JEE MAIN - Physics (2012 - No. 15)
This question 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 center 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.
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 center 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 $${{q\rho } \over {3{\varepsilon _0}}}$$
Statement- $$2:$$ The electric field at a distance $$r\left( {r < R} \right)$$ from the center of the sphere is $${{\rho r} \over {3{\varepsilon _0}}}.$$
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.
Statement- $$1$$ is true, Statement- $$2$$ is true; Statement- $$2$$ is the correct explanation of Statement- $$1$$.
Explanation
The electric field inside a uniformly charged sphere is
= $${{\rho .r} \over {3{ \in _0}}}$$
The electric potential inside a uniformly charged sphere
$$ = {{\rho {R^2}} \over {6{ \in _0}}}\left[ {3 - {{{r^2}} \over {{R^2}}}} \right]$$
$$\therefore$$ Potential difference between center and surface
$$ = {{\rho {R^2}} \over {6{ \in _0}}}\left[ {3 - 2} \right] = {{\rho {R^2}} \over {6{ \in _0}}}$$
$$\Delta U = {{q\rho {R^2}} \over {6{ \in _0}}}$$
= $${{\rho .r} \over {3{ \in _0}}}$$
The electric potential inside a uniformly charged sphere
$$ = {{\rho {R^2}} \over {6{ \in _0}}}\left[ {3 - {{{r^2}} \over {{R^2}}}} \right]$$
$$\therefore$$ Potential difference between center and surface
$$ = {{\rho {R^2}} \over {6{ \in _0}}}\left[ {3 - 2} \right] = {{\rho {R^2}} \over {6{ \in _0}}}$$
$$\Delta U = {{q\rho {R^2}} \over {6{ \in _0}}}$$
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