Page 287 - Electromagnetics

In a good conductor the group velocityis approximatelytwice the phase velocity. We
could have found this relation from the phase velocityusing (4.242). Indeed, noting that

dv p d 2ω 1 2
= =
dω dω µσ 2 ωµσ
and

ωµσ 1 2 1
dv p
β = = ,
dω 2 2 ωµσ 2
we see that
1 1
v p
= 1 − = .
v g 2 2
Note that the phase and group velocities maybe onlysmall fractions of the free-space
7
light velocity. For example, in copper (σ = 5.8 × 10 S/m, µ = µ 0 , = 0 ) at 1 MHz,
we have v p = 415 m/s.
A factor often used to judge the qualityof a conductor is the distance required for a
propagating uniform plane wave to decrease in amplitude bythe factor 1/e. By(4.244)
this distance is given by
1 1
δ = = √ . (4.258)
α π f µσ
We call δ the skin depth. A good conductor is characterized bya small skin depth. For
example, copper at 1 MHz has δ = 0.066 mm.

Power carried by a uniform plane wave. Since a plane wavefront is infinite in
extent, we usuallyspeak of the power density carried bythe wave. This is identical to
the time-average Poynting flux. Substitution from (4.223) and (4.244) gives
∗

ˇ
ˆ
1 1 k × E
ˇ
ˇ
ˇ ∗
S av = Re{E × H }= Re E × . (4.259)
2 2 η
ˇ
Expanding the cross products and remembering that k · E = 0, we get
2
ˇ 2
1 |E| E 0 −2α ˆ k·r
ˆ
ˆ
S av = k Re = k Re e .
2 η ∗ 2η ∗
Hence a uniform plane wave propagating in an isotropic medium carries power in the
direction of wavefront propagation.
Velocity of energy transport. The group velocity(4.237) has an additional interpre-
tation as the velocityof energytransport. If the time-average volume densityof energy
is given by
w em = w e + w m
and the time-average volume densityof energyflow is given bythe Poynting flux density
1
1
ˇ
ˇ
ˇ
ˇ ∗
ˇ ∗
ˇ ∗
S av = Re E(r) × H (r) = E(r) × H (r) + E (r) × H(r) , (4.260)
2 4
then the velocityof energyflow, v e , is defined by
S av = w em v e . (4.261)
© 2001 by CRC Press LLC