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14 Electromagnetism II
Use the vector identify
∇ × (∇ × B) = ∇(∇ · B) − ∇
2 B
∴ ∇ ×
μ 0 ε 0
∂E
∂t
= −∇
2 B (∵ ∇ · B = 0 by(2)
∴ μ 0 ε 0
∂
∂t
(∇ × E) = −μ 0 ε 0
∂
∂t
∂B
∂t
= −∇
2 B
∴ ∇
2 B = μ 0 ε 0
∂ 2 B
∂t 2
14.36 ∇ × B = μ 0 j (Ampere’s law)
(1)
Use the vector identity A · (A × B) = 0. Put A = ∇.
∴ ∇ · (∇ × B) = 0
∴ ∇ · j = 0
( 2 )
More generally, ∇ · j +
∂ρ
∂t
= 0 (continuity equation)
(3)
and ∇ · E = ε 0 ρ (Gauss’ law)
(4)
Combining (3) and (4)
∇ ·
j +
1
ε 0
∂E
∂t
= 0
14.37 ∇ 2 B = μ 0 ε 0
∂ 2 B
∂t 2 (free-space wave equation)
Compare with the standard three-dimensional wave equation
∇
2
=
1
v 2
∂ 2
∂t 2
(5)
∴ v =
1
√
μ 0 ε 0
=
1
(4π × 10 −7 )(8.854 × 10 −12 )
= 2.998 × 10
8 m/s = c
14.38 ω = 2πν =
2π c
λ
=
2π × 3 × 10 8
530 × 10 −9 = 3.55 × 10
15
σ =
1
ρ
=
1
26.5 × 10 −9 = 3.77 × 10
7
δ =
2
μ 0 σ ω
=
2
4π × 10 −7 × 3.77 × 10 7 × 3.55 × 10 15
= 3.45 × 10
−9 m = 3.45 nm