Next: Gauge transformations
Up: Timedependent 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 nonzero.
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 timevarying and non timevarying 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 timevarying fields. Let us substitute Eq. (380) into the field equation (376).
We obtain

(381) 
which can be written

(382) 
We know that a curlfree 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 timevarying magnetic fields is nonconservative, and
is described by the magnetic vector potential .
Next: Gauge transformations
Up: Timedependent Maxwell's equations
Previous: Faraday's law
Richard Fitzpatrick
20060202