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. . .
Suppose we return to the case of electric charges (and ignore for now
other kinds of locally conserved 'stuff'). We can write Ampere's law as
curl(B) - (1/c^2)*dE/dt = ([mu]_0)*j. The LHS of this equation (when
written in 4-vector/4-tensor notation for Minkowski space) is actually 3
(spatial) of the 4 components of the exterior derivative of the dual of
the antisymmetric 2nd rank (2-form) electromagnetic field tensor whose
space-space components are the components of B and whose space-time
components are (up to irrelevant factors of c) the components of E. The
4th (time) component of this 4-tensor equation is expressed by the other
inhomogeneous Maxwell equation i.e. Gauss' law
div(E) = [rho]/[epsilon_0]. Note the current is the source for the
composite expression: curl(B) - (1/c^2)*dE/dt which is a part of a
single 4-tensor expression in Minkowski space. We thus see that it is
not so much that the displacement current is a source for the magnetic
field as that the actual current is a source for a composite
combination of *both* the transverse magnetic field *and* part of the
time-dependent electric field, i.e. the displacement "current" term
when it is not time-steady as well.
David Bowman
dbowman@georgetowncollege.edu