The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Electromagnetic Induction – Lecture 12 (University Physics II)

Overview

This lecture takes Faraday's law from a qualitative rule to a working tool, then uses it to close Maxwell's equations. Magnetic flux can be changed in exactly three ways — by changing $B$ , by changing the area $A$ , or by changing the angle between them — and each route produces a device: the AC generator, the DC generator, and the slide-wire rod. Working through the slide wire in detail shows where the energy comes from and reveals that the induced emf is really the magnetic force separating charge inside the conductor. From there the focus shifts to induced electric fields, which turn out to be non-conservative, and finally to Maxwell's addition of the displacement-current term to Ampère's law — the step that makes electromagnetic waves fall out of the equations.

Key Concepts

Magnetic flux and Faraday's law

Magnetic flux measures how much field passes through a surface: $\Phi_B = BA\cos\phi$ , where $\phi$ is the angle between $\vec{B}$ and the area vector $\vec{A}$ (which points perpendicular to the surface). Flux is maximal when they are parallel, zero when perpendicular, and negative when they oppose.
Faraday's law: a changing flux, not flux itself, drives an emf:

\mathcal{E} = -\frac{d\Phi_B}{dt}

Holding a magnet near a loop produces nothing; moving it produces a current. This surprised Faraday, who was looking for the reverse of Ampère's law — a steady field producing a steady current — and found instead that only the transitions mattered.
Lenz's law is the meaning of the minus sign: the induced current always flows so that its own field opposes the change in flux. Push flux in and the loop fights to expel it; pull flux out and the loop fights to maintain it. It adds no new physics beyond Faraday's law, only the qualitative reading.
Three ways to change the flux: vary $B$ , vary $A$ , or vary $\phi$ . Every example below is one of these three.

AC and DC generators (changing $\phi$ )

A loop of area $A$ spun at constant angular speed $\omega$ in a uniform field has $\phi = \omega t$ , so the flux and emf are:

\begin{aligned} \Phi_B &= AB\cos\omega t \\ \mathcal{E} &= -\frac{d\Phi_B}{dt} = \omega AB\sin\omega t \end{aligned}

The emf is a sine where the flux is a cosine: the emf peaks exactly when the flux passes through zero, because that is where the flux is changing fastest. The current alternates direction each half turn — this is an AC generator, and it is how most electricity is generated.
With $N$ turns instead of one, every term picks up a factor of $N$ : $\mathcal{E} = N\omega AB\sin\omega t$ . More loops or faster spinning are the two ways to raise the output.
A commutator swaps the output leads every half turn, flipping the negative lobes positive. The result is a DC generator, whose emf is $\omega AB\,|\sin\omega t|$ — always one sign, but pulsing.
The average emf of an AC generator is zero and tells you nothing useful. For a DC generator the average is meaningful, and is found as area over base for one hump (see the Worked Example).

Slide-wire generator (changing $A$ )

A rod of length $L$ slides at speed $v$ along conducting rails in a uniform field perpendicular to the circuit. With $x$ the distance swept, $\Phi_B = BLx$ , so:

\begin{aligned} \mathcal{E} &= -\frac{d\Phi_B}{dt} = BLv \\ I &= \frac{\mathcal{E}}{R} = \frac{BLv}{R} \end{aligned}

Energy accounting: the current dissipates power $P = I^2R = B^2L^2v^2/R$ as heat. The rod carries a current in a field, so it feels a force $F = ILB = B^2L^2v/R$ that always opposes the motion — pulling or pushing, the force fights you. The mechanical power you supply is $P = Fv = B^2L^2v^2/R$ , exactly the power dissipated. Conservation of energy holds: the work you do on the rod becomes heat in the circuit.
With zero resistance there is still a current, but no heat and no opposing force — nothing resists the motion, and no work is done.

