Step 1: Method: Set Up Coordinates:
Put the charges on a line at \(x = -a, 0, +a\). Look at the charge at \(x=+a\). Equilibrium means the net force on it is zero.
Step 2: Write the Force on the Right Charge:
The Q at \(-a\) is \(2a\) away and repels it to the right. The charge q at 0 is \(a\) away. The net force to the right is \[ F = \frac{kQ^2}{(2a)^2} + \frac{kQq}{a^2} = \frac{kQ}{a^2}\left(\frac{Q}{4} + q\right) \]
Step 3: Set It to Zero:
Equilibrium needs \(F=0\). \[ \frac{Q}{4} + q = 0 \] \[ q = -\frac{Q}{4} \]
Step 4: Check the Middle Charge:
The forces on q from the two Q charges are equal and opposite for any q, so it is balanced. All three are in equilibrium.
Final Answer:
\[\boxed{q = -\frac{Q}{4}}\]