Question

Using Gauss's law obtain the expression for the electric field due to a uniformly charged thin spherical shell of radius $R$ at a point outside the shell. Draw a graph showing the variation of electric field with $r$, for $r>R$ and $r<R$.

Answer 1

(i) At a point P outside the spherical shell :

Let $\overrightarrow{E}$ be the electric field at P. The electric field through an elemental surface $d\overrightarrow{s}$ is

$d\varphi =\overrightarrow{E}.d\overrightarrow{s}$

$\therefore$ Total electric flux through the gaussian surface is

$\varphi ={\oint }_{S}Eds=E{\oint }_{S}ds=E\times 4\pi r^2$

$\varphi =\frac{q}{{ϵ}_0}$

$E\times 4\pi r^2=\frac{q}{{ϵ}_0}$

$E=\frac{1}{4\pi {ϵ}_0}.\frac{q}{r^2}$ (for $r>R$)

(ii) At a point P inside the spherical shell:

In this case charge enclosed =0

$E\times 4\pi r^2=0$

$E=0$ (for $r<R$)