Flux of an Electric Field — Question 5

Question 5

A point source sends a finite fraction of its flux through a disk

A positive point charge q is at $(0,0,h)$ with $h>0$. A disk of radius R lies in $z=0$ and is oriented downward. Use Coulomb field constant $k=1/(4\pi\epsilon_0)$.

1. Integrate the electric flux through the disk, treating it as rings of radius r.

2. Find R/h for which the disk flux is $\pi kq$, and examine the small-disk and infinite-disk limits.

Question 5 - Solution

A point source sends a finite fraction of its flux through a disk

(a) Method and calculation

$$E_n(r)=\frac{kqh}{(r^2+h^2)^{3/2}},\quad dA=2\pi r\,dr.$$$$\Phi=2\pi kqh\int_0^R\frac{r\,dr}{(r^2+h^2)^{3/2}}=2\pi kq\left(1-\frac h{\sqrt{h^2+R^2}}\right).$$

(b) Interpretation and further calculation

$$\Phi=\pi kq\Rightarrow\frac h{\sqrt{h^2+R^2}}=\tfrac12\Rightarrow R/h=\sqrt3.$$$$R\ll h:\ \Phi\simeq\frac{kq\pi R^2}{h^2};\qquad R\to\infty:\ \Phi\to2\pi kq.$$

Physical check

The downward normal makes the flux positive because the charge field at the disk points away from the source, downward.