12Maxwell's equations

IA Vector Calculus



12.2 Static charges and steady currents
If ρ, j, E, B are all independent of time, E and B are no longer linked.
We can solve the equations for electric fields:
· E = ρ/ε
0
× E = 0
Second equation gives E = −∇ϕ. Substituting into first gives
2
ϕ = ρ/ε
0
.
The equations for the magnetic field are
· B = 0
× B = µ
0
j
First equation gives
B
=
× A
for some vector potential
A
. But the vector
potential is not well-defined. Making the transformation
A 7→ A
+
χ
(
x
)
produces the same
B
, since
×
(
χ
) = 0. So choose
χ
such that
· A
= 0.
Then
2
A = ( · A
|{z}
=0
) × ( × A
| {z }
B
) = µ
0
j.
In summary, we have
Electrostatics Magnetostatics
· E = ρ/ε
0
· B = 0
× E = 0 × B = µ
0
j
2
ϕ = ρ/ε
0
2
A = µ
0
j.
ε
0
sets the scale of electrostatic effects,
e.g. the Coulomb force
µ
0
sets the scale of magnetic effects,
e.g. force between two wires with cur-
rents.