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Unformatted text preview: Electromagnetics (I) Final Examination June 15,2009 . (1) Write down the Maxwell equations in differential form. (2) What is the advantage of the differential—form Maxwell equations over the integral-form ones? Discuss it brieﬂy. (16%) . (1) What is a phasor? Discuss its usefulness in electromagnetic theory brieﬂy. (2) To compute the time-average Poynting vector, we usually introduce a complex Poynting vector. Please write down the expression of that complex Poynting vector. Is it also a phasor? Explain brieﬂy. (16%) . An inﬁnite plane sheet lying in the z = 0 plane carries a time-varying surface current of density .75 (t) = O.lsin(87rx108t+7z/4)ZiyA/m. The region 2 > 0 is a perfect dielectric of a = 980 and . ,u = #0 , whereas the region 2 < 0 is a perfect dielectric of a = 2.2550 and ,u = #0. Find E(z,t) and H(z,t) in both regions 2 > 0 and z < O. (20%) Consider a uniform plane wave with electric ﬁeld E(z,t) = E, cos a)(t alga z)5x propagating in a homogeneous perfect dielectric with permittivity a and permeability ,u . (1) Find the instantaneous energy (electric + magnetic) density u(z, t) associated with this uniform plane wave. (2) Show that E xFI — \/:_c‘zz , and explain its physical meaning. (3) Show that u satisﬁes a #5 ' continuity equation V - (J... a )+ Q— = 0. Discuss the physical meaning brieﬂy. (18%) ,ua at . A steady conduction current is distributed in an inhomogeneous lossy dielectric medium (with conductivity 0'( f) and permittivity 8(7 )) under the inﬂuence of an electrostatic ﬁeld E . (1) Show that in the medium there exist free charges with volume density p f = (Vs — 3 Va) - E . * ‘ or (2) If the volume density of total charges is given by V - 50E , show that in the medium there exist polarization charges with volume density ,0p = [(8—80) Va—VgJ-E . (3) Find the partial a- differential equation satisﬁed by the corresponding electric scalar potential V ( f ). (15%) . The space between two parallel conducting plates each having an area S is ﬁlled with a lossy dielectric whose conductivity varies linearly from 61 at one plate (y=0) to 02 at the other plate (y=d, d<<«/§ ). The permittivity of the dielectric is a constant e. A battery of d-c voltage V0 is applied across the plates, as shown in the following ﬁgure. Determine ( 1) the total resistance between the plates, (2) the surface densities of free charges on both plates, and (3) the volume density of free charges between the plates. (4) Draw an equivalent circuit of this system, neglecting the magnetic effect. (5) If the battery is disconnected, what would happen to this system? Give a qualitative description brieﬂy. (Hint. You may consider ﬁrst the condition of a steady current.) (15%) yr 1 d ...
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