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.