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.

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. 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. The unit of charge is the coulomb (C), and one coulomb is an enormous amount - about 6.24 x 10^18 elementary charges.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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 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.

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.

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.

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.

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.

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.

Dielectrics

Slipping an insulating material (a dielectric) between the plates increases the capacitance by a factor called the dielectric constant kappa (kappa > 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.

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.

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.

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.

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.

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.

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.

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.

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).

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

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.

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 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 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).

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.

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.

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.

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).

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.

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 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.

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.

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.

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.

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).

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.

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.

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.

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).

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.