Two negative charges each of 1 unit and a positive charge q are placed along a straight line. Find the magnitude and position of q for which all the chargews will remain at equilibrium? Is the equilibrium stable, unstable or neutral in this case?

Explanation

The positive charge q has to be placed somewhere between the negative charges so that the attractive forces due to the negative charges may balance each other.
Let the negative charges be placed at A and B and the charge +q be placed at C [Flg, 2.35]

Let AB =r, AC=x
The attractive force on +q due to the negative charge at `A =q/(x^(2))`.
The attractive force on +q due to the negative charge at
`B=q/((r-x)^(2))`
The charge q is at equilibrium, thus
`q/(r^(2))=q/((r-x)^(2))` or, `x=r/2`
Evidently the negative charges at A and B tend to move away from each other due to repulsion but are held at those points by the attractive forces of q.
At equilibrium, the attractive force between q and any negative = the repulsive force between two negative charges.
`therefore(qxx1)/((r)^(2))=(1xx1)/(r^(2))`
or, `q=1/4=0.25` units
This is an unstable equilbrium, because a slight change in the position of any of the charges disturbs the equilibrium.