A charge Q is divided on two bodies. The force between the bodies will be the maximum when the values of the charges on the bodies will be
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Q/2, Q/2
Let the charges be q and (Q − q). By Coulomb's law, at a fixed separation r,
\( \displaystyle F = \frac{1}{4\pi\varepsilon_0}\,\frac{q(Q-q)}{r^{2}} \)
F is maximum when the product \( q(Q-q) \) is maximum:
\( \displaystyle \frac{d}{dq}\left[q(Q-q)\right] = Q - 2q = 0 \)
\( \displaystyle q = \frac{Q}{2} \)
So the charge must be shared equally, Q/2 on each body.
Shortcut: Compare the products: \( \frac{1}{3}\cdot\frac{2}{3} = 0.22 \), \( \frac{1}{4}\cdot\frac{3}{4} = 0.19 \), \( \frac{1}{2}\cdot\frac{1}{2} = 0.25 \), \( \frac{1}{5}\cdot\frac{4}{5} = 0.16 \); the product of two parts with a fixed sum is largest when the parts are equal.