12Maxwell's equations
IA Vector Calculus
12.3 Electromagnetic waves
Consider Maxwell’s equations in empty space, i.e.
ρ
= 0,
j
=
0
. Then Maxwell’s
equations give
∇
2
E = ∇(∇ · E) − ∇ × (∇ × E) = ∇ ×
∂B
∂t
=
∂
∂t
(∇ × B) = µ
0
ε
0
∂
2
E
∂
2
t
.
Define c =
1
√
µ
0
ε
0
. Then the equation gives
∇
2
−
1
c
2
∂
2
∂t
2
E = 0.
This is the wave equation describing propagation with speed
c
. Similarly, we
can obtain
∇
2
−
1
c
2
∂
2
∂t
2
B = 0.
So Maxwell’s equations predict that there exists electromagnetic waves in free
space, which move with speed
c
=
1
√
ε
0
µ
0
≈ 3.00 × 10
8
m s
−1
, which is the speed
of light! Maxwell then concluded that light is electromagnetic waves!