Three point charges each of magnitude `+q` are located at the vertices of an equilateral triangle of side L . What magnitude of charge should be placed at the centroid of the triangle to keep all three charges in equilibrium ?

Explanation

The situation given in question is shown in figure .

Now , `AD = sqrt(AB^(2) - BD^(2))`
`= sqrt([ L^(2) - ((L)/(2))^(2)]) = sqrt((3L^2)/(4)) = (L sqrt3)/(2)`
Also , `AO = (2)/(3) AD = (2)/(3) xx (L sqrt3)/(2) = (L)/(sqrt3)`
Force on A due to charge at B is `F_(1) = (1)/(4 pi epsi_(0)) (q^2)/(L^2) " "` (along B A)
Force on A due to charge at C is
`F_(2) = (1)/(4 pi epsi_(0)) (q^2)/(L^2) " "` (along CA)
Resultant of `F_1` and `F_2`
`F = sqrt(F_(1)^(2) + F_(2)^(2) + 2 F_(1) F_(2)) cos 60^(@)`
`implies = (1)/(4 pi epsi_(0)) sqrt((q^4)/(L^4) + (q^4)/(L^4) + (q^4)/(L^4) ) = (sqrt3 q^2)/(4 pi epsi_(0) L^(2))`
Now , force between charges at A and O
`F. = (1)/( 4pi epsi_0) (q q.)/((L)/(sqrt3))^(2)` (assuming charge at O is q.)
As system is in equilibrium ,
Then, F + F. = 0
`sqrt3 (1)/(4 pi epsi_(0)) (q^(2))/(L^(2)) + (qq.3)/(4 pi epsi_(0) L^(2)) = 0`
`(q sqrt3)/(L^(2)) + (q. xx 3)/(L^(2)) = 0`
`implies q sqrt3 =-q. xx 3`
`implies q = -sqrt3 xx q.`
or `q = -1.732 q.`