Electromagnetic Induction

A Changing Flux Makes a Field

Faraday's law is the first equation in which the electric field stops being electrostatic:

$$\mathcal{E} = -\frac{d\Phi_B}{dt}, \qquad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

where $\Phi_B = \int \mathbf{B}\cdot d\mathbf{A}$. A changing magnetic flux drives an EMF round a loop, and the induced $\mathbf{E}$ has non-zero curl, so no scalar potential describes it, and $\oint\mathbf{E}\cdot d\mathbf{l}$ is no longer zero.

The minus sign is Lenz's law: the induced current opposes the change producing it. This is not an extra postulate but a requirement of energy conservation. If the induced current reinforced the change, a small disturbance would amplify itself without limit and generate energy from nothing.

There are three ways to change the flux, and all appear in real machines: change $B$, change the area, or rotate the loop. The last is how a generator works, giving $\mathcal{E} = NBA\omega\sin\omega t$.

Inductance quantifies flux linkage per unit current. Mutual inductance couples two circuits, $\mathcal{E}_2 = -M\,dI_1/dt$, which is what a transformer exploits. Self-inductance gives $\mathcal{E} = -L\,dI/dt$, opposing changes in a circuit's own current, the reason a switched-off inductor can produce a large spark.

Establishing a current stores energy in the field:

$$U = \tfrac{1}{2}LI^2, \qquad u = \frac{B^2}{2\mu_0}$$

exactly parallel to the electrostatic $\tfrac{1}{2}CV^2$ and $\tfrac{1}{2}\varepsilon_0E^2$.

Common pitfall: thinking the induced current opposes the field. It opposes the change in flux. A loop in a decreasing field carries current that tries to maintain the flux, reinforcing the field, not fighting it.