## Potential Due To An Uniformly

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# Potential due to an uniformly charged non-conducting solid sphere

In a conducting sphere, the entire charge is distributed uniformly in its entire volume. Let R be the radius of the non – conducting sphere. Let q be the total charge on the sphere.Volume charge density = p = q / (4πR

^{3}/ 3)

## (i) Potential at an External Point

Let P be a point distant r from the centre O of the sphere. Divided the sphere into a large number of concentric spherical shells carrying charges q1, q2, q3 ….Potential at P due to } = __

__1____ __

__q____

_{1}the shell of charge q

_{1}}___4πε

_{0}r

The potential V due to the whole sphere is equal to the sum of the potentials due to all the shells.

. : V = _

___1____ __

__q____ + __

_{1}__1____ __

__q2____ + ….

___ 4πε

_{0}_ r 4πε

_{0}r

= _

___1____ (q1 + q2 + q3 + ……)

___ 4πε

_{0}r

But (q1 + q2 + q3 + ……) = q, the charge on the sphere.

. : V = __

__1____ __

__q____

_______4πε0 r

E = - dV / dr =

__- d__(_

___1____ __

__q____) = __

__1____ __

__q____

___________dr 4πε

_{0}r _____4πε

_{0}r

^{2}

Thus the charged sphere behaves toward an external point as if its entire charge were concentrated at its centre.

## (ii) Potential at an Internal Point

Let the point P be inside the sphere at a distance r from the centre O. If we draw a concentric sphere through P, the point P is external for the inner solid sphere is radius r, and internal radius r and external radius R.The charge on the inner solid sphere is 4π r

^{3}p/ 3.

Potential at P due to} = __

__1____

__4/3πr__=

^{3}p__r__

the inner solid sphere}___4πε

^{2}p_{0___}r_____3ε

_{0}

The potential at P due to the whole shell of internal radius r and external radius R is

V

_{2}= ∫

^{r}R

__px dx__=

__p (R__

^{2}– r^{2})._________ε

_{0}____2ε

_{0}

The total potential at P is

V = V

_{1}+ V

_{2}=

__r__+

^{2}P__p (R__=

^{2}– r^{2})__p(3R__

^{2}– r^{2})____________3ε

_{0}___2ε

_{0________}6ε

_{0}

But p = q / (4πR

^{3}/ 3)

V = __

__3q____

__(3R__

^{2}– r^{2})4πR

^{3}6ε

_{0}

. : V = __

__1____

__q (3R__

^{2}– r^{2})4πε

_{0}2R

^{3}

E = - dV / dr = -

__d__[__

__1____

__q (3R__] = __

^{2}– r^{2})__1____

__qr__

_____________dr 4πε

_{0}2R

^{3}___ 4πε

_{0}R

^{3}

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