Radon, a radioactive gas, seeps from the rock below into a basement laboratory…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch The integral theorems: Gauss, Stokes and Green

Problem

Radon, a radioactive gas, seeps from the rock below into a basement laboratory through its floor at a steady rate and spreads through the still air. Let n(r,t)n(\mathbf{r},t) be the number of radon nuclei per unit volume of air, and J(r,t)\mathbf{J}(\mathbf{r},t) the number of nuclei crossing unit area per second (a diffusive flux, caused by the gas spreading out). Each nucleus decays independently at a constant rate λ\lambda (units s−1\mathrm{s^{-1}}), so the number of decays per second inside a small volume equals λ\lambda times the number of nuclei in it. For any fixed volume VV of the air, with closed boundary surface SS, nuclei are gained or lost only by decay or by crossing SS, so ddt∫Vn dV=−λ∫Vn dV−∮SJ⋅dS.\frac{\mathrm{d}}{\mathrm{d}t}\int_V n\,\mathrm{d}V = -\lambda\int_V n\,\mathrm{d}V - \oint_S\mathbf{J}\cdot\mathrm{d}\mathbf{S}. (a) State in words what each of the three terms represents. (b) Derive the local differential equation obeyed by nn and J\mathbf{J} at every point of the air. (c) After a long time the concentration stops changing (∂n/∂t=0\partial n/\partial t = 0 everywhere), with J=−D∇n\mathbf{J} = -D\nabla n for a constant D>0D > 0 (Fick's law). Write the differential equation satisfied by nn in the air in that state. (d) Show that if the floor were sealed, so that no radon entered and none crossed the walls, floor or ceiling, no steady state with any radon present could exist.

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