Two negative point charges each of magnitude 2 esu and another point charge q are lying on a straight line. Each of the charges are in equilibrium. Determine the position, value and the nature of the charge q.

Explanation

Since the two negative charges and the another charge q are in a straight line and in equilibrium, then the charge q must be positive. Also the charge q should be placed in between the two negative charges such that the resultant force acting on q due to the two negative charges becomes zero.
From Fig the two 2 esu charges are placed at points A and B respectively and charge q is placed at point C.

Let `AB=r` and `AC=x`
Now the attractive force acting on `+q` due to the negative charge at `A=(qxx2)/(x^(2))`
and the attractive force acting on `+q` due to the negative charge at `B=(qxx2)/((r-x)^(2))`
Now from the condition of equilibrium of charge q
`(qxx2)/(x^(2))=(qxx2)/((r-x)^(2))` or `x=r/2`
Again the resulsive force between the charge at A and B will try to push them apart but the positive charge at C will attract them and maintain the equilibrium. Then the attractive force acting on any of the negative charge due to the positive charge q, must be equal to the force of repulsion betwen two negative charges.
`:.(qxx2)/((r/2)^(2))=(2xx2)/(r^(2))` or `q=1/2=0.5` esu