A charge q is placed at the mid-point of the line joining two equal charges Q. For what value of q will the system of the three charges be in equlllbrium?

Explanation

Two equal charges Q are placed at A and B
Charge q is placed at O.
Here, `OA=OB=x` (say)
Force on the charge q placed at O,
`F_(1)=(1)/(4piin_(0))[(Qq)/(x^(2))-(Qq)/(x^(2))]=0`
Force on the charge Q placed at A,
`F_(2)=(1)/(4piin_(0))[(Qq)/(x^(2))-(Q^(2))/(4x^(2))]`
If the whole system is in equilibrium,
`F_(2)=F_(1)=0`
`:.(1)/(4piin_(0))[(Qq)/(x^(2))+(Q^(2))/(4x^(2))]=0`
or, `(Qq)/(x^(2))+(Q^(2))/(4x^(2))=0or,(Qq)/(x^(2))=-(Q^(2))/(4x^(2))or,=-(Q)/(4)`