Motional emf, microscopically

Inside the moving rod, each charge carrier shares the rod's velocity $v$ and so feels a magnetic force $qvB$ along the rod's length. Charge piles up at the ends until the electric field it creates exactly cancels the magnetic push:

\begin{aligned} qvB &= qE \\ E &= vB \end{aligned}

The potential difference across the rod is then $\mathcal{E} = EL = vBL$ — the same result Faraday's law gave. The rod acts as a battery, with one end a source of positive charge and the other a sink.
The result is independent of which sign of carrier actually moves: assuming electrons rather than protons reverses both the velocity and the charge, leaving the current direction unchanged.
Magnetic fields do no work — the magnetic force is always perpendicular to the velocity, so it can change a charge's direction but never its speed. In the slide wire the magnetic force separates the charge, and the electric field that separation creates is what does the work you feel as resistance to pulling the rod.
For an arbitrary path, add up the infinitesimal contributions:

\mathcal{E} = \oint (\vec{v} \times \vec{B}) \cdot d\vec{l}

Faraday's paradox

A conducting disk of radius $R$ spinning at $\omega$ in a uniform field appears to have constant flux — its area and the field are both fixed — and so should produce no emf. But integrating the motional emf from centre to rim, with $v = \omega r$ :

\begin{aligned} \mathcal{E} &= \int_0^R vB \, dr = \int_0^R \omega r B \, dr \\ &= \frac{1}{2}\omega B R^2 \end{aligned}

The resolution: Faraday's law is about the flux through the area enclosed by the path the charges actually follow, not through the disk itself. As the disk turns, that circuit — out along a radius, through the external wire, back to the centre — sweeps out a growing area, so its flux really is changing. The paradox is only apparent.

Induced electric fields

A solenoid carrying a changing current, with a separate conducting ring around it, induces an emf in the ring even though the field exists only inside the solenoid and never touches the ring. What the ring responds to is the changing flux through it, not a local field.
Writing the emf as the line integral of the induced electric field gives the second common form of Faraday's law, valid for a stationary path:

\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}

By symmetry $E$ is constant around a ring of radius $r$ , so $E(2\pi r) = |d\Phi_B/dt|$ , giving $E = \dfrac{1}{2\pi r}\left|\dfrac{d\Phi_B}{dt}\right|$ .
The induced field is non-conservative. Carrying a charge once around the loop returns it to the start with a nonzero net change in potential, so Kirchhoff's loop rule fails here, and no static arrangement of charges could reproduce this field. A practical consequence: a voltmeter connected between two points reads a different value depending on which way its leads run around the loop.

Eddy currents

A conductor moving through a localized field has circulating currents induced in it. In a spinning metal wheel passing through a field region, the currents induced on entry and on exit circulate in opposite senses, combining to a single current through the field region whose force opposes the rotation — a magnetic brake.
The same effect slows a copper plate or magnet falling between magnets: the eddy currents dissipate kinetic energy as heat, and the fall looks like it is happening through mud. Cutting slots into the conductor breaks the current paths and largely removes the effect.

Generalized Ampère's law and the displacement current

Ampère's law relates $\oint \vec{B} \cdot d\vec{l}$ to the current through any open surface bounded by the chosen path. For a charging capacitor this breaks: a flat surface cutting the wire encloses the conduction current $i_C$ , while a surface bulging through the gap between the plates encloses no current at all. The same path gives two answers.
Maxwell's fix is a term built from the changing electric flux. Using $q = CV = (\epsilon A/d)(Ed) = \epsilon E A = \epsilon \Phi_E$ :

\begin{aligned} i_D &= \frac{dq}{dt} = \epsilon \frac{d\Phi_E}{dt} \\ \oint \vec{B} \cdot d\vec{l} &= \mu_0 \left( i_C + \epsilon_0 \frac{d\Phi_E}{dt} \right) \end{aligned}

This displacement current is not charge crossing the gap, but it produces a magnetic field exactly as a real current would — a fact confirmed by measuring the field between the plates. It exists only while the field is changing: a charged capacitor sitting idle produces no magnetic field, only one being charged or discharged does.
Together with the Lorentz force law, the four Maxwell's equations — Gauss's law, Gauss's law for magnetism, the generalized Ampère's law, and Faraday's law — account for all of electricity and magnetism. Maxwell did not derive all four, but he supplied the displacement-current term and recognized that the set was complete.

