676
14 Electromagnetism II
14.63 Skin depth δ =
2
2π f σ μ 0 μ
=
2
2π × 20 × 10 3 × 6 × 10 7 × 4π × 10 −7 × 1
= 4.6 × 10
−4 m = 0.46 mm
14.64 δ =
2
2π f σμ 0 μ
=
1
π × 3 × 10 8 × 5.6 × 10 7 × 4π × 10 −7 × 1
= 1.23 × 10
−4 m
= 0.123 mm = 3.88 μm
14.65 Let us begin with the ‘curl H ’ Maxwell’s equation
∇ × H = J +
∂D
∂t
(1)
Assume that all the fields and currents in this equation are sinusoidal at a
single frequency ω. In that case (1) can be replaced by
∇ × H = J + jωD = J + jωεE
(2)
where J is the current density, D = εE is the displacement vector and j is
imaginary
√
−1
.
The ‘curl E’ Maxwell’s equation is obtained in a similar way as (1).
∇ × E = −jωB
(3)
Applying the curl operation to each side of (3)
∇ × (∇ × E) = −jω∇ × B
(4)
Now ∇ × B can be replaced using (2) and the relation H = B/μ:
∇ × (∇ × E) = −jωμ(J + jωεE)
(5)
We now use the vector identity
∇ × ∇ × E = ∇(∇ · E) = −∇
2 E
(6)
Conductive materials do not contain any real charge density because any real
charge that many exist will repel itself and move outwards until it resides on