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, i.e. `q_(1)+q_(2)=Q " " ....(i)`

From Coulomb.s law,
`F="k"(q_(1)q_(2))/(x^(2)) " " [ :. k=(1)/(4pi epsilon_(0))]`
`q_(2)=Q-q_(1)" "`[using Eq. (i)]
`:.F="k"(q_(1)(Q-q_(1)))/(x^(2)) = (kq_(1)Q-kq_(1)^(2))/(x^(2))`
`:.(dF)/(dq_(1)) =(1)/(x^(2)) ["k"(d)/(dq_(1)) (q_(1)Q) -"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`
`:. (k)/(x^(2)) (Q-2q_(1))= 0 rArr Q= 2q_(1)`
`:.q_(1)=(Q)/(2) `
`rArr q_(2)p=Q-q_(1)=(Q)/(2)`