12.7 Potential Due to Uniformly Charged Solid Sphere

  Electrostatics

    1.0 Introduction
    2.0 Electric charge
    3.0 Coulomb's law
    4.0 Principle of superposition
    5.0 Continuous charge distribution
    6.0 Electric field
    7.0 Electric field lines
    8.0 Insulators and conductors
    9.0 Gauss's law
    10.0 Work done
    11.0 Electric potential energy
    12.0 Electric Potential
    13.0 Electric dipole

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12.7 Potential Due to Uniformly Charged Solid Sphere

Derivation of expression for potential due to uniformly charged solid sphere of radius $R$ and total charge $Q$ $($ distributed in volume $),$ at a point which is at a distance $r$ from center for the following situations.

$(i)$ $r \geqslant \;R$ $(ii)$ $r \leqslant \;R$

Consider and elementary shell of radius $X$ and $dx$

$(i)$ For $r \geqslant \;R:$ $$V = \int\limits_0^R {\frac{{K.4\pi {X^2}dx\rho }}{r}} = \frac{{KQ}}{r}$$ $(ii)$ for $r \leqslant \;R$ $$V = \int\limits_0^r {\frac{{K.4\pi {X^2}dx\rho }}{r}} + \int\limits_0^R {\frac{{K4\pi {X^2}dx\rho }}{x}} $$ $$ = \frac{{KQ}}{{2{R^3}}}\left( {3{R^2} - {r^2}} \right)$$ $$ \Rightarrow \quad \left( {\rho = \frac{Q}{{\frac{4}{3}\pi {R^3}}}} \right)$$

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From definition of potential:

$(i)$ For $r \geqslant \;R$ $$V = \int_\infty ^r {\frac{{KQ}}{{{r^2}}}} \widehat r.\;dr = \frac{{KQ}}{r}$$

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$(ii)$ For $r \leqslant \;R$ $$V = - \int\limits_\infty ^R {\frac{{KQ}}{{{r^2}}}} .dr - \int\limits_R^r {\frac{{KQr}}{{{R^3}}}} \;dr$$ $$V = \frac{{KQ}}{R} - \frac{{KQ}}{{2{R^3}}}[{r^2} - {R^2}] = \frac{{KQ}}{{2{R^3}}}[2{R^2} - {r^2} + {R^2}] = \frac{{KQ}}{{2{R^3}}}(3{R^2} - {r^2})$$