4Electrodynamics

IB Electromagnetism 4.4 Displacement currents
Recall that the Maxwell equations are
· E =
ρ
ε
0
· B = 0
× E =
B
t
× B = µ
0
J + ε
0
E
t
So far we have studied all the equations apart from the
µ
0
ε
0
E
t
term. Historically
this term is called the displacement current.
The need of this term was discovered by purely mathematically, since people
discovered that Maxwell’s equations would be inconsistent with charge conserva-
tion without the term.
Without the term, the last equation is
× B = µ
0
J.
Take the divergence of the equation to obtain
µ
0
· J = · ( × B) = 0.
But charge conservation says that
˙ρ + · J = 0.
These can both hold iff
˙ρ
= 0. But we clearly can change the charge density
pick up a charge and move it elsewhere! Contradiction.
With the new term, taking the divergence yields
µ
0
· J + ε
0
·
E
t
= 0.
Since partial derivatives commute, we have
ε
0
·
E
t
= ε
0
t
( · E) = ˙ρ
by the first Maxwell’s equation. So it gives
· J + ˙ρ = 0.
So with the new term, not only is Maxwell’s equation consistent with charge
conservation it actually implies charge conservation.