Drude theory of AC transport. The Drude theory can be extended to study the response of...

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Drude theory of AC transport. The Drude theory can be extended to study the response of a metal to a time-dependent electric field E(t) = Ewe-it. From this we can understand why a metal is shiny, and also find a surprising result in the ultraviolet. (a) Using the ansatz Pav = Pweit, solve the Drude equation for pw. Hence obtain the current j(t) je it and show that jw = o(w) Ew, where do is the DC (i.e. w = 0) conductivity. Hint: You can no longer use the steady-state approximation. Why not? o(w) Jo 1- = = (b) The AC conductivity is imaginary. We can understand this by considering the real part of the current Re{j(t)} = Re{jwe-it). Show that the imaginary AC conductivity implies that this current lags the real part of the electric field by a time d = arctan(TW). Hint: You can assume Ew to be real. (c) In the presence of a current density j (but with vanishing charge density p), Maxwell's equations take the form V.E=0, V.B=0, where is the permeability and c is the speed of light. Using these equations and the results of par (a), show that the AC electric field satisfies 7Ew == where e(w) is the dielectric function. w VxE=- 5 (W) Ew c B t " V x B= oj + (w) = 1 + 1 JE c t io (w) EDW (1 Drude theory of AC transport. The Drude theory can be extended to study the response of a metal to a time-dependent electric field E(t) = Ewe-it. From this we can understand why a metal is shiny, and also find a surprising result in the ultraviolet. (a) Using the ansatz Pav = Pweit, solve the Drude equation for pw. Hence obtain the current j(t) je it and show that jw = o(w) Ew, where do is the DC (i.e. w = 0) conductivity. Hint: You can no longer use the steady-state approximation. Why not? o(w) Jo 1- = = (b) The AC conductivity is imaginary. We can understand this by considering the real part of the current Re{j(t)} = Re{jwe-it). Show that the imaginary AC conductivity implies that this current lags the real part of the electric field by a time d = arctan(TW). Hint: You can assume Ew to be real. (c) In the presence of a current density j (but with vanishing charge density p), Maxwell's equations take the form V.E=0, V.B=0, where is the permeability and c is the speed of light. Using these equations and the results of par (a), show that the AC electric field satisfies 7Ew == where e(w) is the dielectric function. w VxE=- 5 (W) Ew c B t " V x B= oj + (w) = 1 + 1 JE c t io (w) EDW (1

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