Two unit negative charge and a positive charge `+q` are along the same line . What should be the magnitude of charge q and in which position must it be placed so that the three charges remain in equilibrium state ?

Explanation

Let the charge q be at distances `r_1` and `r_2` from the two charges at point P and Q respectively as shown in the figure given below .

For equilibrium of q , the force on it exerted by P and Q must be co-linear , equal and opposite i.e.,
`E_(qP) = (q)/(4 pi epsi_(0) r_(1)^(2)) " "` (towards P)... (i)
and `F_(q Q) = (q)/(4 pi epsi_(0) r_(2)^(2)) " "` (towards Q) ... (ii)
From Eqs. (i) and (ii) , we get
`therefore |F_(qP) | = |F_(qQ)| implies (q)/(r_(1)^(2)) = (q)/(r_(2)^(2))`
`implies r_(1) = r_(2) = r `( say)
Hence , charge q should be equidistant from P and Q .
For the system to be in equilibrium , the charge at point P and Q must also be in equilibrium .
Now ,
`F_(qP) = (q)/(4 pi epsi_(0) e^(2)) " "` (towards q) ..... (i)
`F_(PQ) = (1)/(4pi epsi_(0) (2r)^(2)) ` (away from P and Q) ... (ii)
From Eqs. (i) and (ii) , we get
`therefore (q)/(r^(2)) = (1)/(4r^(2)) implies q = (1)/(4)`
Similarly for charge at Q , we would get `q = (1)/(4)`
Thus , q = `1/4` in magnitude of either charge at point P or Q .