# PS9 - explain why 9.3H Electromagnetic waves propagating in...

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Phys601/F11/Problem Set 9 due Nov 7 2011 Subject to upgrade 9.1H Find all the roots of the algebraic equation ε 2 x 6 - ε x 4 ± x 3 + 8 = 0 by perturbative methods assuming ε << 1. The roots may be complex. Obtain the roots correct to first non-vanishing order (i.e., get x = x 0 + x 1 , at least, for both roots) and check for self-consistency (i.e., | x 1 <<| x 0 | ). 9.2H Solve the ODE below by perturbative methods if ε << 1. ε y’’ + (1+x)y’ + y = 0. Assume we are only interested in solutions valid in the range x 0. Obtain both solutions correct to first order (i.e., y = y 0 + y 1 ). Check each for self-consistency of solution. Make a sketch of both solutions in the domain x = [0,2] , assuming ε = 1/10. [You may find an unexpected form for y 1 for the WKB solution. Try y 2 . If that’s also similar, try to
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Unformatted text preview: explain why.] 9.3H Electromagnetic waves propagating in a medium with dielectric ε (x) satisfy the wave equation ε (x) ∂ 2 ψ / ∂ t 2 = c 2 ∂ 2 ψ / ∂ x 2 , ψ = ψ (x,t). Suppose ε (x) = (1 + x 2 /2L 2 ) 2 . The medium is disturbed at x=0 by an antenna with frequency ω . If ω L/c >> 1, find the WKB solution for x > 0 for waves propagating to the right, correct to 1 st order. By demanding that |k 1 | << |k |, obtain the condition that the WKB solution is a good approximation. At what x/L do you find the worst violation? Make a sketch of the real part of ψ , to first order, over a scale of length a few L’s. Find x(t), the location of the wavepacket moving with the local group velocity if x(0) = 0. From the constancy of ω , find k(t)....
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