

Because an inductor resists changes in current, it acts as a choke that smooths a fluctuating current, passing steady flow while blocking rapid wiggles. Paired with capacitors it builds filters that separate signals by frequency, a role as common in radios and power supplies as the resistor networks of earlier lessons.

Two coils close together share flux, so a changing current in one induces a voltage in the other, an effect called mutual inductance. This is how a transformer couples its windings, and how a phone charges wirelessly on a pad: the changing field of one coil induces current in the other with no wires between them. Every one of these uses rests on the single idea that a changing current makes an opposing voltage.

Key idea: Inductors smooth currents as chokes, build frequency filters with capacitors, and couple coils through mutual inductance in transformers and wireless chargers.

Worked example: inductor voltage

Given: a 0.50 H inductor carries a current that increases at a rate of 4.0 A/s. Find: the induced voltage.

Solution: |EMF| = L (change in I)/(time) = (0.50)(4.0) = 2.0 V.

Worked example: stored energy

Given: a 0.20 H inductor carries 3.0 A. Find: the energy stored.

Solution: U = (1/2) L I^2 = (0.5)(0.20)(3.0)^2 = (0.5)(0.20)(9.0) = 0.90 J.

Worked example: an RL time constant

Given: a 0.40 H inductor in series with a 20 ohm resistor and a battery. Find: the time constant and roughly how long until the current is nearly steady.

Solution: tau = L/R = 0.40/20 = 0.020 s. The current reaches about 63% of its final value in 0.020 s and is essentially steady after about five time constants, roughly 0.10 s.

The duality with the capacitor

Inductor and capacitor are mirror images, and lining them up makes both easier to remember. A capacitor stores energy (1/2) C V^2 in an electric field and opposes changes in voltage; an inductor stores (1/2) L I^2 in a magnetic field and opposes changes in current.

Swap voltage for current, C for L, and electric field for magnetic field, and every statement about one becomes a true statement about the other. This deep symmetry is why the two so often appear together, and why an LC circuit oscillates: energy poured into the electric field of the capacitor flows into the magnetic field of the inductor and back again, endlessly.

Key idea: The inductor is the capacitor's dual, swapping current for voltage and magnetic field for electric, which is why the two together oscillate.

Common misconceptions

• An inductor blocks all current. It only opposes changes; a steady current passes with no induced voltage.
• Energy in an inductor is stored in the wire. It is stored in the surrounding magnetic field, at density B^2/(2 mu_0).
• Inductance depends on the current. Like capacitance, it is fixed by geometry and the core, not by the current.
• Opening a switch simply stops the current. The current's resistance to change can spike the voltage and cause a spark.
• Inductors and capacitors behave alike. One opposes current change, the other voltage change; they are duals.

Recap

• An inductor opposes changes in current, with EMF = -L (change in I)/(time) in henries.
• Inductance is set by geometry, growing with the square of the turns and with an iron core.
• An inductor stores U = (1/2) L I^2 in its magnetic field, at density B^2/(2 mu_0).
• An RL circuit's current rises exponentially with time constant tau = L/R.
• With a capacitor, an inductor oscillates, trading energy between magnetic and electric fields.

Sources

1. OpenStax. (2016). 14.2 Self-inductance and inductors. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 14.3 Energy in a magnetic field. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 14.4 RL circuits. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 14.1 Mutual inductance. In University Physics Volume 2. Rice University. openstax.org
5. OpenStax. (2022). 23.9 Inductance. In College Physics 2e. Rice University. openstax.org
6. Nave, R. (n.d.). Inductor concepts. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
7. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 16: Induced currents. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu

By the 1860s the separate laws of electricity and magnetism were known. James Clerk Maxwell wrote them as four equations, spotted a missing piece, and in adding it discovered something breathtaking: light itself is an electromagnetic wave. This synthesis is one of the great achievements in the history of science.

This capstone lesson gathers every idea in the course. The charge and field of the first module, the potential and current of the middle, and the magnetism and induction of the last all fold into four compact statements. From them springs the prediction of light, tying the whole subject together and reaching far beyond it.

It is worth pausing to appreciate the scale of the achievement. Before Maxwell, electricity, magnetism, and optics were three separate sciences with three separate sets of experts. After him, they were one theory, written in four lines, that not only explained what was known but predicted a whole spectrum of invisible waves no one had yet detected. Few unifications in science have been so complete.

Key idea: Maxwell unified electricity and magnetism into four equations and, by completing them, showed that light is an electromagnetic wave.

