Derive the formula for the electric potential energy of system of two charges.

Figures $(a)$ and $(b)$ show the field lines of a positive and negative point charge respectively

$(a)$ Give the signs of the potential difference $V_{ P }-V_{ Q } ; V_{ B }-V_{ A }$

$(b)$ Give the sign of the potential energy difference of a small negative charge between the points $Q$ and $P ; A$ and $B$.

$(c)$ Give the sign of the work done by the field in moving a small positive charge from $Q$ to $P$.

$(d)$ Give the sign of the work done by the external agency in moving a small negative charge from $B$ to $A$.

$(e)$ Does the kinetic energy of a small negative charge increase or decrease in going from $B$ to $A?$

The displacement of a charge $Q$ in the electric field $E = {e_1}\hat i + {e_2}\hat j + {e_3}\hat k$ is $\hat r = a\hat i + b\hat j$. The work done is

A test charge $q$ is made to move in the electric field of a point charge $Q$ along two different closed paths as per 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 ?

A circle of radius $R$ is drawn with charge $+ q$ at the centre. A charge $q_0$ is brought from point $B$ to $C$, then work done is