Page 111 - Electromagnetics

and where 2τ is the temporal duration of the pulse. At t =−8 µs the leading edge of
the pulse is at z = 233 m, while at −4 µs the pulse has traveled a distance z = vt =
7
−6
(3.33 × 10 ) × (4 × 10 ) = 133 m in the −z-direction, and the leading edge is thus at
100 m. At t =−1 µs the leading edge strikes the conductor and begins to induce a
current in the conductor surface. This current sets up the reflected wave, which begins
to travel in the opposite (+z) direction. At t =−0.5 µs a portion of the wave has begun
to travel in the +z-direction while the trailing portion of the disturbance continues to
travel in the −z-direction. At t = 1 µs the wave has been completely reflected from
the surface, and thus consists only of the component traveling in the +z-direction. Note
that if we plot the total field in the z = 0 plane, the sum of the forward and backward
traveling waves produces the pulse waveform (2.338) as expected.
Using the expressions for E and H we can determine many interesting characteristics
of the wave. We see that the terms f (t ± z/v) represent the components of the waves
traveling in the ∓z-directions, respectively. If we were to isolate these waves from each
other (by, for instance, measuring them as functions of time at a position where they do
not overlap) we would find from (2.336) and (2.337) that the ratio of E to H forawave
traveling in either direction is

E(z, t) 1/2
= vµ = (µ/
) ,

H(z, t)

independent of the time and position of the measurement. This ratio, denoted by η and
carrying units of ohms, is called the intrinsic impedance of the medium through which
the wave propagates. Thus, if we let E 0 = ηH 0 we can write

E 0 z E 0 z
E(z, t) = ˆ p f t − − ˆ p f t + . (2.339)
2 v 2 v
We can easily determine the current induced in the conductor by applying the boundary
condition (2.200):

J s = ˆ n × H| z=0 = ˆ z × [H 0 ˆ q f (t)] =−ˆ pH 0 f (t). (2.340)

We can also determine the pressure exerted on the conductor due to the Lorentz force
interaction between the fields and the induced current. The total force on the conductor
5
can be computed by integrating the Maxwell stress tensor (2.288) over the xy-plane :

¯
F em =− T em · dS.
S
The surface traction is
1

¯ ¯
t = T em · ˆ n = (D · E + B · H)I − DE − BH · ˆ z.
2
Since E and H are both normal to ˆ z, the last two terms in this expression are zero. Also,
the boundary condition on E implies that it vanishes in the xy-plane. Thus
1 µ 2
t = (B · H)ˆ z = ˆ z H (t).
2 2
5 We may neglect the momentum term in (2.291), which is small compared to the stress tensor term. See
Problem 2.20.

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