7.2 Equation of Wave Motion
If we further confine ourselves to the case where neither electric charge nor electric
current is present in a uniform dielectric medium, we can readily obtain equations of
wave motion regarding the electric and magnetic fields. That is,
div D ¼ 0,
ð7:26Þ
div B ¼ 0,
ð7:27Þ
rot E þ
∂B
∂t
¼ 0,
ð7:28Þ
rot H À
∂D
∂t
¼ 0:
ð7:29Þ
The relations (7.27) and (7.28) are identical to (7.2) and (7.3), respectively.
Let us start with a formula of vector analysis. First, we introduce a grad operator.
We have
grad f ¼ — f ¼
∂f
∂x
e 1 þ
∂f
∂y
e 2 þ
∂f
∂z
e 3 :
That is, the grad operator transforms a scalar to a vector. We have a following
formula:
rot rot V ¼ grad div V À —
2 V:
ð7:30Þ
The operator —
2 has already appeared in (1.24). To show (7.30), we compare a xcomponent of both sides of (7.30). That is
rot rot V
½
Š x ¼
∂
∂y
∂V y
∂x
2
∂V x
∂y
2
∂
∂z
∂V x
∂z
2
∂V z
∂x
:
ð7:31Þ
grad div V À —
2 V
Â
Ã
x
=
∂
∂x
∂V x
∂x
þ
∂V y
∂y
þ
∂V z
∂z
À
∂
2 V x
∂x
2
À
∂
2 V x
∂y
2
À
∂
2 V x
∂z
2
=
∂
∂x
∂V y
∂y
þ
∂V z
∂z
À
∂
2 V x
∂y
2
À
∂
2 V x
∂z
2
:
ð7:32Þ
Again assuming that
∂
2 V y
∂y∂x
¼
∂
2 V y
∂x∂y
and
∂
2 V z
∂z∂x
¼
∂
2 V z
∂x∂z
, we have
rot rot V
½
Š x ¼ grad div V À ∇
2
V
Â
Ã
x
:
ð7:33Þ
Regarding y- and z-components, we have similar relations as well. Thus (7.30) holds.
276
7 Maxwell’s Equations