Question

A charge Q is divided into two parts q and (Q-q). What is q in terms of Q when the repulsion between them is maximum?

• A. Q/2
• B. Q
• C. 2Q
• D. 2/Q

Answer 1

The correct option is A Q/2

From Coulomb's Law,

$F=\frac{Kq(Q-q)}{R^2}$

$F=\frac{K}{R^2}(Qq-q^2)$

Adding and subtracting $(Q/2{)}^2$ and rearranging,

$F=\frac{K}{R^2}(-\{q^2-Qq+\left(Q/2{)}^2\right\}+\left(Q/2{)}^2\right)$

$F=\frac{K}{R^2}(\frac{Q^2}{4}-{(q-\frac{Q}{2})}^2)$

$F\le \frac{K{Q}^2}{4R^2}$ and maximum value occurs when second term is zero.

Hence, $q=Q/2$ for maximum force.