Continuum Electromechanics Exam Help
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Problem: 1
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A potential sheet, , is located at x = 0 between two dielectric media with dielectric permittivity . A sheet of surface charge, , is located at x=d between two dielectric media with dielectric permittivity . All materials are lossless. The system is of infinite extent in the y direction. a) Find the complex amplitudes VÖ 0 and )Ö 0 in terms of V 0 and )0 . b) What is the complex amplitude of the potential )Ö (x d) along the sheet of surface charge at x = d ? c) What is the complex amplitude of the surface charge density along the potential sheet at x=0 ? (Express your answer in terms of VÖ 0 , )Ö 0 , and )Ö (x d) ). d) What is the space average force per unit area in the x and z directions on the sheet of surface charge at x=d ? Use the results of part (a) to greatly simplify your answer. Hint: The Maxwell Stress tensor will be the easiest way to solve for the space average free charge and polarization forces in the x and z directions.
Problem: 2 An infinitely long cylinder of radius R , conductivity V , and magnetic permeability P is placed Ö in a uniform magnetic field . The region r ! R is free space. liveexamhelper.com
The governing equation is the magnetic diffusion equation.
There is no surface current on the r R interface. a) Assume that
and show that the governing equation is
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b) Solve part (a) for Hz r for r What boundary conditions must the solution satisfy? Ö ( ) R . Hint 1: The solution to Bessel’s Equation
Is
where J n is called a Bessel function of the first kind of order n and Yn is called the nth order Bessel function of the second kind. Note that J n (0) is finite while Yn (0) is infinite. Note also that k has two solutions. Which solution for k can be used to solve for Ö z H ( )r ? Hint 2: c) What is the current density J ( )r ? Hint d) Plot H r z ( ,t J (r,t 0) for G / R 0.05 , 0.1, 0.25 , 0.5, 0.75 , 1, f where the skin-depth. liveexamhelper.com
is
Solution: 1 a) b)
Figure 1: A three dielectric layer system with potential constrained at x = 0 and surface charge density constrained at x = d
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c)
d)
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Solution: 2 a)
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b) Bessel’s Equation: Multiply (a) by
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Since J0(kr) has a series form of even powers of (kr), either sign can be used for k. c)
d) Mathematica Program to Plot Clear[Hzn, Jphin] Hzn[rn, deltan] = Re[BesselJ[0,(1 − I) ∗ rn/deltan]/BesselJ[0,(1 − I)/deltan]]
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Jphin[rn, deltan] = Re[−(1 − I) ∗ BesselJ[1,(1 − I) ∗ rn/deltan]/BesselJ[0,(1 − I)/deltan]]
Plot[{Hzn[rn, 0.05], Hzn[rn, 0.1], Hzn[rn, 0.25], Hzn[rn, 0.5], Hzn[rn, 0.75], Hzn[rn, 1], Hzn[rn,Infinity]}, {rn, 0, 1},PlotRange→ {−.2, 1}]
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Plot[{Jphin[rn, 0.05], Jphin[rn, 0.1], Jphin[rn, 0.25], Jphin[rn, 0.5], Jphin[rn, 0.75], Jphin[rn, 1], Jphin[rn,Infinity]}, {rn, 0, 1},PlotRange→ {−.2, 1}]
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