 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-2/10d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Substitute

x = p c / (k_{B} T)

ϵ = \frac{4 π k _{B}^{4}}{h ^{3} c ^{3}} T^{4} \int_{0}^{\infty} \frac{x ^{3}}{e ^{x} + 1} d x .

The integral converges, is positive, and is independent of temperature. Thus $ϵ = α T^{4}$ , with

α

equal to the displayed constant integral. The given normalization is for the specified degeneracy; further internal degrees of freedom multiply it.

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