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Unformatted text preview: ECE604 Homework 6
Out: Tuesday, February 24, 2004 (Session 13)
Due: Tuesday, March 2, 2004 (Session 15)
[Students with tape delays: due according to session number] Problem numbers are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994). 1)
2)
3)
4) 5) 3.70 3.11c
3.18a
3.18b 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: V x H = J
Assuming that this approximation holds, show that Maxwell’s equations for a spatially
uniform medium with ﬁnite conductivity 0 lead to the magnetic diffusion equation, written VZH =u6211
6t Note: You may assume J = GE and B = pH. 6) A spherical COnductor of radius a is exposed to a unifonn time—varying magnetic field
in the z—direction: B(t) = B0 sin out 2 It may be assumed that the radius a of the sphere is very small compared to the skin
depth 5, given by San/(ﬂue) where o is the conductivity, f is the frequency, and u is the permeability, assumed to be
no everywhere. Derive an expression for the timeaverage power dissipated in the sphere. 2 Conducting sphere
of radius a HIM 7) Consider a simple parallel plate capacitor connected to a sinusoidal voltage source
V(t) = V0 sin (0t, as shown. ‘d’ 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 s,
and permeability no . Let d<<a<<?t, where A is the wavelength that a freely propagating wave at frequency to 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, 03, a, d, 8, no and/or 7L. State all assumptions and
approximations that you are making. (b) A timevarying D ﬁeld is associated with a circulating magnetic ﬁeld. Show that
avowr a
cos cut (1)
2d where ti) is the unit vector in the :1) 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. the vector ﬁeld H in the region between the plates is given by H = (WM)
(WE)
magnetic ﬁeld energy and W5 is the total electric ﬁeld energy within the capacitor. Write your answer as a function of no, a, 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 ((1) Determine the instantaneous Poynting vector within the dielectric medium. Discuss
the meaning of its direction and its dependence ont and r. ...
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 Spring '08
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 Electromagnet, Permittivity, Dielectric, Maxwell’s equations

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