Example 4.1: Electric field of a uniformly charged sphere

Question: An insulating sphere of radius $a$ carries a total charge $Q$ which is uniformly distributed over the volume of the sphere. Use Gauss' law to find the electric field distribution both inside and outside the sphere.
 
Solution: By symmetry, we expect the electric field generated by a spherically symmetric charge distribution to point radially towards, or away from, the center of the distribution, and to depend only on the radial distance $r$ from this point. Consider a gaussian surface which is a sphere of radius $r$, centred on the centre of the charge distribution. Gauss' law gives

$$A(r)\,E_r(r) = \frac{q(r)}{\epsilon_0},$$

where $A(r) = 4\pi\, r^2$ is the area of the surface, $E_r(r)$ the radial electric field-strength at radius $r$, and $q(r)$ the total charge enclosed by the surface. It is easily seen that

$$q(r) = \left\{\begin{array}{lcl} Q&\mbox{\hspace{1cm}}&r\geq a\\ [0.5ex] Q\,(r/a)^3&&r<a \end{array}\right..$$

Thus,

$$E_r(r) = \left\{\begin{array}{lcl} \frac{Q}{4\pi\epsilon_0\,... ...5ex] \frac{Q\,r}{4\pi\epsilon_0\,a^3}&&r<a \end{array}\right..$$

Clearly, the electric field-strength is proportional to $r$ inside the sphere, but falls off like $1/r^2$ outside the sphere.