Question

Find the electric field intensity due to a uniformly charged spherical shell at a point (i) outside the shell and (ii) inside the shell. Plot the graph of electric field with distance from the centre of the shell.

Answer 1

Consider a thin spherical shell of radius R with a positive charge q distributed uniformly on the surface. As the charge is uniformly distributed, the electric field is symmetrical and directed radially outward.
(i) Electric field outside the shell:
For point $r>R$; draw a spherical gaussian surface of radius r.
Using gauss law, $\oint E.ds=\frac{q_{end}}{q_0}$
Since $\overrightarrow{E}$ is perpendicular to gaussian surface, angle betwee $\overrightarrow{E}$ is 0.
Also $\overrightarrow{E}$ being constant, can be taken out of integral.
So, $E\left(4\pi r^2\right)=\frac{q}{q_0}$

So, $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}$
Thus electric field outside a uniformly charged spherical shell is same as if all the charge q were concentrated as a point charge at the center of the shell.
(ii) Inside the shell:
In this case, we select a gaussian surface concentric with the shell of radius r $(r>R)$.
So, $\oint E.ds=E\left(4\pi r^2\right)$
According to gauss law,
$E\left(4\pi r^2\right)=\frac{Q_{end}}{{\epsilon }_0}$
Since charge enclosed inside the spherical shell is zero.
So, $E=0$
Hence, the electric field due to a uniformly charged spherical shell is zero at all points inside the shell.