Faraday's Law and Lenz's Law

Faraday’s Law

A changing magnetic field induces an electric field.

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

Explanation¹

In 1831, Faraday announced the results of experiments as follows:

1. A wire loop placed in a magnetic field was pulled to the right. A current flowed through the loop.
2. With the wire loop fixed in the magnetic field, a magnet was pushed to the left. A current flowed through the loop.
3. Keeping the wire loop and the magnet stationary, the current flowing through the coil, which was used as an electromagnet, was changed to alter the strength of the magnetic field. A current flowed through the loop.

1. is about electromotive force due to motion. Considering the movement of the wire loop and the magnet, the content of 2. is essentially no different from 1. Thus, in the scenarios of 1. and 2., an electromotive force is generated as below, causing a current to flow.

$$ \begin{equation} \mathcal{E} = -\dfrac{d \Phi}{d t} \label{1} \end{equation} $$

As such, when the loop (or magnet) moves, magnetic forces create electromotive force, but it’s a different story when the wire and the magnet are stationary as in 3.. However, all three experiments showed the same result, and a common explanation was necessary. Thus, Faraday thought that “a changing magnetic field induces an electric field”. That is, a change in magnetic field flux through the wire causes a current to flow. The magnetic flux passing through the wire loop changes in all three experiments.

In 1. 2. 3., the cause of the change in the magnetic field is different. However, the same result can be obtained. Therefore, whenever the magnetic flux passing through the loop changes, regardless of the reason, an electromotive force like $(1)$ is always generated in the loop. The direction of the current flowing through the loop is determined by the following rule:

This is called Lenz’s law. Simply put, if the flux inside the loop increases, a current flows in a direction to decrease the flux, and if the flux inside the loop decreases, a current flows in a direction to increase the flux. The current flowing to increase (decrease) the flux in the loop again decreases (increases) the loop’s flux. It seems as if the current flows to reduce the change in flux. This can be understood as inertia in electromagnetism.

Induction

The electromotive force is $\mathcal{E} = \displaystyle \oint \mathbf{E} \cdot d\mathbf{l}$, and according to the results of the experiment, if this is equal to the rate of change of the magnetic flux, the following equation holds:

$$ \oint \mathbf{E} \cdot d\mathbf{l} = -\dfrac{d \Phi}{dt} $$

By the definition of flux, substituting $\displaystyle \Phi=\int \mathbf{B} \cdot d\mathbf{a}$ gives

$$ \oint \mathbf{E} \cdot d\mathbf{l} = -\int \dfrac{\partial \mathbf{B}} {\partial t} \cdot d \mathbf{a} $$

The above equation is called Faraday’s Law in integral form.

Stokes’ Theorem

$$ \int_{\mathcal{S}} (\nabla \times \mathbf{v} )\cdot d\mathbf{a} = \oint_{\mathcal{P}} \mathbf{v} \cdot d\mathbf{l} $$

Applying Stokes’ Theorem to the left side of Faraday’s Law in integral form gives the equation below.

$$ \int (\nabla \times \mathbf{E}) \cdot d\mathbf{a} = \oint \mathbf{E} \cdot d\mathbf{l} = -\int \dfrac{\partial \mathbf{B}} {\partial t} \cdot d \mathbf{a} $$

Therefore, the differential form of Faraday’s Law can be derived as follows.

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

If the magnetic field $\mathbf{B}$ is constant, then $\dfrac{\partial \mathbf{B}}{\partial t} = \mathbf{0}$. This matches the result from electrostatics that the curl of the electric field is $\mathbf{0}$.

$$ \nabla \times \mathbf{E} = \mathbf{0} $$

■