Page 412 - Engineering Electromagnetics, 8th Edition

394 ENGINEERING ELECTROMAGNETICS

The time-average power loss is easily obtained, since the average value of the cosine-
squared factor is one-half,

1
2
P L = J bLδ (89)
x0
4σ
Comparing (88) and (89), we see that they are identical. Thus the average power
loss in a conductor with skin effect present may be calculated by assuming that the
total current is distributed uniformly in one skin depth. In terms of resistance, we may
say that the resistance of a width b and length L of an infinitely thick slab with skin
effect is the same as the resistance of a rectangular slab of width b, length L, and
thickness δ without skin effect, or with uniform current distribution.
We may apply this to a conductor of circular cross section with little error,
provided that the radius a is much greater than the skin depth. The resistance at
a high frequency where there is a well-developed skin effect is therefore found by
considering a slab of width equal to the circumference 2πa and thickness δ. Hence

L L
R = = (90)
σ S 2πaσδ
A round copper wire of 1 mm radius and 1 km length has a resistance at direct
current of
10 3
R dc = = 5.48
7
π10 (5.8 × 10 )
−6
At 1 MHz, the skin depth is 0.066 mm. Thus δ a, and the resistance at 1 MHz is
found by (90),
10 3
R = = 41.5
−3
7
2π10 (5.8 × 10 )(0.066 × 10 )
−3
D11.7. A steel pipe is constructed of a material for which µ r = 180 and
6
σ = 4 × 10 S/m. The two radii are 5 and 7 mm, and the length is 75 m. If
the total current I(t) carried by the pipe is 8 cos ωt A, where ω = 1200π rad/s,
find: (a) the skin depth; (b) the effective resistance; (c) the dc resistance; (d)
the time-average power loss.

Ans. 0.766 mm; 0.557
; 0.249
; 17.82 W

11.5 WAVE POLARIZATION

In the previous sections, we have treated uniform plane waves in which the electric
and magnetic field vectors were assumed to lie in fixed directions. Specifically, with
the wave propagating along the z axis, E was taken to lie along x, which then required
H to lie along y. This orthogonal relationship between E, H, and S is always true for
a uniform plane wave. The directions of E and H within the plane perpendicular to a z