Question

Find electric field due to charged spherical shell. At a point (i) outside the shell (ii) on the shell and (iii) inside the shell.

Answer 1

A spherical shell is a hollow sphere of infinitesimally small thickness. Consider a spherical shell of radius R. Let q be charged on the shell. Let us find electric field at a point P at a distance r from the center O of the spherical shell.

Case I. When point P is lying outside the shell (i.e r > R)
From the point P, draw a Gaussian surface which will be a sphere of radius r. Let dS be small area element on the Gaussian surface P as shown in Fig. below.

          .....(1)

i.e. the electric field outside the spherical shell behaves as if whole charge is concentrated at the centre of the spherical shell.

In terms of surface charge density

If σ is the surface density, then

q = σ 4πR²

∴ From eq. (1), we have^.

E = \(\frac{1}{4\piɛ_o} \frac{\sigma4\pi R^2}{r^2}\)

= \(\frac{\sigma}{ɛ_o} \frac{R^2}{r^2}\) ........................(2)

Case II. When point P lies on the surface of spherical shell
On the surface of the shell. Fig.

r = R

So from equation (1), we have,

E = \(\frac{1}{4\piɛ_o} \frac{q}{R^2}\) ........................(3)

And from equation (2)

E = \(\frac{\sigma}{ɛo}\) ........................(4)

Case III When point P lies inside the spherical shell
From point P draw a gaussian surface which will be a sphere of radius r. [Fig.]

From the Gauss's theorem

\(∮_s\vec E. \vec {dS}= \frac{0}{ɛ_o} = 0\)

[∵ No charge exists inside the sphereical shell]

or E = 0 .........................(5)

i.e. electric field inside the charged spherical shell is zero.

Variation of electric field E with distance

Fig. shows the variation of electric field with distance from the center of charged spherical shell.