The field magnitude is proportional to the distance $r$ of the field point

from the center of the sphere $($see the graph of $E$ versus $r$ in Fig. $22\mathrm{.}22).$

To find $E$ outside the sphere, we take $r>R.$ This surface encloses

the entire charged sphere, so $Q_{\text{e}\text{n}\text{c}\text{l}}=Q,$ and Gauss's law gives

$4r^{2}E=\frac{Q}{lo{n}_0}$ and

$E=\frac{1}{4lo{n}_0}\frac{Q}{r^2}(field$ outside a uniformly charged sphere$)$

The field outside any spherically symmetric charged body varies as

$\frac{1}{r^2},$ as though the entire charge were concentrated at the center. This

is graphed in Fig. $22\mathrm{.}22.$

If the charge is negative, vec$(E)$ is radially inward and in the expressions

for $E$ we interpret $Q$ as the absolute value of the charge.

EVALUATE Notice that if we set $r=R$ in either expression for $E,$ we

get the same result $E=\frac{Q}{4}lo{n}_0{R}^2$ for the magnitude of the field at

the surface of the sphere. This is because the magnitude $E$ is a con$-$

tinuous function of $r.$ By contrast, for the charged conducting sphere of

Example $22\mathrm{.}5$ the electric$-$field magnitude is discontinuous at $r=R($it

jumps from $E=0$ just inside the sphere to $E=\frac{Q}{4}lo{n}_0{R}^2$ just outside

the sphere$).$ In general, the electric field vec$(E)$ is discontinuous in magni$-$

tude, direction, or both wherever there is a sheet of charge, such as at

the surface of a charged conducting sphere $($Example $22\mathrm{.}5),$ at the sur$-$

face of an infinite charged sheet $($Example $22\mathrm{.}7),$ or at the surface of a

charged conducting plate $($Example $22\mathrm{.}8).$

The approach used here can be applied to any spherically symmetric

distribution of charge, even if it is not radially uniform, as it was here.

Figure $22\mathrm{.}22$ The magnitude of the electric field of a uniformly

charged insulating sphere. Compare this with the field for a conduct$-$

ing sphere $($see Fig. $22\mathrm{.}18).$

Such charge distributions occur within many atoms and atomic nuclei,

so Gauss's law is useful in atomic and nuclear physics.

KEYCONCEPT For any spherically symmetric distribution of

charge, the field at a distance $r$ from the center of the distribution is

the same as if all the charge interior to $r$ were concentrated in a point at

the center. The charge at distances greater than $r$ from the center has no

effect on the field at $r.$