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hw7solv2

# hw7solv2 - Homework#7 Solutions Question 1 K K Chapter 4...

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Homework #7 Solutions Question 1) K+K, Chapter 4, Problem 8. See Figure (1) for the setup. Figure 1: The situation before (top) and after (bottom) a temperature-floating shield is placed between a hot surface (at temperature T u ) and a cold surface (at temperature T l ). We expect J 1 = J 2 in equilibrium, where J 1 = σ B ( T 4 u - T 4 m ) and J 2 = σ B ( T 4 m - T 4 l ). Solve for T m : T m = T 4 u + T 4 l 2 1 / 4 (1) Now the heat current is given by J = σ B ( T 4 u - T 4 m ) = σ B T 4 u - T 4 l 2 = J u 2 (2) This is half the original heat current. Note that we would get the same answer if we calculated the flux between the inserted plate and the low temperature plate (up to a possible sign difference). This has to be so in equilibrum for a black body due to energy conservation. The general case of N floating heat shields reduces the heat current by a factor of (N+1). This is the principle behind super-insulation, in which many layers of Al-coated mylar are wrapped around a cold object to insulate it from radiative heating from nearby warm objects. 1

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Question 2) K+K, Chapter 4, Problem 11 (First part only). Starting with Eq. (41) in K+K we can write the energy of phonons in a solid as U = 9 4 ( k B θ D ) 3 Z k B θ D 0 d x x 3 e x - 1 (3) We are working in the limit where the upper limit of the integral is small: k B θ D 1. Hence x remains small in the integrand, and we can expand the denominator as e x - 1 = 1 + x + · · · - 1 x . So, U 9 4 ( k B θ D ) 3 Z k B θ D 0 d xx 2 (4) The integration is now simple and gives k 3 B θ 3 D / 3 τ 3 . Simplifying the expression we get U = 3 (5) Differentiating this with respect to temperature we get the Dulong-Petit law for the heat capacity of a solid at high temperature ( θ D T ): C V = ∂U ∂T U = 3 Nk B for θ D T (6) Question 3) K+K, Chapter 4, Problem 12. The phonon specific heat at 1 K is C phonon V = 12 π 4 Nk B 5 T θ D 3 since T θ D . (7) For photons, we know from Eqs. (25-26) in K+K
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hw7solv2 - Homework#7 Solutions Question 1 K K Chapter 4...

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