The four Maxwell equations, in words

1. Gauss's law (electric): electric charges create electric fields; the flux out of a closed surface is set by the enclosed charge.
2. Gauss's law (magnetic): there are no magnetic monopoles; magnetic field lines always form closed loops, so the net magnetic flux through any closed surface is zero.
3. Faraday's law: a changing magnetic field creates an electric field (this is induction).
4. Ampere-Maxwell law: both currents and changing electric fields create magnetic fields. The "changing electric field" part is Maxwell's addition.

Read as a set, these four say something remarkable. Charges make electric fields, there are no magnetic charges, and each kind of field can be created by the other one changing. That last symmetry, between Faraday's law and the Ampere-Maxwell law, is the engine of everything that follows.

Key idea: The four equations describe how charges and currents make fields and how each changing field creates the other, a symmetry at the heart of light.

The missing piece: displacement current

Maxwell's great insight was the term he added to Ampere's law. Faraday had shown that a changing magnetic field makes an electric field; Maxwell reasoned that nature should be symmetric, so a changing electric field should make a magnetic field. He called this source the displacement current.

The clue came from a charging capacitor. Current flows in the wires but stops at the gap between the plates, yet a magnetic field still circles the gap. What threads the gap is not charge in motion but a changing electric field, and Maxwell's new term accounts for exactly that. Without it, the equations contradict themselves; with it, they predict waves.

Key idea: Maxwell added that a changing electric field creates a magnetic field, the displacement current, completing the equations and making waves possible.

A wave that carries itself

Faraday's and the Ampere-Maxwell laws together allow a remarkable loop: a changing electric field makes a magnetic field, and that changing magnetic field makes an electric field, and so on. The two fields regenerate each other and travel through empty space as an electromagnetic wave - no medium required. In the wave, E and B are perpendicular to each other and both perpendicular to the direction of travel (a transverse wave).

The two fields rise and fall together, in step, each sustaining the other as the wave advances. Because the wave makes its own fields as it goes, it needs no material to travel through, which is why starlight crosses the vacuum of space to reach us. This was a shock in Maxwell's day, when every known wave needed a medium.

Key idea: Changing E and B fields regenerate each other, forming a self-sustaining transverse wave that needs no medium to travel.

The speed of light falls out

Maxwell's equations predict the wave speed from two constants you have already met:

c = 1 / sqrt(mu_0 epsilon_0) = 3.00 x 10^8 m/s.

This number matched the measured speed of light exactly - the proof that light is electromagnetic. All electromagnetic waves travel at this speed c in vacuum, related to frequency and wavelength by c = f lambda.

Pause on what this means. The permittivity epsilon_0 came from Coulomb's law about static charges, and the permeability mu_0 came from the magnetic force between wires. Neither had anything obvious to do with light, yet combined they give its exact speed. That coincidence was the decisive clue that optics, electricity, and magnetism are one subject.

Key idea: The wave speed c = 1/sqrt(mu_0 epsilon_0) equals the measured speed of light, proving that light is an electromagnetic wave.

What the wave carries

An electromagnetic wave carries energy, which is how sunlight warms your skin and powers a solar panel across ninety-three million miles of vacuum. The energy travels in the fields themselves, at densities set by the same epsilon_0 E^2 and B^2/mu_0 expressions met with capacitors and inductors.

It also carries momentum, so light exerts a tiny push called radiation pressure when it strikes or reflects off a surface. The effect is faint but real: it helps shape a comet's tail and could drive a solar sail through space. In a wave, the electric and magnetic field strengths stay locked in the ratio E = c B.

The rate at which a wave delivers energy per unit area is its intensity. A wave spreading outward from a small source thins over an ever-larger sphere, so its intensity falls off as the inverse square of distance, which is why a distant lamp looks dim and a nearby one bright. The same inverse-square geometry that governed the point-charge field governs the reach of radiated light.

Key idea: Electromagnetic waves carry energy and momentum, with fields locked at E = cB, and a spreading wave's intensity falls off as the inverse square of distance.

How the waves are made

Where do such waves come from? From accelerating charges. A charge moving steadily makes static fields, but a charge that speeds up, slows down, or oscillates sends ripples of field radiating outward. Shake a charge and you launch a wave.

This is exactly what a radio antenna does: electrons driven back and forth along it accelerate constantly, broadcasting a wave at their oscillation frequency. The same principle, at far higher frequencies, produces light when electrons jump within atoms. From radio masts to glowing filaments, radiation traces back to charges that accelerate.

Key idea: Accelerating charges radiate electromagnetic waves, from electrons oscillating in an antenna to electrons shifting within atoms that emit light.

Polarization

The direction in which a wave's electric field oscillates is its polarization. Light from the sun or a bulb is unpolarized, a jumble of all directions, but a polarizer passes only the component along one axis, producing a wave that oscillates in a single plane.

