Two point charges having a total charge Q are at a distance x apart. Find the value of each charge, so that the force they exert on one another is maximum.

Explanation

Consider two point charges `q_1 and q_2` are at a distance x apart.
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From Coulomb.s law,
` " "F = k (q_1q_2)/(x^(2)) " " ( " where , " k = (1)/(4piin_0)) `
` q_2 = Q -q_1 " " ` [ using Eq. (i) ]
` therefore " " F =k (q_1 (Q-q_1))/(x^(2)) = (kq_1Q-kq_1^(2))/(x^(2))`
` therefore " " (dF)/(dq_1) = (1)/(x^(2)) [k= (d)/(dq_1) (q_1Q) - k (d)/(dq_1) (q_1^(2))]`
` rArr " "(dF)/(dq_1) = (k)/(x^(2)) (Q- 2q_1) `
For maxima or minima ,` (dF)/(dq_1) =0`
` therefore " "(k)/(x^(2)) (Q- 2q_1) =0 rArr Q = 2q_1`
` therefore " " q_1 = (Q)/(2) rArr q_2 = Q - q_1 = Q - (Q)/(2) = (Q)/(2) `