Faraday's law

Faraday’s law of induction states that a changing magnetic field induces an electric field. Alternatively, we say that if the magnetic flux through a loop changes, then an emf appears in the loop.¹ In its integral form, it is given by:

$$\epsilon ={\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d{\Phi }_B}{dt}=-{\iint }_S\frac{\partial \vec{B}}{\partial t}d\vec{S}$$

where $ϵ$ is the emf and ${\Phi }_B$ is the magnetic flux. With Stokes’ theorem, we can convert it to differential form:

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

In the electrostatic case, the closed line integral for the emf reduces down to 0, as expected. The intuition behind the direction of induced field is given by Lenz’s law.

If we examine a coil with $N$ number of turns, an induced EMF appears in every turn, and the total induced EMF is the sum of the individually induced EMFs.

$$\epsilon =-N\frac{d{\Phi }_B}{dt}$$

Footnotes

1. These are actually two mechanisms that underly this. One has the emf induced by the Lorentz force law (a resulting current) and the other is the electric field (induced by the changing magnetic field). David Griffiths, in Introduction to Electrodynamics, describes this as a “peculiar accident” if we look from the view of classical electrodynamics. ↩