Faraday's Law of Induction

Equivalent forms

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

\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}

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

The minus sign (Lenz's law) is nature's way of enforcing energy conservation — induced effects always fight their cause.

Symbol	Name	SI unit	Dimension	Range
$ε$	Induced EMFoutput Electromotive force induced in the loop	V	M·L²·T⁻³·I⁻¹	-100 – 100
$Φ_B$	Magnetic flux Total magnetic flux through the loop	Wb	M·L²·T⁻²·I⁻¹	0 – 1
$N$	Number of turns Number of loops in the coil	dimensionless	1	1 – 1000

Unit systems

Symbol	SI	CGS	Imperial
$ε$	V	statV	V
$Φ_B$	Wb	Mx	Wb
$t$	s	s	s

Where it holds

Universally valid in classical electromagnetism — one of Maxwell's four equations. The integral form applies to any closed loop, moving or stationary. For moving conductors, both motional and transformer EMF must be considered.

Dimensional analysis

$[\varepsilon ] = [\Phi _B]/[t] \to Wb/s = (V\cdot s)/s = V \checkmark$

Discovery

Michael Faraday · 1831

Faraday discovered that a changing magnetic field induces an electric current by moving a magnet through a coil. He had no formal mathematics — Maxwell later expressed it as an equation.

Try this

How does shaking a magnet inside a flashlight make it light up without batteries?

A circular loop of radius 0.1 m is in a magnetic field that changes from 0.5 T to 0 T in 0.02 s. Find the induced EMF.

Research status: stable

Real-world applications

Electric generators and alternators in power plants
Transformers for voltage conversion in power grids
Induction charging pads for smartphones
Magnetic stripe card readers and RFID systems

Common misconceptions

A static magnetic field does NOT induce a voltage — only changes in flux matter
The induced EMF opposes the change in flux, not the flux itself (Lenz's law)
Flux can change by changing B, the area, or the angle between B and the surface normal

Experimental verification

Faraday's original ring experiment (1831): a current pulse in one coil induced a transient current in a nearby coil. Modern verification includes precision measurements of AC generator output and magnetic braking experiments.

Derivation

From experimental observation:

\varepsilon \propto d\Phi _B/dt

.

The minus sign (Lenz's law) follows from energy conservation.

In differential form:

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

, obtained by applying Stokes' theorem to the integral form.

Limiting cases

d\Phi _B/dt = 0

⟶

\varepsilon = 0

A constant magnetic flux induces no voltage — you need change.

N \to

large⟶

\varepsilon

scales linearly with NMore turns multiply the EMF — the principle behind transformers.

Rapid flux change⟶

\varepsilon

→ largeFaster changes produce larger voltages — used in spark ignition systems.

What if…

What if the loop has 50 turns?

EMF multiplies by 50: $\varepsilon = 50 \times 0.785 = 39.25\,\mathrm{V}$ . More turns = more flux linkage per unit flux change.

What if the field changes

10\times

faster?

EMF increases $10\times$ : $\varepsilon = 7.85\,\mathrm{V}$ . Faster changes induce larger voltages — the basis of spark ignition in car engines.

What if the loop area shrinks instead of B changing?

Same effect — flux still changes. A loop collapsing from A to 0 in the same time gives the same EMF. Faraday's law cares about flux change, not its cause.

1

EMF from a collapsing field

Given ·

R loop:: 0.1
B initial:: 0.5
B final:: 0
dt:: 0.02

Find · Induced EMF

Steps

Loop area: $A = \pi R^{2} = \pi \times 0.1^{2} = 0.0314\,\mathrm{m}^{2}$
Initial flux: $\Phi _i = B_i \times A = 0.5 \times 0.0314 = 0.01571\,\mathrm{Wb}$
Final flux: $\Phi _f = 0\,\mathrm{Wb}$
Rate of change: | $\Delta \Phi /\Delta t| = 0.01571 / 0.02 = 0.785\,\mathrm{V}$
By Lenz's law, the induced current flows to maintain the original flux direction

Result · |

\varepsilon | = |\Delta \Phi _B/\Delta t| = |\Delta (BA)/\Delta t| = |0 - 0.5 \times \pi \times 0.01| / 0.02 = 0.785\,\mathrm{V}

2

Generator EMF (rotating coil)

Given ·

N:: 100
A:: 0.05
B:: 0.2
omega:: 377

Find · Peak EMF

Steps

Flux through rotating coil: $\Phi = BA \cos (\omega t)$
EMF: $\varepsilon = -Nd\Phi /dt = NBA\omega \sin (\omega t)$
Peak EMF: $\varepsilon$ _peak $= NBA\omega$
Substitute: $100 \times 0.2 \times 0.05 \times 377 = 377\,\mathrm{V}$
$\omega = 377\,\mathrm{rad/s}$ corresponds to 60 Hz (US power frequency)

Result ·

\varepsilon

_peak

= NBA\omega = 100 \times 0.2 \times 0.05 \times 377 = 377\,\mathrm{V}

Ampère's Law (with Maxwell's correction)→Lorentz Force Law→inductance