Question

For the given figure potential at center is (nKq)/a, find n​

Answer 1

Figure has been attached.

• It is a solid sphere with uniform charge present whole throughout the bulk.'a' is radius of sphere.

To find:

Value of 'n' for which potential at centre is (nkq)/a?

Calculation:

We will use pure calculus to solve this problem.

When r < a (at a point inside the sphere):

$\rm dV = - E \: dr$

Definite Integration on both sides:

$\rm \implies\displaystyle \rm \int_{0}^{V} dV = - \int_{ \infty}^{r} E \: dr$

Now, limits have to be divided as follows:

$\rm \implies\displaystyle \rm V = - \int_{ \infty}^{a} \frac{kq}{ {r}^{2} } \: dr - \int_{a}^{r}\frac{kqr}{ {a}^{3} } \: dr$

$\rm \implies\displaystyle \rm V = \frac{kq}{a} - \frac{kq}{2 {a}^{3} } ( {r}^{2} - {a}^{2} )$

For centre of sphere, value of r = 0:

$\rm \implies\displaystyle \rm V = \frac{kq}{a} - \frac{kq}{2 {a}^{3} } ( 0- {a}^{2} )$

$\rm \implies\displaystyle \rm V = \frac{kq}{a} + \frac{kq}{2a }$

$\rm \implies\displaystyle \rm V = \frac{3kq}{2a}$

$\rm \implies\displaystyle \rm V = \dfrac{3}{2} \times \frac{kq}{a}$

So, value of 'n' = 3/2.