]> 29.1 Ampere's Law and its Consequences

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

4 π c S ( j · n ^ ) d S = δ S B · d l

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

S ( ( × B ) · n ^ ) d S = δ S B · d l

we get

S ( × B 4 π j c ) · n ^ d S = 0

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

× B = 4 π j c

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

· j = ρ t

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

0 = · ( × B ) = 4 π · j c = 4 π c ρ t

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