The correct option is (A) Q=−4q9
Step 1, Given data
Charges =q and 4q distance =d
third charge =Q
distance from charge q =x
Step 2, Finding the value of Q
We can write
Since the charges are in equilibrium
So,
charges =q
Kqx2=K(4q)(L−x)2
or, x=L3⟶(1)
This is the position where the third charge should be placed. Now, we have to determine the nature and magnitude of it.
For this just try to satisfy the condition of equilibrium on any one of the remaining two charges. Let's take +q charges. The force on it due to +4q charge is directed leftwards so the force on it due to the third charge should be directed rightwards. This implies that the force between the +q charge and the third charge should be of attractive nature. Thus the nature of the 3rd charge is (−ve).
Taking the third charge to be −Q (say) and then on applying the condition of equilibrium on +q charge
Forces will be equal
So,
KQx2=−K(4q)L2
Putting value of x from equation 1
KQ(L/3)2=−K(4q)L2
Or,
⇒9KQL2=−4KqL2
⇒9Q=−4q
Or, Q=−4q9
Hence the value of Q is Q=−4q9