# Faraday's law with non-induced electric fields

Griffiths shows that the emf induced in a moving circuit due to the change in magnetic flux $\Phi$ through the circuit is

$$\mathcal{E} = - \frac{\mathrm{d} \Phi}{\mathrm{d} t}.$$

He then cites Faraday's hypothesis (and empirical finding) that "a changing magnetic field induces an electric field", in which case the above expression still holds. By the definition of emf,

$$\oint \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac{\mathrm{d} \Phi}{\mathrm{d} t},$$

where $\mathbf{E}$ is the induced electric field. However, what then happens if one inserts an external emf, like a battery, to one's circuit? It does not seem to follow that the same expression holds for the total electric field, and Griffiths does not comment on it as far as I can tell.

I might be misreading it, but the Wikipedia page for the Maxwell-Faraday equation also seems to indicate that the general relation is, well, a generalisation of Faraday's law and so cannot be derived from it.

If this is the case, what is the justification for extending Faraday's law?

The relation

$$\oint \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac{\mathrm{d} \Phi}{\mathrm{d} t}$$

where $\mathbf E$ is total electric field is indeed believed to be valid generally. Written in differential form, this law states

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

Since this equation is simple and elegant, and since it works so well in all practical cases, it is usually assumed that it is valid in every situation we may prepare, even such as circuit fed by a battery while in changing magnetic field.

Students may have one apparent problem with this - if magnetic field determines induced emf in every case, how come emf is sometimes non-zero even if there is no induced electric field, such as if we have a circuit fed by a battery?

The answer to this question is that there are different kinds of emf; the Faraday law given above only gives the emf due to induced electric field.

The others kinds of emf are :

1) emf due to motion of wire in static magnetic field (the emf is due to wire pushing on deflected charge carriers while losing mechanical energy)

2) emf due to electrochemical forces in a battery (such forces are usually directed against electric field inside the battery)

3) emf due to thermoelectric phenomena

In more detail: electrochemical force (let us denote its vector per unit charge as $\mathbf E^*$) is pushing charges through the battery (against the electric field). The Faraday law is still believed to be true, the battery is not supposed to change the general relation between the electric and magnetic field. In other words, the idea of superposition of independent forces is assumed; if there are two independent sources of electromotive force (electric field in the wire, electrochemical force in the battery), total force moving the charge is simply their vector sum. Then, total emf is sum of emf due to macroscopic electric field and emf due to electrochemical force in the battery:

$$\oint \mathbf E\cdot d\mathbf l + \oint \mathbf E^* \cdot d\mathbf l$$

(the second integral has contribution only inside the battery, so could be written as integral over finite nonclosed path). This probably corresponds well to the actual phenomena. I do not know of any study that would deal with this question experimentally though - I think most people just assume this is valid since it corresponds to the idea of addition of independent forces and there is no prominent counterexample. Of course, it may be that magnetic field changes something in the battery that changes the electrochemical emf - but that is hard to track back to some violation of the Faraday law, because the convention is that all complicated behaviour is due to matter, not due to some violation of the Maxwell equations.

• So the Maxwell-Faraday equation does in fact relate the total $\mathbf{E}$ and $\mathbf{B}$ fields. But then you say that "the Faraday law given above only gives the emf due to induced electric field". I don't understand why those statements are not contradictory. Also, are you saying that superposition is sufficient to extend Faraday's law for an induced emf to a general electric field? Because that still doesn't seem to tell you anything about the relationship between the magnetic flux change and the non-emf field. Commented Apr 12, 2018 at 20:09
• They are not contradictory. The Faraday law is general in the sense it is always correct, but the circulation integral of E gives only one kind of emf; there are other kinds of emf that are not given by this integral and hence the Faraday law says nothing about them. Commented Apr 12, 2018 at 21:55
• "the Faraday law given above only gives the emf due to total electric field" would be more correct, but usually one talks about the induced part only, because the static part has zero contribution for closed loop. Commented Apr 12, 2018 at 22:13