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Topic 6 of 9

Electromagnetic Induction

A changing magnetic flux induces an EMF — this is Faraday's law, the basis of generators, transformers, and electric motors. Lenz's law tells you the direction: the induced current always opposes the change causing it. Self-inductance stores energy in the magnetic field, just as capacitance stores energy in the electric field.

Key Concepts

Faraday's Law

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

. The EMF around a loop equals minus the rate of change of magnetic flux through it. In differential form:

\nabla\times\vec E=-\partial\vec B/\partial t

Lenz's Law

The induced EMF drives a current that creates a magnetic field opposing the change in flux. This is the minus sign in Faraday's law — nature resists changes in flux.

Self-Inductance

L=N\Phi/I

(in henrys). Voltage across an inductor:

\mathcal{E}=-L\,dI/dt

. Energy stored:

U=\frac{1}{2}LI^2

. For a solenoid:

L=\mu_0n^2 V_{\rm core}

Mutual Inductance

M_{12}=N_2\Phi_{12}/I_1

. EMF induced in coil 2 by current in coil 1:

\mathcal{E}_2=-M\ dI_1/dt

. Neumann's formula:

M_{12}=M_{21}

— mutual inductance is symmetric.

Key Equations

Faraday's law

\mathcal{E} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}\int_S\vec{B}\cdot d\vec{a}

Induced EMF equals negative rate of flux change.

Inductance of solenoid

L = \mu_0 n^2 \ell A = \mu_0 n^2 V

L for a solenoid of volume V = ℓA, n turns/m.

Magnetic energy

U = \frac{1}{2}LI^2 = \frac{1}{2\mu_0}\int B^2\,d\tau

Energy stored in the magnetic field of an inductor.

Worked Example

EMF from a Changing Flux

Problem

A rectangular loop ( $0.20\times0.30$ m) lies in the $xy$ -plane in a field $B_z=0.50t$ T (increasing). Find the induced EMF and current direction ( $R=2.0\,\Omega$ ).

Solution

Flux: $\Phi_B=B\cdot A=0.50t\times0.06=0.030t$ Wb.

\mathcal{E} = -\frac{d\Phi_B}{dt} = -0.030\text{ V}

Magnitude: 30 mV. By Lenz's law, the current flows to oppose increasing $+z$ flux — counterclockwise when viewed from $+z$ .

$I=|\mathcal{E}|/R=0.030/2.0=0.015$ A.

Answer

|\mathcal{E}|=30

mV;

I=15

mA, counterclockwise.

Practice

Exercises

7 problems

1 of 7

Watch the B arrows grow inside the loop — Lenz's law drives a counter-current around the rim. For a loop with $R = 0.15$ m and $dB/dt = 2.0$ T/s, find the induced |EMF| (in mV).

Faraday's law — circular loop

Rate of change dB/dt 2.0 T/s → 141.4 mV

Increasing flux induces a current that fights the change (Lenz's law). The induced EMF = rate of flux change = A·dB/dt. At R=0.15m, dB/dt=2.0 T/s, find EMF.

Induced EMF

|EMF| = mV

2 of 7

The solenoid current decreases — the self-induced EMF fights the change. First find $L = \mu_0 n^2 \ell A$, then $|\mathcal{E}| = L|dI/dt|$. For $n=1000$/m, $\ell=0.30$ m, $A=5\times10^{-4}$ m², $|dI/dt|=2.0$ A/s, find |EMF| (in mV).

Self-inductance — solenoid EMF

A solenoid resists current changes — the self-induced EMF opposes dI/dt. First find L, then |EMF| = L|dI/dt|.

Self-induced EMF

|EMF| = mV

n = 1000/m, ℓ = 0.30 m, A = 5.0×10⁻⁴ m², dI/dt = 2.0 A/s

3 of 7

Energy stored in an inductor: $L=0.050$ H, $I=4.0$ A (in J).

Unlock Exercise 3