This is everyday physics. Polarized sunglasses cut glare by blocking the horizontally polarized light that reflects off roads and water. Every liquid-crystal display steers polarized light to form its image, and photographers rotate a polarizing filter to darken skies. Polarization is possible only because the wave is transverse, with its field free to point across the direction of travel.

Key idea: Polarization is the direction of a wave's electric field, exploited in sunglasses and LCD screens and possible only because the wave is transverse.

The electromagnetic spectrum

Electromagnetic waves differ only in frequency (and wavelength). From lowest frequency to highest: radio, microwave, infrared, visible light, ultraviolet, X-rays, gamma rays. Visible light is a thin band in the middle, from red (longer wavelength, ~700 nm) to violet (shorter, ~400 nm). They also carry energy and momentum; the energy of one photon is E = h f, with higher-frequency waves (like X-rays) carrying more energetic photons than radio waves.

Each band earns its keep. Radio and microwaves carry broadcasts and heat food; infrared is felt as warmth and seen by night-vision cameras; visible light is the narrow slice our eyes evolved to catch. Beyond it, ultraviolet tans and sterilizes, X-rays image bone, and gamma rays, the most energetic, arise from nuclear and cosmic events. It is one continuous family, differing only in frequency.

Key idea: The spectrum runs from radio to gamma rays by rising frequency, with visible light a thin central band and photon energy growing with frequency as E = hf.

Worked example: wavelength of an FM station

Given: an FM radio station broadcasts at 100 MHz (1.0 x 10^8 Hz). Find: the wavelength.

Solution: lambda = c / f = (3.00 x 10^8) / (1.0 x 10^8) = 3.0 m. So the radio wave is about 3 meters long, which is why FM antennas are sized on the order of a meter.

Worked example: a microwave oven

Given: a microwave oven works at 2.45 GHz (2.45 x 10^9 Hz). Find: the wavelength.

Solution: lambda = c/f = (3.00 x 10^8)/(2.45 x 10^9) = 0.12 m. The 12-centimeter waves are just the right size to shake water molecules in food, and the same c = f lambda relation ties every band of the spectrum together.

The legacy: relativity and the photon

Maxwell's result did more than unify three subjects; it reshaped physics. The equations say light travels at one fixed speed c, and asking how that speed could look the same to every observer led Einstein to special relativity in 1905, overturning old ideas of absolute space and time.

A second revolution followed. The wave picture is complete for many purposes, yet light also comes in tiny packets of energy called photons, each carrying E = h f. That grainy, quantized side of light launched quantum mechanics. This course's classical fields are thus a doorway to the two great theories of modern physics.

Key idea: Maxwell's fixed speed of light led to Einstein's relativity, and light's photon nature launched quantum mechanics, making this classical theory a gateway to modern physics.

Common misconceptions

• Electromagnetic waves need a medium. They make their own fields and travel through vacuum; starlight proves it.
• Different colors travel at different speeds in vacuum. All electromagnetic waves move at the same speed c in vacuum.
• E and B point along the wave's motion. They are perpendicular to the motion and to each other, a transverse wave.
• Radio waves and light are different things. They are the same phenomenon at different frequencies.
• Static charges radiate. Only accelerating charges radiate electromagnetic waves.

Recap

• Maxwell's four equations unify electricity and magnetism and predict electromagnetic waves.
• His added displacement current, a changing electric field making a magnetic field, completes the set.
• E and B regenerate each other into a self-sustaining transverse wave needing no medium.
• The wave speed c = 1/sqrt(mu_0 epsilon_0) is the speed of light, with c = f lambda.
• The spectrum from radio to gamma rays is one family, differing only in frequency.

Sources

1. OpenStax. (2016). 16.1 Maxwell's equations and electromagnetic waves. In University Physics Volume 2. Rice University. openstax.org
2. OpenStax. (2016). 16.2 Plane electromagnetic waves. In University Physics Volume 2. Rice University. openstax.org
3. OpenStax. (2016). 16.3 Energy carried by electromagnetic waves. In University Physics Volume 2. Rice University. openstax.org
4. OpenStax. (2016). 16.5 The electromagnetic spectrum. In University Physics Volume 2. Rice University. openstax.org
5. Nave, R. (n.d.). Maxwell's equations. In HyperPhysics. Georgia State University, Department of Physics and Astronomy. hyperphysics.phy-astr.gsu.edu
6. Feynman, R. P., Leighton, R. B., & Sands, M. (2013). Chapter 18: The Maxwell equations. In The Feynman Lectures on Physics, Volume II (New Millennium ed., online). California Institute of Technology. feynmanlectures.caltech.edu
7. National Institute of Standards and Technology. (n.d.). CODATA value: speed of light in vacuum. The NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov
8. Maxwell, J. C. (1865). VIII. A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of London, pp. 459-512. doi.org/10.1098/rstl.1865.0008