Homework2-3 - Homework 2 and 3 Answers, 95.658 Spring 2011,...

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Homework 2 and 3 Answers, 95.658 Spring 2011, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell Problem 1 Consider the general case where the wave number is complex-valued and the permittivity is complex- valued (but the material is non-magnetic). As outlined in class, expand the dispersion relation for plane waves into complex components and match up real and imaginary parts. Invert these equations to derive the explicit expressions for the wave number's complex components in terms of the permittivity's complex components. Carefully sketch a three-dimensional plot for both of these expressions, setting the real part of the permittivity as the x-axis, the imaginary part as the y-axis, and the wave number component as the z-axis. Mark several points on each plot and label with a physical description of what is going on at those points. SOLUTION: The dispersion relation for plane waves in uniform, linear, non-magnetic material was found to be: k =  0 If we expand each side into complex components, k = β + i α /2, and = ' i '' , we have:  i / 2 = ' i '' 0 Square both sides: 2 − 2 / 4 2 i / 2 = ' 0 2 i '' 0 2 Match up real and imaginary parts: 2 − 2 / 4 = ' 0 2 = '' 0 2 ' = 2 − 2 / 4 0 2 and '' =  0 2 Now we need to invert these equations. Solve the second equation for alpha and plug into the first equation: = '' 0 2 2 '' 0 2 2 / 4 = ' 0 2 4 − ' 0 2 2 1 4 '' 2 0 2 4 = 0
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Use the quadratic equation: 2 = −− ' 0 2 ± − ' 0 2 2 4 − 1 4 '' 2 0 2 4 2 2 = ' ± ' 2  '' 2 2 0 2 =± 0 2 ' ± ' 2  '' 2 Plug this back in to solve for alpha: 2 / 4 =− ' 0 2 ' ± ' 2  '' 2 2 0 2 =± 2 0 2 − ' ± ' 2  '' 2 Now we realize that because ε ' and ε '' are just the real and imaginary parts of a complex number ε , the square root of the sum of their squares is just the distance of the complex number from the origin: =± 0 2 ' ±∣∣ =± 2 0 2 − ' ±∣∣ By definition, alpha and beta are the real components of a complex number. To keep them real, we must choose the upper sign inside the square root: =± 0 2 ∣∣ ' =± 2 0 2 ∣∣− ' Expressing the wave with the complex wave vector expanded gives us some understanding of what the signs mean: E = E 0 e −/ 2 x e i  x − t The positive-valued solution for beta is for right-handed waves that arise in standard materials, and the
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Homework2-3 - Homework 2 and 3 Answers, 95.658 Spring 2011,...

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