Prerequisites: Curl: Rotation and Circulation Electric Fields Magnetic Fields

Faraday’s Law

This is an early draft. Content may change as it gets reviewed.

The third of Maxwell’s equations:

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

In words: a changing magnetic field creates a curling electric field.

This is the law of electromagnetic induction — the principle behind every electric generator, transformer, and wireless charger.

The discovery

Michael Faraday found this experimentally in 1831. Move a magnet near a wire loop and current flows — even though no battery is connected. The changing magnetic field induces an electric field that pushes charges around the loop.

The key word is changing. A static magnetic field produces no electric field. But the moment $\mathbf{B}$ starts changing in time ($\partial \mathbf{B} / \partial t \neq 0$), an electric field appears — and it curls.

What the equation says

The left side, $\nabla \times \mathbf{E}$, is the curl of the electric field — its tendency to circulate.
The right side, $-\partial \mathbf{B} / \partial t$, is the rate at which the magnetic field is changing.
The minus sign is Lenz’s law: the induced electric field opposes the change that created it. Nature resists.

Try It: Electromagnetic Induction

A magnetic field (into the screen, shown by ⊗) is changing in strength. Watch the electric field curl in response. The faster the change, the stronger the curl.

dB/dt = 1.0 (positive = B increasing into screen)

The integral form

Integrate over a surface bounded by a loop:

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

The electromotive force (EMF) around a closed loop equals the negative rate of change of the magnetic flux through the loop. This is the form you use to calculate the voltage in a generator.

What’s revolutionary

Before Faraday, electricity and magnetism were separate subjects. This equation links them: a time-varying magnetic phenomenon creates an electric phenomenon.

But notice the asymmetry. In static situations ($\partial \mathbf{B}/\partial t = 0$), this reduces to $\nabla \times \mathbf{E} = 0$ — the electric field has no curl, meaning it’s conservative (you can define a voltage). But when things change in time, the electric field swirls. Voltage becomes path-dependent. This is fundamentally different from electrostatics.

Equation #3 of 4

What we know so far
$\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$	Charges create E (divergence)
$\nabla \cdot \mathbf{B} = 0$	No magnetic charges (no divergence)
$\nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t$	Changing B creates curling E
???	What creates curling B?

The last equation completes the picture — and contains Maxwell’s greatest insight.