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

Ampere's and Faraday's Laws – Lecture 11 (University Physics II)

Overview

Biot–Savart gets the right answer, but it makes you integrate. This lecture takes Ampère's law as the starting point instead and reworks the standard geometries — straight wire, solid cylinder, solenoid, toroid — in a fraction of the labor, exactly the way Gauss's law replaced direct Coulomb integration for electric fields. From there the chapter finishes its real subject, sources of magnetic field, by descending to the atom: the magnetic moment of an orbiting electron, the Bohr magneton, and the paramagnetic, diamagnetic, and ferromagnetic behavior those moments produce. Two loose ends from the previous chapter — the Hall effect and the DC motor — close it out. The lecture then opens the next chapter with Faraday's law, the statement that a changing magnetic flux drives an emf.

Key Concepts

Gauss's law for magnetism. The magnetic flux through any closed surface is always zero:

\begin{aligned} \oint \vec B \cdot d\vec A &= 0 \end{aligned}

Unlike its electric counterpart, this relates the field to nothing — there is no magnetic monopole, no enclosed "magnetic charge" for it to point at. Every north comes paired with a south, so the flux in always equals the flux out. Useful as a constraint, useless as a tool for finding $\vec B$ .

Ampère's law. The law that does tie the magnetic field to its source. For any closed path, the line integral of $\vec B$ around it equals $\mu_0$ times the current passing through any surface bounded by that path:

\begin{aligned} \oint \vec B \cdot d\vec l &= \mu_0 I_\text{enc} \end{aligned}

Why it works. Walk a circular path of radius $r$ around a long straight wire, in the same sense as $\vec B$ . Then $dl\cos\theta = r\,d\theta$ , and substituting the known straight-wire field gives

\begin{aligned} \oint \vec B \cdot d\vec l &= \oint \frac{\mu_0 I}{2\pi r}\,r\,d\theta \\ &= \frac{\mu_0 I}{2\pi}\oint d\theta = \frac{\mu_0 I}{2\pi}(2\pi) = \mu_0 I \end{aligned}

The radius cancels, which is precisely why the shape of the path does not matter.

Currents outside the path contribute nothing. The algebra is identical, but the bounds change: a path that does not enclose the wire sweeps up to some maximum angle and back down again, so the total $\oint d\theta$ is zero. Only enclosed current counts.
Ampère's law and Biot–Savart are logically equivalent — either can be derived from the other. From here on, Ampère's law is the one worth remembering.

Choosing the path. The whole art is picking a path on which $\vec B$ is constant in magnitude and parallel to $d\vec l$ , so the dot product collapses to $B\,dl$ and $B$ pulls out of the integral. Symmetry tells you which path that is. Always walk it in the same sense as $\vec B$ to keep the signs positive.

Long straight wire. Take a circle of radius $r$ centered on the wire. Then $B(2\pi r) = \mu_0 I$ :

\begin{aligned} B &= \frac{\mu_0 I}{2\pi r} \end{aligned}

The same result the Biot–Savart integration produced, in three lines instead of a page.

Solid cylindrical conductor. A real wire is a cylinder of radius $R$ carrying total current $I$ spread uniformly over its cross section. Outside ( $r > R$ ) the enclosed current is all of $I$ , so the field is the straight-wire result above. Inside ( $r < R$ ) a circle of radius $r$ encloses only the fraction of the current matching the fraction of the area, $I_\text{enc} = I\,(\pi r^2)/(\pi R^2)$ :

\begin{aligned} B(2\pi r) &= \mu_0 I\,\frac{r^2}{R^2} \\ B &= \frac{\mu_0 I r}{2\pi R^2} \end{aligned}

So the field climbs linearly from zero at the axis (a vanishing circle encloses no current) to a maximum of $\mu_0 I / 2\pi R$ at the surface, then falls off as $1/r$ beyond it.

Solenoid. For a tightly wound coil, take a rectangular path with one side of length $L$ running along the axis inside. The two perpendicular sides contribute nothing ( $\vec B \perp d\vec l$ ), and the outside leg contributes nothing because the external field is negligible. If $n$ is the number of turns per unit length, that path encloses $nL$ turns each carrying $I$ :

\begin{aligned} BL &= \mu_0 (nL) I \\ B &= \mu_0 n I \end{aligned}

Note what is absent: the radius of the coil and its total length. Only the turn density matters.

