hw7solutions - sa/wl—fon ECE604 Homework 7 Out Tuesday...

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Unformatted text preview: sa/wl—fon ECE604 Homework 7 Out: Tuesday, March 1, 2005 Due: Tuesday, March 8, 2005 Problems given by number are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994). 1. 3.18b 2. See below 3. See below 4. See below 5. 5.2c 6. 5.2g (Note: for your solution you may assume that the material between the inner and outer conductors is air.) 5 7. 5.4a 8. 5.5a 9. 5.5b 2) In the magnetoquasistatic approximation to Maxwell’s equations, the displacement current is assumed to be much less than the conduction current, so that Ampere’s Law may be written: VXH=J Assuming that this approximation holds, show that Maxwell’s equations for a spatially uniform medium with finite conductivity 6 lead to the magnetic diffusion equation, written an V2H= o— ” at Note: You may assume J = GE and B = uH. 3) A spherical conductor of radius a is exposed to a uniform time—varying magnetic field in the z-direction: B(t) = B0 sin cut 2 It may be assumed that the radius a of the sphere is very small compared to the skin depth 5, given by 8: 1/(1cfuo) where 0 is the conductivity, f is the frequency, and u is the permeability, assumed to be ”0 everywhere. Derive an expression for the time-average power dissipated in the sphere. Conducting sphere of radius a -Hm 4) Consider a simple parallel plate capacitor connected to a sinusoidal voltage source V(t) = V0 sin out, as shown. The radius of each of the round conducting plates is a, and the separation between the plates is d. The medium between the plates is a perfect insulator with permittivity 8, and permeability no . Let d<<a<<}t, where A is the wavelength that a freely— propagating wave at frequency 0) would have in this dielectric medium. (a) Using one or more of Maxwell’s equations (in integral or differential form, your choice) determine the D field in the region between the plates. Express your answer in terms of V0, (1), a, d, 8, no and/or k. State all assumptions and approximations that you are making. (b) A time-varying D field is associated with a circulating magnetic field. Show that aVOwr the vector field H in the region between the plates is given by H = cos on E) where (i) is the unit vector in the ¢ direction, as shown on the diagram above. Show the contour, surface, or volume that allows you to make this determination. State all assumptions and approximations that you are making. (WM> <WE> magnetic field energy and WE is the total electric field energy within the capacitor. Write your answer as a function of Mo, 8, a and k. Discuss the relative magnitude of the ratio. Under what conditions should your results be regarded as valid? (c) Determine the ratio of the time—averaged energies , where WM is the total (d) Determine the instantaneous Poynting vector within the dielectric medium. Discuss the meaning of its direction and its dependence on t and r. a» gem! we‘va‘” #2 4 3 '0) ‘ .2 ”a ‘ .....TA .45 7511/44? wF§w ’5. (4.7 34/0135“) 42 52?. A 5303/9... 74W - .. . ..m. rwficoi’ama .t. .49}. 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