HW-2 - Theory of Solids HW-2 Solution By Qifeng Shan...

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1 Theory of Solids HW-2 Solution By Qifeng Shan Ashcroft and Mermin’s book 1-4 Helicon Waves Suppose that a metal is placed in a uniform magnetic field H along the z -axis. Let an AC electric field E e i ω t be applied perpendicular to H . (a) If the electric field is circularly polarized ( E y = ± i E x ) show that Eq. (1.28) must be generalized to 0 , , 1 0 z x y x c x j j i j E i j (1.61) (b) Show that, in conjunction with (1.61), Maxwell’s equations (1.31) have a solution 0 , , 0 z x y t kz i x E E i E e E E , (1.62) provided that k 2 c 2 = ε ω 2 , where / 1 1 2 i c p (1.63) (c) Sketch ε ( ω ) for ω > 0 (choosing the polarization E y = i E x ) and demonstrate that solutions to k 2 c 2 = ε ω 2 exist for arbitrary k at frequencies ω > ω p and ω < ω c . (Assume the high field condition ω c τ >> 1, and note that even for hundreds of kilogauss, ω p / ω c >> 1.) (d) Show that when ω << ω c the relation between k and ω for the low frequency solution is 2 2 2 p c c k . (1.64)
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