W ¼
ε
2
LE
2
¼
μ
2
LH
2
:
ð9:145Þ
Representing an energy per unit volume as f
W e , g
W m , and e
W, we have
f
W e ¼
ε
2
E
2 sin
2
ωt, g
W m ¼
ε
2
E
2 cos
2
ωt, e
W ¼
ε
2
E
2
¼
μ
2
H
2
:
ð9:146Þ
In Chap. 2, we treated motion of a harmonic oscillator. There, we had
x t
ð Þ ¼
v 0
ω
sin ωt ¼ x 0 sin ωt:
ð2:7Þ
Here, we have defined an amplitude of the harmonic oscillation as x 0 (>0)
x 0
v 0
ω
:
Then, momentum is described as
p t
ð Þ ¼ m _
x t
ð Þ ¼ mωx 0 cos ωt:
Defining
p 0 mωx 0 ,
we have
p t
ð Þ ¼ p 0 cos ωt:
Let a kinetic energy and potential energy of the oscillator be K and V, respectively. Then, we have
K ¼
1
2m
p t
ð Þ
2 ¼
1
2m
p 0
2 cos
2
ωtx t
ð Þ,
V ¼
1
2
mω
2 x t
ð Þ
2 ¼
1
2
mω
2 x 0
2 sin
2
ωtx t
ð Þ,
W ¼ K þ V ¼
1
2m
p 0
2
¼
1
2
mv 0
2
¼
1
2
mω
2 x 0
2
:
ð9:147Þ
Comparing (9.146) and (9.147), we recognize the following relationship in energy
between the electromagnetic fields and harmonic oscillator motion [15]:
ffiffiffi ffi
m
p ωx 0 ⟷
ffiffi ffi
ε
p
E and p 0 =
ffiffiffi ffi
m
p ⟷
ffiffiffi
μ
p H:
ð9:148Þ
9.6 Mechanical System
377