29.1 Ampere's Law and its Consequences

According to Ampere's law, the flux of electric current through a straight wire produces a "circulation" of magnetic field, B, on a circular path around the wire.
In terms of symbols, we get

If we combine this statement, generalized to hold for any surface, with Stokes' theorem applied to the vector B

we get

for any surface S.

Physicists draw the conclusion that the integrand must be more or less 0 everywhere and claim the following differential law holds everywhere, for steady current magnetic fields

We have already seen that when there is not steady current, there will still be conservation of charge, which as we have seen, obeys the equation

Taking the divergence of both sides of the previous equation, we see that it cannot be true when , the charge density, is time dependent. We get

which, if true, would imply that charge density could never change.