Toroidal solenoid. Windings wrapped around a donut, with $N$ turns total. Take a circle of radius $r$ concentric with the hole. Symmetry makes $B$ constant along it, so the result depends entirely on what current the circle encloses:

In the hollow center: no current is enclosed, so $B = 0$ . A constant value summing to zero over a closed path must itself be zero.
Within the windings: the circle cuts all $N$ turns, so $B(2\pi r) = \mu_0 N I$ , giving

\begin{aligned} B &= \frac{\mu_0 N I}{2\pi r} \end{aligned}

Outside the toroid: each winding is crossed twice, once going in and once coming out, so the enclosed currents cancel and $B = 0$ again.

Magnetic dipole moment (review). For a current loop of area $\vec A$ carrying current $I$ :

\begin{aligned} \vec \mu &= I\vec A, \qquad \vec \tau = \vec \mu \times \vec B, \qquad U = -\vec \mu \cdot \vec B \end{aligned}

The area vector points perpendicular to the loop, in the sense given by the right hand rule applied to the current. Intuitively $\mu$ measures how strongly the loop responds to an external field — how hard it is twisted into alignment with it.

The Bohr magneton. Model an electron of charge magnitude $e$ orbiting at radius $r$ with speed $v$ . It is a moving charge, hence a current: one electron's worth of charge per period $T = 2\pi r / v$ . So $I = ev/2\pi r$ , and

\begin{aligned} \mu &= IA = \frac{ev}{2\pi r}\,\pi r^2 = \frac{evr}{2} \\ &= \frac{e}{2m}\,(mvr) = \frac{e L}{2m} \end{aligned}

Multiplying top and bottom by the electron mass $m$ turns $mvr$ into the angular momentum $L$ — and that substitution is the whole point, because angular momentum is quantized. It comes only in integer multiples of $h/2\pi$ , where $h = 6.626 \times 10^{-34}\ \text{J}\cdot\text{s}$ is Planck's constant. Setting that integer to $1$ defines the natural unit of atomic magnetic moment:

\begin{aligned} \mu_\text{B} &= \frac{eh}{4\pi m} = \frac{e\hbar}{2m} \approx 9.274\times10^{-24}\ \text{A}\cdot\text{m}^2 \end{aligned}

An electron's intrinsic spin contributes very nearly one Bohr magneton. An atom with a single unpaired electron therefore carries a moment of that order; electrons paired with opposite spins produce opposite moments that cancel, leaving the atom magnetically neutral. This orbiting-ball picture is a model, not a literal description — spin is not rotation — but it predicts the right magnitudes.

Magnetic materials. Three behaviors, distinguished by what the atomic moments do in an applied field:

Paramagnetic — the atomic moments tend to align with the field, slightly strengthening it. Remove the field and thermal agitation randomizes the moments again, so nothing persists.
Diamagnetic — the material induces a moment opposing the applied field, weakening it. The reason only becomes clear once Faraday's law is in hand. Superconductors are the extreme case.
Ferromagnetic — atoms with large moments whose alignment holds neighbors in line, so the ordering survives removal of the field. This is what makes a permanent magnet. Heat it enough and the thermal jiggling wins, and the magnetization is destroyed.

The Hall effect. Run a current through a wide flat conductor sitting in a field $\vec B$ perpendicular to it. The magnetic force pushes the moving charges toward one edge, and they pile up there until the transverse electric field they create exactly cancels that force; after that, carriers travel straight through. Two consequences:

Measuring the resulting transverse voltage measures $B$ — the basis of Hall sensors.
The sign of that voltage reveals which charge carrier is actually moving. Positive and negative carriers produce the same conventional current but accumulate on opposite edges. This is how we know that conduction in an ordinary metal is carried by electrons.

DC motors. A current loop between the poles of a magnet feels opposite forces on its two sides, producing a torque. But once the loop swings through the plane where the forces line up with it, the torque vanishes and then reverses — the loop merely rocks. The fix is the commutator: brushes that reverse the current direction each time the loop passes that point, so the torque always drives rotation the same way. Real motors use many loops at staggered angles for smooth torque.

Magnetic flux. How much field passes through a surface:

\begin{aligned} \Phi_B &= \int \vec B \cdot d\vec A = BA\cos\phi \end{aligned}

where $\phi$ is the angle between $\vec B$ and the area vector — the normal to the surface, not the surface itself. A surface face-on to the field catches the most flux; edge-on it catches none.

