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This questions has statement$-1$ and statement$-2$. Of the four choices given after the statements, choose the one that best describe the two statements.
An insulating solid sphere of radius $R$ has a uniformly
positive charge density $\rho$. As a result of this uniform charge distribution there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at a point out side the sphere. The electric potential at infinite is zero.
Statement$ -1$ : When a charge $q$ is take from the centre of the surface of the sphere its potential energy changes by $\frac{{q\rho }}{{3{\varepsilon _0}}}$
Statement$ -2$ : The electric field at a distance $r(r < R)$ from centre of the sphere is $\frac{{\rho r}}{{3{\varepsilon _0}}}$
Statement$-1$ is true, Statement$-2$ is true; Statement$-2$ is the correct explanation of Statement$-1$.
Statement$-1$ is true, Statement$-2$ is true; Statement$-2$ is not the correct explanation of Statement$-1$.
Statement$-1$ is true, Statement$-2$ is false.
Statement$-1$ is false, Statement$-2$ is true.
Solution
The electric field inside a uniformly charged sphere is
$\frac{\rho r}{3 \epsilon_{0}}$
The electric potential inside a uniformly charged sphere
$=\frac{\rho R^{2}}{6 \epsilon_{0}}\left[3-\frac{r^{2}}{R^{2}}\right]$
$\therefore $ Potential difference between centre and surface
$=\frac{\rho R^{2}}{6 \epsilon_{0}}[3-2]=\frac{\rho R^{2}}{6 \epsilon_{0}}$
$\Delta \mathrm{U}=\frac{q \rho R^{2}}{6 \epsilon_{0}}$
