W ¼
1
2
E Á D þ H Á B
ð
Þ ,
ð8:72Þ
where the first and second terms are pertinent to the electric and magnetic fields,
respectively. Note in (8.72) that the dimension of E Á D is [
V
m Á
C
m 2 ] ¼ [
J
m 3 ] and that the
dimension of H Á B is [
A
m Á
VÁs
m 2 ] ¼ [
WÁs
m 3 ] ¼ [
J
m 3 ]. Using (7.7) and (7.10), we have
W ¼
1
2
εE
2
þ μH
2
À
Á :
ð8:73Þ
As in the above case, estimating a time-averaged energy density W, we get
W ¼
1
2
1
2
εE
2
þ
1
2
μH
2
=
1
4
εE
2
þ
1
4
μH
2
:
ð8:74Þ
We also get this relation by integrating (8.73) over a wavelength λ at a time of t ¼ 0.
Using (7.60) and (7.61), we have
εE
2
= μH
2
:
ð8:75Þ
This implies that the energy density resulting from the electric field and that due to
the magnetic field have the same value. Thus, rewriting (8.74) we have
W ¼
1
2
εE
2
=
1
2
μH
2
:
ð8:76Þ
Moreover, using (7.43), we have for an impedance
Z ¼ E=H ¼
ffiffiffiffiffiffiffi ffi
μ=ε
p
¼ μv or E ¼ μvH:
ð8:77Þ
Using this relation along with (8.75), we get
S ¼
1
2
vεE
2 e 3 ¼
1
2
vμH
2 e 3 :
ð8:78Þ
Thus, we have various relations among amplitudes of electromagnetic waves and
related physical quantities together with constant of dielectrics.
Returning to Examples 8.1 and 8.2, let us further investigate the reflection and
transmission properties of the electromagnetic waves. From (8.51) to (8.55) as well
as (8.59) to (8.62), we get in both the cases of TE and TM waves
ÀR
⊥
E R
⊥
H þ T
⊥
E T
⊥
H
cos ϕ
cos θ
¼ 1,
ð8:79Þ
310
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media
1
2
E Á D þ H Á B
ð
Þ ,
ð8:72Þ
where the first and second terms are pertinent to the electric and magnetic fields,
respectively. Note in (8.72) that the dimension of E Á D is [
V
m Á
C
m 2 ] ¼ [
J
m 3 ] and that the
dimension of H Á B is [
A
m Á
VÁs
m 2 ] ¼ [
WÁs
m 3 ] ¼ [
J
m 3 ]. Using (7.7) and (7.10), we have
W ¼
1
2
εE
2
þ μH
2
À
Á :
ð8:73Þ
As in the above case, estimating a time-averaged energy density W, we get
W ¼
1
2
1
2
εE
2
þ
1
2
μH
2
=
1
4
εE
2
þ
1
4
μH
2
:
ð8:74Þ
We also get this relation by integrating (8.73) over a wavelength λ at a time of t ¼ 0.
Using (7.60) and (7.61), we have
εE
2
= μH
2
:
ð8:75Þ
This implies that the energy density resulting from the electric field and that due to
the magnetic field have the same value. Thus, rewriting (8.74) we have
W ¼
1
2
εE
2
=
1
2
μH
2
:
ð8:76Þ
Moreover, using (7.43), we have for an impedance
Z ¼ E=H ¼
ffiffiffiffiffiffiffi ffi
μ=ε
p
¼ μv or E ¼ μvH:
ð8:77Þ
Using this relation along with (8.75), we get
S ¼
1
2
vεE
2 e 3 ¼
1
2
vμH
2 e 3 :
ð8:78Þ
Thus, we have various relations among amplitudes of electromagnetic waves and
related physical quantities together with constant of dielectrics.
Returning to Examples 8.1 and 8.2, let us further investigate the reflection and
transmission properties of the electromagnetic waves. From (8.51) to (8.55) as well
as (8.59) to (8.62), we get in both the cases of TE and TM waves
ÀR
⊥
E R
⊥
H þ T
⊥
E T
⊥
H
cos ϕ
cos θ
¼ 1,
ð8:79Þ
310
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media