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.