Two identical point charges Q are kept at a distance r from each other. A third point charge is placed on the line joining the above two charges such that all the three charges are in equilibrium. The third charge
(a) should be of magnitude q=………..
(b) should be of sign .......
(c ) should be placed ........ .

Explanation

Let a charge q be placed at distance x from charge Q, as shown in the figure .

For charge q to be in equilibrium, the forces acting on it should be equal and opposite . `F_(CA)=F_(CB)`
`rArr 1/(4piepsilon_0)(Qq)/x^2 =(Qq)/(4piepsilon_0(d-x)^2)`
`rArr (d-x)/x=1`
`rArr d/x-1 =+1`
`rArr x=d/2`
Now, force between charges placed at positions A and B are repulsive in nature. Thus for charge Q to A to be in equilibrium , the force of attraction between charges placed at A and C should be equal to force of repulsion between charges placed at A and B . `(F_(AB)=F_(CB))`
`rArr Q^2/(4piepsilon_0d^2)=(Qq)/(4piepsilon_0x^2)`
`rArr q=(Qx^2)/d^2`
As `x=d/2`
`rArr q=(Q(d/2)^2)/d^2 =Q/4`
`rArr q=-Q/4`
Charge q must be negative and its magnitude should be `Q/4` and for the system of the three charges to be in equilibrium charge q should be placed at the midpoint of A and B.