Page 120 - Electromagnetics

From (2.354) and (2.355), the time-rate of change of stored energy is
1 ∂ 2 2
P sphere (t) = [
E (r,θ, t) + µH (r,θ, t)] dV
2 ∂t V
1 ∂ 2π θ 2 dθ r 2 1 2 r 1 1 2 r 2
= dφ
A t − + µ A t − r dr
2 ∂t 0 θ 1 sin θ r 1 r 2 v r (vµ) 2 v
2
∂ r 2 2 r
= 2π
F A t − dr
∂t v
r 1
where

tan(θ 2 /2)
F = ln .
tan(θ 1 /2)
Putting u = t − r/v we see that
∂ t−r 2 /v 2
P sphere (t) =−2π
F A (u)v du.
∂t t−r 1 /v
An application of Leibnitz’ rule for differentiation (A.30) gives
2π 2 r 2 2 r 1
P sphere (t) =− F A t − − A t − . (2.357)
η v v
Next we find the Poynting flux term:

P sphere (t) =− (E × H) · dS
S
2π θ 2 1 r 1 1 1 r 1 dθ
ˆ
ˆ
=− dφ A t − θ × A t − φ · (−ˆ r)r 1 2 −
0 θ 1 r 1 v r 1 µv v sin θ
2π θ 2 1 1 1 dθ
ˆ
ˆ
− dφ A t − r 2 θ × A t − r 2 φ · ˆ rr 2 2 .
0 θ 1 r 2 v r 2 µv v sin θ
The first term represents the power carried by the traveling wave into the volume region
by passing through the spherical surface at r = r 1 , while the second term represents
the power carried by the wave out of the region by passing through the surface r = r 2 .
Integration gives
2π 2 r 2 2 r 1
P sphere (t) =− η F A t − v − A t − v , (2.358)
which matches (2.357), thus verifying Poynting’s theorem.
It is also interesting to compute the total energy passing through a surface of radius
r 0 . From (2.358) we see that the flux of energy (power density) passing outward through
the surface r = r 0 is
2π 2 r 0
P (t) = FA t − .
sphere
η v
The total energy associated with this flux can be computed by integrating over all time:
we have
2π ∞ 2 r 0 2π ∞ 2
E = F A t − dt = F A (u) du
η v η
−∞ −∞
after making the substitution u = t −r 0 /v. The total energy passing through a spherical
surface is independent of the radius of the sphere. This is an important property of
spherical waves. The 1/r dependence of the electric and magnetic fields produces a
power density that decays with distance in precisely the right proportion to compensate
2
for the r -type increase in the surface area through which the power flux passes.

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