Question

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 $$r\left( {r < R} \right)$$ from the centre of the sphere is a $$\frac{{\rho r}}{{3{\varepsilon _0}}}.$$

A. Statement | is true, Statement 2 is true; Statement 2 is not the correct explanation of statement 1.

B. Statement 1 is true Statement 2 is false.

C. Statement 1 is false Statement 2 is true.

D. Statement 1 is true, Statement 2 is true, Statement 2 is the correct explanation of Statement 1

Answer : Statement 1 is false Statement 2 is true.

Solution :
The electric field inside a uniformly charged sphere is $$\frac{{\rho .r}}{{3{ \in _0}}}$$
The electric potential inside a uniformly charged sphere $$ = \frac{{\rho {R^2}}}{{6{ \in _0}}}\left[ {3 - \frac{{{r^2}}}{{{R^2}}}} \right]$$
$$\therefore $$ Potential difference between centre and surface
$$\eqalign{ & = \frac{{\rho {R^2}}}{{6{ \in _0}}}\left[ {3 - 2} \right] = \frac{{\rho {R^2}}}{{6{ \in _0}}} \cr & \Delta U = \frac{{q\rho {R^2}}}{{6{ \in _0}}} \cr} $$

Two point charges $$ + q$$ and $$ - q$$ are held fixed at $$\left( { - d,o} \right)$$ and $$\left( {d,o} \right)$$ respectively of a $$x-y$$ coordinate system. Then

A. The electric field $$E$$ at all points on the $$x$$-axis has the same direction

B. Electric field at all points on $$y$$-axis is along $$x$$-axis

C. Work has to be done in bringing a test charge from $$\infty $$ to the origin

D. The dipole moment is $$2qd$$ along the $$x$$-axis

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