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 / (4πR³ / 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__ __q₁__
the shell of charge q₁}___4πε₀      r

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

. :    V = __1__ __q₁__ + __1__ __q2__ + ….
        ___ 4πε₀    _ r             4πε₀      r
   
    = __1__ (q1 + q2 + q3 + ……)
___ 4πε₀r

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

. :    V = __1__ __q__
_______4πε0     r

E = - dV / dr = - d (__1__ __q__) = __1__ __q__
___________dr    4πε₀     r    _____4πε₀     r²

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 r³ p/ 3.

Potential at P due to}    = __1__ 4/3πr³ p = r²p
the inner solid sphere}___4πε_{0___}r_____3ε₀

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

    V₂ = ∫^rR px dx = p (R² – r²).
_________ε₀    ____2ε₀

The total potential at P is
    V = V₁ + V₂ = r²P + p (R² – r²) = p(3R² – r²)
____________3ε₀     ___2ε_{0________}6ε₀

But    p = q / (4πR³ / 3)
   
V = __3q__ (3R² – r²)
         4πR³          6ε₀

. :     V = __1__ q (3R² – r²)
               4πε₀        2R³

    E = - dV / dr = - d [__1__ q (3R² – r²)] = __1__ qr
_____________dr   4πε₀     2R³       ___ 4πε₀   R³



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