References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Problem 7.12.

The relation between emf and change of magnetic flux turns out to be a special case of a more general law discovered by Michael Faraday and called Faraday’s law. It turns out that any change of magnetic flux through a loop, whatever the cause, results in an emf being generated around the loop. This can be caused by moving the loop relative to a fixed magnet, moving the magnet relative to a fixed loop, or keeping both loop and magnet fixed and varying the field strength.

Actually, it may seem surprising that the first two cases are treated separately; surely the relative motion of loop and magnet means they are both the same? Special relativity would, of course, confirm this, and it was partly this aspect of electromagnetism that inspired Einstein to think about relative motion. However, in the 19th century, motion was always considered relative to a fixed reference frame so the two cases were quite different. In particular, if the loop is considered fixed and the magnet moves, then the charges in the loop are at rest, so should not feel any magnetic force from the Lorentz force law. The fact that an emf is generated in this case as well led Faraday to postulate that a changing magnetic field produces an electric field. Faraday’s law states that the emf generated in a loop is minus the rate of change of magnetic flux through the loop. That is

$\displaystyle \mathcal{E}=\oint\mathbf{E}\cdot d\boldsymbol{\ell}=-\frac{d\Phi}{dt}$

Since ${\Phi=\int\mathbf{B}\cdot d\mathbf{a}}$ then if the area enclosed by the loop stays the same, we have

$\displaystyle \oint\mathbf{E}\cdot d\boldsymbol{\ell}=-\frac{d\Phi}{dt}=-\int\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{a}$

From Stokes’s theorem, we have

$\displaystyle \oint\mathbf{E}\cdot d\boldsymbol{\ell}=\int\left(\nabla\times\mathbf{E}\right)\cdot d\mathbf{a}=-\int\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{a}$

so in differential form, we have

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

This is the generalization of the electrostatic condition ${\nabla\times\mathbf{E}=0}$ which applied in the absence of magnetic fields.

As a simple example, suppose we have solenoid in which the current in the wire wrapped around it varies with time according to ${I\left(t\right)=I_{0}\cos\omega t}$, so that the magnetic field inside the solenoid is

$\displaystyle \mathbf{B}\left(t\right)=\hat{\mathbf{z}}B_{0}\cos\omega t$

A circular loop of wire with resistance ${R}$ and radius ${a/2}$ is placed inside the solenoid so that its axis is coincident with the solenoid’s axis. The flux through the loop is then

$\displaystyle \Phi\left(t\right)=\frac{\pi a^{2}B_{0}}{4}\cos\omega t$

so the emf generated in the loop is

$\displaystyle \mathcal{E}=\frac{\pi a^{2}B_{0}\omega}{4}\sin\omega t$

and the current is

$\displaystyle I=\frac{\mathcal{E}}{R}=\frac{\pi a^{2}B_{0}\omega}{4R}\sin\omega t$