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solutions 11

Modern Physics

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PHY251 Homework Set 11 Reading: Chapter 12, 14 (all sections except 6) Homework: Chapter 12, Problems 32, Chapter 14, Problems 2, 6(Li, Al, Cu only!),9(Li, Al, Cu only!),10,17,18,31 Hints and Solutions Problem XII.32 (15 points) Consider a free-electron gas at a temperature T such that kT << E F . Write down an expression for the electron number density N / V for electrons that have an energy in excess of E F . Show, by making the change of variables ( E - E F )/ kT = x , that the number density is proportional to T . Calculate an expression for N / V under these circumstances, making use of the fact that 0 (e x + 1) - 1 dx = ln 2. Hints: In working out the integral over E the integrand is such that ( x + E F ) ½ 2245 E F ½ . Solution: From notes or book: n e ( E > E F ) = ( N e / V ) = E F n e ( E ) dE = E F g ( E ) dE F FD ( E ; T ) = = (8 π 2 m e 3/2 / h ³) E F E dE (e x + 1) - 1 2245 (8 π 2 m e 3/2 / h ³) E F E F kT dx (e x + 1) - 1 = = (8 π 2 m e 3/2 / h ³) kT E F ln2 = [ 2 m e 3/2 /( π ² h ³)] kT E F ln2 = (6.85 nm - 3 ) kT E F ln2. In turn, E F can be expressed in terms of the total electron density of the metal... Problem XIV.2 (10 points) Assuming that an electron under the influence of a drag force starts from rest at t =0, show that the solution of Equation (14-1) is v ( t ) = ( eE τ / m ) [1 - e - t / τ ].

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