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
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.