 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/section-b/b2/c/solution

Write

b = π k_{B}^{2} / (3ℏ c)

, so

U = b L T^{2}

. At fixed length,

d S = C_{L} d T / T

S = 2 b L T

with the entropy constant fixed by the unique photon vacuum. The thermal Helmholtz free energy and generalized pressure follow:

S = \frac{2 π L k _{B}^{2} T}{3ℏ c}, A = U - TS = - \frac{π L k _{B}^{2} T ^{2}}{3ℏ c}, P = - (\frac{\partial A}{\partial L})_{T} = \frac{π k _{B}^{2} T ^{2}}{3ℏ c} = \frac{U}{L} .

The entropy constant can also be checked from the exact discrete photon partition function; the continuum approximation should not be extrapolated blindly below its level-spacing regime. In one dimension

P = U / L

, whereas the isotropic three-dimensional relation is

p = U / (3 V)

. Both follow from the number of spatial directions carrying momentum flux.

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