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  4. A point charge +Q is held at rest at a point P. Another point charge -q, whose mass is m, moves at a constant velocity v in a circular orbit of radius R1 around P. The work required to increase the radius of revolution of -q from R1 to another orbit R2 is (R2 > R1)

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Q. A point charge $+Q$ is held at rest at a point $P$. Another point charge $-q,$ whose mass is m, moves at a constant velocity $v$ in a circular orbit of radius $R_1$ around $P$. The work required to increase the radius of revolution of $-q$ from $R_1$ to another orbit $ R_2$ is $(R_2 > R_1)$

KEAMKEAM 2018Electrostatic Potential and Capacitance

Solution:

The force due to point charge $+Q$ on charge $-q$ is
$F=\frac{k Q(-q)}{R^{2}}$
Therefore work required to increase the radius of revolution of $-q$ from $R_{1}$ to $R_{2}$ is
$W =-\int\limits_{R_{1}}^{R_{2}} F . d R$
$=-\int\limits_{R_{1}}^{R_{2}}-k \frac{Q q}{R^{2}} d R=k Q q\left[\frac{1}{R}\right]_{R_{1}}^{R_{2}}$
$=-k Q q\left[\frac{1}{R_{2}}-\frac{1}{R_{1}}\right]$