Faraday's law

Faraday’s law states that a time-varying magnetic flux through a loop induces an electromotive force (emf) around the loop:

$V_{\text{emf}}=-\frac{d\Phi }{dt}\text{(V)}.$

For a coil with $N$ turns enclosing the same flux per turn:

$V_{\text{emf}}=-N\frac{d\Phi }{dt}.$

The minus sign is Lenz’s law — the induced emf drives a current whose magnetic field opposes the change in flux. This is energy conservation in disguise: without it you could extract infinite energy from a changing flux.

Sign convention (often glossed over): pick a direction $\hat{\mathbf{n}}$ for the surface element $d\mathbf{s}=\hat{\mathbf{n}}ds$ used to compute $\Phi$. Then traverse the loop in the right-hand-rule sense — fingers curl in the loop direction, thumb in $\hat{\mathbf{n}}$. With this paired choice, $V_{\text{emf}}=-d\Phi /dt$ as written; reverse one of the two choices and the sign flips. The physics doesn’t change, only the bookkeeping.

A stationary loop in a time-varying $\mathbf{B}$ field. The induced emf drives a current $I$ through the external resistor; ${\mathbf{B}}_{\text{ind}}$ from this current opposes the original change.

The differential form, one of Maxwell’s equations:

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}.$

A changing magnetic field creates a curling (non-conservative) electric field. In time-varying problems, $\mathbf{E}$ is no longer the gradient of a scalar — the closed-loop integral of $\mathbf{E}$ is nonzero.

Three ways flux changes

The flux $\Phi ={\int }_S\mathbf{B}\cdot d\mathbf{s}$ can change in three distinct ways. Each gives a named contribution to the emf:

1. Transformer emf $V_{\text{emf}}^{\text{tr}}$: loop is stationary, $\mathbf{B}(t)$ varies in time.

$V_{\text{emf}}^{\text{tr}}=-{\int }_S\frac{\partial \mathbf{B}}{\partial t}\cdot d\mathbf{s}.$

This is the mechanism behind transformers: a time-varying current in the primary creates a time-varying $\mathbf{B}$ that links the secondary, inducing $V_{\text{emf}}$ there.

2. Motional emf $V_{\text{emf}}^{\text{m}}$: $\mathbf{B}$ is static, but the loop moves or changes shape so the linked area changes.

$V_{\text{emf}}^{\text{m}}={\oint }_C(\mathbf{u}\times \mathbf{B})\cdot d\mathbf{l},$

with $\mathbf{u}$ the velocity of the conductor element $d\mathbf{l}$. This is the mechanism behind generators: rotating a coil in a static field creates motional emf.

3. Combined: loop moves through time-varying field. Both contributions add:

$V_{\text{emf}}=V_{\text{emf}}^{\text{tr}}+V_{\text{emf}}^{\text{m}}.$

The decomposition above is for a conducting loop where the EMF is what drives charges around the wire. For a purely mathematical (non-conducting) loop, the rigorous statement is $V_{\text{emf}}=-d\Phi /dt$ computed in the lab frame, with $\Phi$ accounting for both the changing $\mathbf{B}$ field and the moving/deforming surface. The conducting-loop split into transformer + motional pieces follows from the total derivative and matches what you’d measure with a voltmeter.

Worked example: stationary inductor in oscillating field

A circular loop of $N$ turns, radius $a$, in the $xy$-plane, connected to a resistor $R$. External field $\mathbf{B}=(2\hat{\mathbf{y}}+3\hat{\mathbf{z}})B_0\sin \omega t$.

Only the $\hat{\mathbf{z}}$ component flows through the loop (the $\hat{\mathbf{y}}$ component is parallel to the loop plane). Flux per turn:

$\Phi ={\int }_S\mathbf{B}\cdot \hat{\mathbf{z}}ds=3B_0\sin \omega t\cdot \pi a^2.$

Transformer emf:

$V_{\text{emf}}^{\text{tr}}=-N\frac{d\Phi }{dt}=-3\pi N{a}^2{B}_0\omega \cos \omega t.$

For $N=10$, $a=10$ cm, $B_0=0.2$ T, $\omega =10^3$ rad/s: $V_{\text{emf}}=-188.5\cos (10^{3}t)$ V.

Induced current: $I=V_{\text{emf}}/R=0.19\cos (10^{3}t)$ A for $R=1$ kΩ.

Why the minus sign

If the induced current’s ${\mathbf{B}}_{\text{ind}}$ reinforced the change in flux, the current would grow without bound — an infinite-energy violation. Lenz’s law’s sign ensures induced effects always oppose the cause, like inertia.

Operationally: if $\Phi$ is increasing in $+\hat{\mathbf{z}}$, the induced current flows clockwise when viewed from $+\hat{\mathbf{z}}$ (right-hand rule), producing ${\mathbf{B}}_{\text{ind}}$ in $-\hat{\mathbf{z}}$ inside the loop — opposing the increase.

In the integral Maxwell’s equation

${\oint }_C\mathbf{E}\cdot d\mathbf{l}=-{\int }_S\frac{\partial \mathbf{B}}{\partial t}\cdot d\mathbf{s}.$

This is the integral form of Faraday’s law for a stationary loop. For a moving or deforming loop, the left side picks up the motional emf contribution — Maxwell’s equations are written for a fixed frame, but the line integral around a moving loop in that frame includes the motion effects automatically.

The closed-loop integral of $\mathbf{E}$ being nonzero is what distinguishes dynamic from electrostatic problems. In statics, $\oint \mathbf{E}\cdot d\mathbf{l}=0$ everywhere and the electric field is a gradient. In dynamics, $\mathbf{E}$ acquires a “rotational” piece that’s not a gradient.