Can the potential function have a maximum or minimum in free space?
No, The absence of atmosphere around conductor prevents the phenomenon of electric discharge or potential leakage and hence, potential function do not have a maximum or minimum in free space.
A test charge $q$ is made to move in the electric field of a point charge $Q$ along two different closed paths [figure first path has sections along and perpendicular to lines of electric field. Second path is a rectangular loop of the same area as the first loop. How does the work done compare in the two cases?
As electric field is conservative, work done will be zero in both the cases.
Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.
Let's assume contradicting statement that the potential is not same inside the closed equipotential surface. Let the potential just inside the surface is different to that of the surface causing in a potential gradient $\left(\frac{d V}{d r}\right)$. Consequently electric field comes into existence, which is given by as $E=-\frac{d V}{d r}$.
Consequently field lines pointing inwards or outwards from the surface. These lines cannot be again on the surface, as the surface is equipotential. It is possible only when the other end of the field lines are originated from the charges inside.
This contradict the original assumption. Hence, the entire volume inside must be equipotential.
A capacitor has some dielectric between its plates and the capacitor is connected to a DC source. The battery is now disconnected and then the dielectric is removed . State whether the capacitance, the energy stored in it, electric field, charge stored and the voltage will increase, decrease or remain constant.
The capacitance of the parallel plate capacitor, filled with dielectric medium of dielectric constant $K$ is given by
$$C=\frac{K \varepsilon_0 A}{d} \text {, where signs are as usual. }$$
The capacitance of the parallel plate capacitor decreases with the removal of dielectric medium as for air or vacuum $K=1$.
After disconnection from battery charge stored will remain the same due to conservation of charge.
The energy stored in an isolated charge capacitor $=\frac{q^2}{2 C}$; as $q$ is constant, energy stored $\propto$ $1 / C$ and $C$ decreases with the removal of dielectric medium, therefore energy stored increases. Since $q$ is constant and $V=q / C$ and $C$ decreases which in turn increases $V$ and therefore $E$ increases as $E=V / d$.
Prove that, if an insulated, uncharged conductor is placed near a charged conductor and no other conductors are present, the uncharged body must intermediate in potential between that of the charged body and that of infinity.
Let us take any path from the charged conductor to the uncharged conductor along the direction of electric field. Therefore, the electric potential decrease along this path.
Now, another path from the uncharged conductor to infinity will again continually lower the potential further. This ensures that the uncharged body must be intermediate in potential between that of the charged body and that of infinity.