A charge q is placed at the center of the line joining two equal charges Q. Show that the system of three charges will be in equilibrium if q = -Q/4.

Here,

OA = AB = x/2

OB = x

Force on the charge at B due to the charge at O is,

F1 = kQ2/x2

Force on the charge at B due to the charge at A is,

F2 = kQq/(x/2)2

For the system to be in equilibrium,

F1 + F2 = 0

=> kQ2/x2 + kQq/(x/2)2 = 0

=> q = -Q/4

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 You have to just find you the individual force on each charge by two other charges is zero.

Calculate force on each charge by two other charges and then add conventionally, you would find it 0, hence you can say that the system is in equillibrium.

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