Next: Gauge transformations Up: Time-dependent Maxwell's equations Previous: Faraday's law

Electric scalar potential?

Now we have a problem. We can only write the electric field in terms of a scalar potential (i.e., ) provided that . However, we have just found that in the presence of a changing magnetic field the curl of the electric field is non-zero. In other words, is not, in general, a conservative field. Does this mean that we have to abandon the concept of electric scalar potential? Fortunately, no. It is still possible to define a scalar potential which is physically meaningful.

Let us start from the equation

 (379)

which is valid for both time-varying and non time-varying magnetic fields. Since the magnetic field is solenoidal, we can write it as the curl of a vector potential:
 (380)

So, there is no problem with the vector potential in the presence of time-varying fields. Let us substitute Eq. (380) into the field equation (376). We obtain
 (381)

which can be written
 (382)

We know that a curl-free vector field can always be expressed as the gradient of a scalar potential, so let us write
 (383)

or
 (384)

This is a very nice equation! It tells us that the scalar potential only describes the conservative electric field generated by electric charges. The electric field induced by time-varying magnetic fields is non-conservative, and is described by the magnetic vector potential .

Next: Gauge transformations Up: Time-dependent Maxwell's equations Previous: Faraday's law
Richard Fitzpatrick 2006-02-02