Electromagnetic waves and the spectrum

In empty space with no charge and no conduction current, the two curl equations reduce to a pair in which each field's change sources the other:

\begin{aligned} \oint \vec{B} \cdot d\vec{l} &= \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} \\ \oint \vec{E} \cdot d\vec{l} &= -\frac{d\Phi_B}{dt} \end{aligned}

A changing $\vec{B}$ produces $\vec{E}$ , and a changing $\vec{E}$ produces $\vec{B}$ , so the disturbance propagates — with no medium required. Maxwell called these electromagnetic waves, and light was eventually identified as one.
The spectrum, from highest frequency down: gamma rays, X-rays, ultraviolet, visible light, infrared, microwaves, radio. Higher frequency means higher photon energy and momentum, which is why gamma rays and X-rays are the ionizing, cancer-causing end while microwaves and radio are not. Microwave ovens work not by photon energy but by resonance — microwave frequencies match the natural jiggling frequency of water.
Alpha and beta radiation are particles, not electromagnetic waves, and carry far more kinetic energy than a single photon. Earth's magnetic field deflects these charged particles from the solar wind, trapping them into the spiraling motion that produces the auroras; it has no effect on photons.

Superconductors

Magnetic field cannot exist inside a superconductor. As a magnet approaches, the growing flux induces an eddy current in the surface which, with no resistance to damp it, generates exactly the field needed to cancel the incoming one — perfectly and indefinitely.
The consequence is levitation: the magnet is held off the surface by the field its own approach induces. Move it closer and a larger current cancels a larger flux, so the opposition scales to match. Cooling a material through its transition temperature expels the field lines that were passing through it, which then route around the object instead.

Worked Example

Average emf of a DC generator. The DC generator's output is $\mathcal{E} = \omega AB\,|\sin\omega t|$ — a train of identical positive humps. Because every hump is the same, the long-run average equals the average over one of them, and the average height of a function is its area divided by the width of its base.

One hump runs from $t = 0$ to $t = \pi/\omega$ (at which point $\sin\omega t = \sin\pi = 0$ ). Averaging the $\sin\omega t$ factor over that interval:

\begin{aligned} \langle \sin\omega t \rangle &= \frac{1}{\pi/\omega}\int_0^{\pi/\omega} \sin\omega t \, dt \\ &= \frac{\omega}{\pi} \left[ -\frac{\cos\omega t}{\omega} \right]_0^{\pi/\omega} \\ &= \frac{1}{\pi}\left( -\cos\pi + \cos 0 \right) \\ &= \frac{1}{\pi}(1 + 1) = \frac{2}{\pi} \end{aligned}

Averaging the sine alone is enough, since the amplitude $\omega AB$ is a constant multiplier. So the average output is:

\mathcal{E}_{\text{av}} = \frac{2}{\pi}\,\omega A B

To design for a target voltage, solve for the required spin rate:

\omega = \frac{\pi \, \mathcal{E}_{\text{av}}}{2AB}

With $N$ coils the factor carries through as $\mathcal{E}_{\text{av}} = \dfrac{2}{\pi}N\omega AB$ , so doubling the number of coils halves the spin rate needed for the same average voltage.

Summary

Only a changing magnetic flux drives an emf, $\mathcal{E} = -d\Phi_B/dt$ ; the sign is Lenz's law, which says the induced current always opposes the change producing it.
Flux can be changed three ways — by $B$ , by area, or by angle — giving the AC generator ( $\mathcal{E} = \omega AB\sin\omega t$ ), the DC generator (same, commutated, averaging $\tfrac{2}{\pi}\omega AB$ ), and the slide wire ( $\mathcal{E} = BLv$ ).
In the slide wire the opposing force $B^2L^2v/R$ makes the mechanical power supplied exactly equal the heat dissipated; the magnetic force separates the charge but the resulting electric field does the work, since magnetic forces never do work.
Induced electric fields are non-conservative: Kirchhoff's loop rule fails, and a voltmeter's reading depends on the path its leads take around the loop.
Maxwell's displacement current $\epsilon_0 \, d\Phi_E/dt$ removes the surface ambiguity in Ampère's law and produces a real magnetic field wherever the electric flux is changing.
With a changing $\vec{B}$ sourcing $\vec{E}$ and a changing $\vec{E}$ sourcing $\vec{B}$ , the fields propagate through empty space as electromagnetic waves — of which visible light is one band.