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284 ENGINEERING ELECTROMAGNETICS
at t = 1 µs; (c) find the value of the closed line integral of E around the peri-
meter of the given surface.
5
Ans. −20 000 sin 10 t cos 10 −3 ya z V/m; 0.318 mWb; −3.19 V
D9.2. With reference to the sliding bar shown in Figure 9.1, let d = 7 cm,
B = 0.3a z T, and v = 0.1a y e 20y m/s. Let y = 0at t = 0. Find: (a) ν(t = 0);
(b) y(t = 0.1); (c) ν(t = 0.1); (d) V 12 at t = 0.1.
Ans. 0.1 m/s; 1.12 cm; 0.125 m/s; −2.63 mV
9.2 DISPLACEMENT CURRENT
Faraday’s experimental law has been used to obtain one of Maxwell’s equations in
differential form,
∂B
∇× E =− (15)
∂t
which shows us that a time-changing magnetic field produces an electric field. Re-
membering the definition of curl, we see that this electric field has the special property
of circulation; its line integral about a general closed path is not zero. Now let us turn
our attention to the time-changing electric field.
We should first look at the point form of Amp`ere’s circuital law as it applies to
steady magnetic fields,
∇× H = J (16)
and show its inadequacy for time-varying conditions by taking the divergence of each
side,
∇ · ∇× H ≡ 0 =∇ · J
The divergence of the curl is identically zero, so ∇ · J is also zero. However, the
equation of continuity,
∂ρ ν
∇ · J =−
∂t
then shows us that (16) can be true only if ∂ρ ν /∂t = 0. This is an unrealistic limitation,
and (16) must be amended before we can accept it for time-varying fields. Suppose
we add an unknown term G to (16),
∇× H = J + G
Again taking the divergence, we have
0 =∇ · J +∇ · G
Thus
∂ρ ν
∇ · G =
∂t