Chapter 17.8, Problem 39E

Gauss’ Law for electric fields The electric field due to a point charge Q is $E=\frac{Q}{4\pi {\epsilon }_0}\frac{r}{{\left|r\right|}^3}$, where r = 〈x, y, z〉, and ε₀ is a constant.

a.    Show that the flux of the field across a sphere of radius a centered at the origin is ${\displaystyle {\iint }_{S}E\cdot ndS}=\frac{Q}{{\epsilon }_0}$.

b.    Let S be the boundary of the region between two spheres centered at the origin of radius a and b with a < b. Use the Divergence Theorem to show that the net outward flux across S is zero.

c.    Suppose there is a distribution of charge within a region D Let q(x, y, z) be the charge density (charge per unit volume). Interpret the statement that

${\displaystyle \underset{S}{\iint }E\cdot ndS}=\frac{1}{{\epsilon }_0}{\displaystyle \underset{D}{\iiint }q\left(x,y,z\right)dV.}$

d.    Assuming E satisfies the conditions of the Divergence Theorem on D. conclude from part (c) that $\nabla \cdot E\text{=}\frac{q}{{\epsilon }_0}$.

e.    Because the electric force is conservative, it has a potential function ϕ. From part (d). conclude that ${\nabla }^2\phi =\nabla \cdot \nabla \phi =\frac{q}{{\epsilon }_0}$.