# (a) Two point charges   and are placed r distance apart. Obtain the expression for the amount of work done to place a third charge at the midpoint of the line joining the two charges.(b) At what distance from charge  on the line joining the two charges (in terms of and r) will this work done be zero.

a)

$\\W_{Q_1q_3}=\frac{kQ_1q_3}{r/2}=2\frac{kQ_1q_3}{r}\\\\W_{Q_2q_3}=-\frac{kQ_1q_3}{r/2}=-2\frac{kQ_1q_3}{r}\\\\Total \ work=2\frac{kQ_1q_3}{r}-2\frac{kQ_1q_3}{r}=\frac{q_3(Q_1-Q_2)}{2\pi\epsilon_0r}$

b)

Let x be the required distance

$\\\frac{kQ_1}{x}-\frac{kQ_2}{x-r}=0\\\\x=\frac{Q_1r}{Q_1-Q_2}$

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