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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
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