Chapter 24, Problem 60EAP

A sphere of radius R has total charge Q. The volume charge

density (C/m³) within the sphere is

$\rho ={\rho }_0(1-\frac{r}{R})$

This charge density decreases linearly from ${\rho }_0$ at the center to

zero at the edge of the sphere.

a. Show that ${\rho }_0=3Q/\pi R^3$ .

b. Show that the electric field inside the sphere points radially

outward with magnitude

$E=\frac{Qr}{4\pi {\in }_0{R}^3}\left(4-3\frac{r}{2}\right)$

c. Show that your result of part b has the expected value at r = R.