Faraday's law. The experiment: a magnet held near a coil drives no current, but a magnet moving toward or away from it does, and reversing the motion reverses the current. It is not the field, and not the flux, but the change in flux that matters:

\begin{aligned} \mathcal{E} &= -\frac{d\Phi_B}{dt}, \qquad \mathcal{E} = -N\frac{d\Phi_B}{dt} \ \ \text{for } N \text{ turns} \end{aligned}

Lenz's law is the minus sign. The induced emf drives a current whose own field opposes the change that produced it: increase the flux and the induced current fights the increase; decrease it and the induced current tries to prop it up. It is the law that tries to preserve the flux.
The law says nothing about the material of the loop. A wooden loop develops the same emf as a copper one; its enormous resistance simply makes the resulting current negligible, since $\mathcal{E} = IR$ .
The same effect appears whether the flux changes because a permanent magnet moves, because an electromagnet's current is ramped up or down, or because the loop itself rotates.

AC generator. Spin a coil at constant rate in a uniform field and the flux varies sinusoidally. The emf is the negative slope of that curve, so it is a quarter cycle out of phase with the flux: when the flux is at its maximum the emf is zero, and when the flux passes through zero the emf peaks. A motor consumes current to produce rotation; a generator consumes rotation to produce current.

Worked Example

A loop in a strengthening field. A single loop of area $A = 120\ \text{cm}^2 = 0.012\ \text{m}^2$ lies face-on in a field that is increasing at $dB/dt = 0.02\ \text{T/s}$ . The total circuit resistance is $5\ \Omega$ . Find the induced emf and current.

Because the field is perpendicular to the plane of the loop, $\phi = 0$ and $\Phi_B = BA$ . The area is fixed, so only $B$ carries the time dependence:

\begin{aligned} \mathcal{E} &= -\frac{d\Phi_B}{dt} = -A\,\frac{dB}{dt} \\ &= -(0.012\ \text{m}^2)(0.02\ \text{T/s}) = -2.4\times10^{-4}\ \text{V} \\ I &= \frac{\mathcal{E}}{R} = \frac{-2.4\times10^{-4}\ \text{V}}{5\ \Omega} = -4.8\times10^{-5}\ \text{A} \end{aligned}

The minus sign is read against the chosen area vector: it says the current runs opposite the sense the right hand rule assigns to $\vec A$ — which is Lenz's law, the induced current opposing the growing flux. Replacing the loop with an insulator changes nothing about the emf; Faraday's law never mentions the material. Only the resistance changes, driving the current essentially to zero.

A tilted coil in a weakening field. A coil of $N = 500$ turns and radius $4\ \text{cm}$ sits with its plane at $60^\circ$ to a uniform field, so the angle between $\vec B$ and the normal is $\phi = 30^\circ$ . The field is decreasing at $dB/dt = -0.2\ \text{T/s}$ . Find the induced emf.

\begin{aligned} \mathcal{E} &= -N\frac{d}{dt}\left(BA\cos\phi\right) = -NA\cos\phi\,\frac{dB}{dt} \\ &= -(500)\,\pi(0.04\ \text{m})^2\cos 30^\circ\,(-0.2\ \text{T/s}) \\ &= -(500)(5.03\times10^{-3}\ \text{m}^2)(0.866)(-0.2\ \text{T/s}) \\ &= 0.435\ \text{V} \end{aligned}

The positive result means the current runs with the area vector. That is the expected direction: the flux is dropping, so the induced current acts to sustain it.

Summary

Ampère's law, $\oint \vec B \cdot d\vec l = \mu_0 I_\text{enc}$ , relates the magnetic field to the current that produces it, and does for magnetism what Gauss's law does for electricity. Only enclosed current counts; the shape of the path is irrelevant.
Its power comes from symmetry: choose a path along which $B$ is constant and parallel to $d\vec l$ , and the integral collapses to $B$ times the path length.
The standard results follow in a few lines each — $\mu_0 I/2\pi r$ outside a wire, $\mu_0 I r/2\pi R^2$ inside a solid conductor, $\mu_0 n I$ in a solenoid, and $\mu_0 N I/2\pi r$ within a toroid, with zero in the toroid's hollow center and outside it.
Atomic magnetism traces to moving charge: an orbiting electron has $\mu = eL/2m$ , and because $L$ is quantized in units of $h/2\pi$ , moments come in multiples of the Bohr magneton $eh/4\pi m$ . Whether those moments align with a field, oppose it, or lock in permanently distinguishes paramagnetic, diamagnetic, and ferromagnetic materials.
The Hall effect measures field strength and, through the sign of its transverse voltage, identifies which charge carrier actually moves. A DC motor needs a commutator to keep its torque one-directional.
Faraday's law, $\mathcal{E} = -N\,d\Phi_B/dt$ , is driven by the rate of change of flux, not by the flux itself; the minus sign is Lenz's law, and it makes every induced current oppose the change that created it.