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Chapter 7: Q. 7.66 (page 323)
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The heat capacity of liquid below is proportional to , with the measured value. This behavior suggests that the dominant excitations at low temperature are long-wavelength photons. The only important difference between photons in a liquid and photons in a solid is that a liquid cannot transmit transversely polarized waves-sound waves must be longitudinal. The speed of sound in liquid is , and the density is . From these numbers, calculate the photon contribution to the heat capacity ofin the low-temperature limit, and compare to the measured value.
Starting from equation 7.83, derive a formula for the density of states of a photon gas (or any other gas of ultra relativistic particles having two polarisation states). Sketch this function.
The speed of sound in copper is . Use this value to calculate its theoretical Debye temperature. Then determine the experimental Debye temperature from Figure 7.28, and compare.
Use the formula to show that the pressure of a photon gas is 1/3 times the energy density (U/V). Compute the pressure exerted by the radiation inside a kiln at 1500 K, and compare to the ordinary gas pressure exerted by the air. Then compute the pressure of the radiation at the centre of the sun, where the temperature is 15 million K. Compare to the gas pressure of the ionised hydrogen, whose density is approximately 105 kg/m3.
Problem 7.69. If you have a computer system that can do numerical integrals, it's not particularly difficult to evaluate .
(a) As usual when solving a problem on a computer, it's best to start by putting everything in terms of dimensionless variables. So define ,. Express the integral that defines , equation 7.22, in terms of these variables. You should obtain the equation
(b) According to Figure
the correct value of when is approximately . Plug in these values and check that the equation above is approximately satisfied.
(c) Now vary , holding fixed, to find the precise value of for . Repeat for values of ranging from up to , in increments of . Plot a graph of as a function of temperature.
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