#### Faraday's Law
emd.Fl.Fl

Around 1831, Faraday performed a number of experiments pertaining to the effects of time-dependent fields.

The first experiment that Faraday performed (1831) involved two metal coils wound on opposite sides of a metal ring. When a current was turned on through the first coil, it generated a transient current in the second coil (as measured by a galvanometer). Within the next few months he had come up with lots of variations on this idea. Faraday observed transient current in a circuit when:

- a steady current flowing in an adjacent circuit was turned on or off;
- an adjacent circuit with current was moved relative to the first;
- a permanent magnet was thrust through circuit.

Faraday's big insight was to summarize these effects by noticing that \[ \boxed{ \mbox{A changing magnetic field induces an electric field.} } \] Empirically: the changing magnetic field induces an electric current around the circuit. This current is really driven by an electric field having a component along the wire. The line integral of this field is called the

Electromotive force (or electromotance),

- Gr (7.9)

\[ {\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}. \tag{elmofo}\label{elmofo} \]

You can think of the emf in different ways. It's the energy accumulated as a unit charge is moved around the circuit; altenatively, if you cut the wire, it would be the voltage you would measure between the two ends.

The precise statement associated to Faraday's observations is that the electromotive force is proportional to the rate of change of the magnetic flux,

- Gr (7.14)

\[ {\cal E} = \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} \tag{Fl_flux}\label{Fl_flux} \] so we obtain

Faraday's law (integral form N.B.: for a stationary loop)

- Gr (7.15)

\[ \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a} \tag{Fl_int}\label{Fl_int} \]

Note that Faraday's law is valid for any loop (on a wire or not). Using Stokes' theorem, \[ \oint_{\cal P} {\bf E} \cdot d{\bf l} = \int_{\cal S} ({\boldsymbol \nabla} \times {\bf E}) \cdot d{\bf a}, \] we obtain

- Gr (7.16)

Faraday's law (differential form) \[ {\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t} \tag{Fl}\label{Fl} \]

Right-hand rule always sorts signs out. Easier rule: Lenz's law, which states that physical systems naturally resist a change in flux. This is in fact just Le Châtelier's principle of any action at an equilibrium point leading to an opposing counter-reaction.

Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42