The electric force between two charges can be calculated with the formula below (Coulomb's law):
[tex]F_e=\frac{K_e\cdot|q_1|\cdot|q_2|}{d^2}[/tex]
Where Ke is the Coulomb's constant, q1 and q2 are the charges and d is the distance between them.
So, for Fe = F, q1 = q2 = Q and d = L, we have:
[tex]F=\frac{K_e\cdot Q^2}{L^2}[/tex]
Now, after the addition of a positive charge in the middle of the charges, each negative charge will suffer another force, acting on the opposite direction of force F:
Since the new resulting force on the negative charges is F/2, the new force created by the positive charge addition is also F/2, so we have:
[tex]\begin{gathered} \frac{F}{2}=\frac{K_e\cdot Q\cdot q}{(L/2)^2} \\ F=\frac{2\cdot K_e\cdot Q\cdot q}{L^2/4} \\ F=\frac{8\cdot K_e\cdot Q\cdot q}{L^2} \\ \frac{K_e\cdot Q^2}{L^2}=\frac{8\cdot K_e\cdot Q\cdot q}{L^2} \\ Q=8q \\ q=\frac{Q}{8} \end{gathered}[/tex]
Therefore the correct option is the fourth one: q = Q/8.
