13.4 Induced electric fields

The fact that emfs are induced in circuits implies that work is being done on the conduction electrons in the wires. What can possibly be the source of this work? We know that it’s neither a battery nor a magnetic field, for a battery does not have to be present in a circuit where current is induced, and magnetic fields never do work on moving charges. The answer is that the source of the work is an electric field $\overrightarrow{E}$ that is induced in the wires. The work done by $\overrightarrow{E}$ in moving a unit charge completely around a circuit is the induced emf ε; that is,

where ${\displaystyle \oint }$ represents the line integral around the circuit. Faraday’s law can be written in terms of the induced electric field    as

There is an important distinction between the electric field induced by a changing magnetic field and the electrostatic field produced by a fixed charge distribution. Specifically, the induced electric field is nonconservative because it does net work in moving a charge over a closed path, whereas the electrostatic field is conservative and does no net work over a closed path. Hence, electric potential can be associated with the electrostatic field, but not with the induced field. The following equations represent the distinction between the two types of electric field:

Our results can be summarized by combining these equations: