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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.

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