692
14 Electromagnetism II
14.83 Using Maxwell’s equations
∇ · E = 0
( 1 )
∇ · B = 0
( 2 )
∇ × E = −
∂ B
∂t
(3)
(a) E = E 0 e
i(ωt−k.r+φ)
(4)
B = B 0 e
i(ωt−k.r+φ)
(5)
Let the wave be propagated in the z-direction. Then
∇ · E = −i k · E = 0
and ∇ · B = −i k · B = 0
This shows that both E and B are perpendicular to k, which is the direction of propagation. Therefore both E and B are transverse oscillations.
(b) B and E are in phase
(c) Using (3)
∇ × E = −i k × E = −iω B
∴ B =
k × E
ω
=
1
c
k × E
k
∴ B =
1
c
ˆ
s × E
(6)
where ˆ
s = k/k is a unit vector in the direction of propagation. The three
vectors E, B and k form a right-handed rectangular coordinate system.
From (6) we obtain B = E/c.
14.84 E =
σ
2ε r ε 0
∴ σ = 2ε 0 ε r E = 2 × 8.85 × 10
−12
× 6 × 2 × 10
3
= 2.124 × 10
−7 C/m
2
14.3.3 Phase Velocity and Group Velocity
14.85 Using the result of prob. 14.93
1
V g
−
1
V p
=
V p − V g
V g V p
=
V p − V g
c 2 − =
ω
c
dn
dω
14 Electromagnetism II
14.83 Using Maxwell’s equations
∇ · E = 0
( 1 )
∇ · B = 0
( 2 )
∇ × E = −
∂ B
∂t
(3)
(a) E = E 0 e
i(ωt−k.r+φ)
(4)
B = B 0 e
i(ωt−k.r+φ)
(5)
Let the wave be propagated in the z-direction. Then
∇ · E = −i k · E = 0
and ∇ · B = −i k · B = 0
This shows that both E and B are perpendicular to k, which is the direction of propagation. Therefore both E and B are transverse oscillations.
(b) B and E are in phase
(c) Using (3)
∇ × E = −i k × E = −iω B
∴ B =
k × E
ω
=
1
c
k × E
k
∴ B =
1
c
ˆ
s × E
(6)
where ˆ
s = k/k is a unit vector in the direction of propagation. The three
vectors E, B and k form a right-handed rectangular coordinate system.
From (6) we obtain B = E/c.
14.84 E =
σ
2ε r ε 0
∴ σ = 2ε 0 ε r E = 2 × 8.85 × 10
−12
× 6 × 2 × 10
3
= 2.124 × 10
−7 C/m
2
14.3.3 Phase Velocity and Group Velocity
14.85 Using the result of prob. 14.93
1
V g
−
1
V p
=
V p − V g
V g V p
=
V p − V g
c 2 − =
ω
c
dn
dω