 Solution

ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2016/paper-2/section-c/c12/a/solution

The collision flux contains an extra normal velocity

v cos θ

: a faster atom sweeps out a longer approach cylinder per unit time. Integrating the Maxwell-Boltzmann distribution over the incoming hemisphere gives

J = n A \int_{0}^{\infty} v^{3} e^{- α v^{2}} d v \int_{0}^{π /2} sin θ cos θ d θ \int_{0}^{2 π} d ϕ = n A π I_{3} .

Therefore

J = n \frac{k _{B} T}{2 πm} = \frac{1}{4} n \overline{v} .

Here

J

is the number striking unit area per unit time and

\overline{v} = 8 k_{B} T / (πm)

is the bulk mean speed. A simple factor of one half would count the incoming hemisphere but miss the projection factor and velocity weighting.

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