Gauss' Law and Coulomb's Law — Question 8

Question 8

Approaching a cube corner exposes a subtle boundary-charge limit

A cube spans $0\le x,y,z\le L$. A positive charge q is at $(\eta,\eta,\eta)$ with $0<\eta<L$. Consider the limit $\eta\to0^+$, keeping the charge strictly inside until the limit is taken. Call the faces at coordinate L the three far faces.

1. Find the total cube flux for every positive $\eta$. Use a larger cube $[-L,L]^3$ with a centered charge to infer the limiting flux through each far face.

2. Find the limiting total flux through the three near faces. Explain why assigning a charge exactly on a mathematical boundary requires a separate convention or limiting prescription.

Question 8 - Solution

Approaching a cube corner exposes a subtle boundary-charge limit

(a) Method and calculation

$$\Phi_{\mathrm{cube}}(\eta)=q/\epsilon_0\quad(0<\eta<L).$$$$\Phi_{\mathrm{large\ face}}=q/(6\epsilon_0),\quad\Phi_{\mathrm{far\ face}}\longrightarrow\tfrac14\Phi_{\mathrm{large\ face}}=q/(24\epsilon_0).$$

(b) Interpretation and further calculation

$$\Phi_{\mathrm{three\ far}}\to q/(8\epsilon_0),\quad\Phi_{\mathrm{three\ near}}\to7q/(8\epsilon_0).$$$$\Phi_{\mathrm{total}}\to q/\epsilon_0.$$The near-face field becomes singular and its finite integrated contribution cannot be discarded. A source exactly on the boundary is outside the ordinary nonsingular-surface statement used here.

Physical check

Each far face is one quarter of a large-cube face by reflection symmetry; the singular limit only affects the near-face reasoning.