Question

Electric potential at the centre of a uniformly charged conducting shell

Answer 1

Electrical potential at the centre of a uniformly charged conducting shell is $\frac{KQ}{R}$ .

Explanation :

Consider a uniformly charged conducting spherical shell of radius R .

Let the charge on the shell = Q

Consider a Gaussian surface of radius r , so from Gauss Law we have :

$\int \vec E. \vec {ds} = \frac{Q}{\epsilon_0}$

Or, $E 4 \pi r^2= \frac{Q}{\epsilon_0}$

Or , $E = \frac{Q}{4\pi \epsilon_0r^2} =\frac{KQ}{r^2}$      ( k is electrostatic constant )

Since electric potential is defined as the work done to move a charge from infinity to the required point . Since E inside the shell is zero , so work done for moving it from the surface to centre is zero .

Hence , the electric potential at the centre will be given as :

$W =- \int\limits^R_{\infty} \, dw$

    =$-\int\limits^R_{\infty} E. dr$

    =$-\int\limits^R_{\infty} \frac{KQ}{r^2} \, dr$

   =$-KQ\int\limits^R_{\infty} \frac{dr}{r^2}$

   =$\frac{KQ}{R}$

Hence , electric potential at the centre of a uniformly charged conducting shell is $\frac{KQ}{R}$ .