Physics Friday 127

Part 7: Electromagnetic Induction

Moving beyond electrostatics and megnetostatics into the study of time-varying fields, we begin with Faraday’s Law of Induction. Developed independently by Faraday and Henry (though Faraday was first to publish; see here), in simple terms, it states that a change in magnetic flux through a closed circuit will induce a current in that circuit. Specifically, given a circuit with magnetic flux Φ in SI units, one has an electromotive force  (in Gaussian/cgs units, the equation is ).
Note that electromotive force is a misnomer; the quantity it refers to is not a force, but has units of electric potential (energy per unit charge, or, alternately, electric field times distance). The minus sign is given by Lenz’s law, which holds that the current induced by the change in flux is in a direction such that the field it produces (via the Biot-Savart law) opposes the change in flux.
An important part of Faraday’s law is that the change in flux may be due to motion of the circuit through a spatially-varying field, or due to a change in the field at a stationary circuit. Classically, these are very different phenomena. The identical mathematical description of these was one of the steps that led Einstein to develop special relativity (PDF).
In terms of vector calculus, if our circuit is along the curve C, the magnetic flux is , where S is any simple surface bounded by the curve C; and the orientation of the surface normal is determined by the orientation of C via the right-hand rule. Similarly, the electromotive force for induction is the line integral over the circuit of the electric field in the frame of the circuit, denoted here by E
;
,
where the time derivative on the right-hand side is a total derivative.

Let’s consider a frame where the circuit is stationary. Then E‘=E, and so we are defining the electric and magnetic field in the same frame. Further, the surface S is stationary, so the total time derivative is equivalent here to the partial derivative, and will commute with the surface integral. so then we have
.
Using the Kelvin-Stokes theorem, we can convert the line integral into a surface integral:
.
Since this must hold for any stationary surface in space, we see the integrands must be equal, and we obtain the differential form of Faraday’s Law:

in SI units, or

in Gaussian units.

Note that this means that, in electrodynamics, the electric field is no longer irrotational, and thus we can no longer find scalar field φ such that .