Consider two conducting spheres of radii $R_1$ and $R_2$ with $R_1>R_2$. If the two are at the same potential, the larger sphere has more charge than the smaller sphere. State whether the charge density of the smaller sphere is more or less than that of the larger one.
Since, the two spheres are at the same potential, therefore
$$\frac{k q_1}{R_1}=\frac{k q_2}{R_2} \Rightarrow \frac{k q_1 R_1}{4 \pi R_1^2}=\frac{k q_2 R_2}{4 \pi R_2^2}$$
$$\begin{gathered} \text{or}\quad\sigma_1 R_1=\sigma_2 R_2 \Rightarrow \frac{\sigma_1}{\sigma_2}=\frac{R_2}{R_1} \\ R_2>R_1 \end{gathered}$$
This imply that $\sigma_1>\sigma_2$.
The charge density of the smaller sphere is more than that of the larger one.
Do free electrons travel to region of higher potential or lower potential?
The free electrons experiences electrostatic force in a direction opposite to the direction of electric field being is of negative charge. The electric field always directed from higher potential to lower travel. Therefore, electrostatic force and hence direction of travel of electrons is from lower potential to region of higher potential .
Can there be a potential difference between two adjacent conductors carrying the same charge?
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.