Question

1. Obtain the expression for the electric field intensity due to a uniformly charged spherical shell of radius R at a point distant r from the centre of the shell outside it.
2. Draw a graph showing the variation of electric field intensity E with r, for r > R and r < R.

Answer 1

Electric field due to a uniformly charged thin spherical shell:

i. When point P lies outside the spherical shell: Suppose that we have calculate the field at point P at a distance r(r > R) from its centre. Draw the Gaussian surface through point P so as to enclose the charged spherical shell. The gaussian surface is a spherical surface of radius r and centre O.

Let `vecE` be the electric field at point P, then the electric flux through the area element of area `vec(ds)` is given by dφ = `vecE.vec(ds)`

Since `vec(ds)` is also along normal to the surface dφ = E dS

∴ Total electric flux through the Gaussian surface is given by

φ = `oint  Eds = E point  ds`

Now, `oint` ds = 4πr²  ...(i)

= E × 4πr²  

Since the charge enclosed by the Gaussian surface is q, according to Gauss’s theorem,

φ = `q/∈_0`  ...(ii)

From equations (i) and (ii) we obtain

E × 4πr² = `q/∈_0`

E = `1/(4π∈_0) . q/r^2`  ...(For r > R)

ii. A graph showing the variation of electric field as a function of r is shown below.