Solution:

To find the magnitude of the third charge that will keep the system in equilibrium, follow these steps:

Setup:
- Two charges \( Q \) are placed at a distance \( d \) apart.
- A third charge \( q \) is placed at the midpoint of the line joining the two charges.

For the system to be in equilibrium, the net force on all the charges must be zero.

Forces on the third charge \( q \) at the midpoint:
- The two charges \( Q \) exert forces on \( q \) from opposite directions.
- These forces will be equal in magnitude but opposite in direction, so we need to balance them by choosing the right value for \( q \).

Using Coulomb's law, the force between \( Q \) and \( q \) (at a distance \( d/2 \)) is given by:

\[
F = k \frac{|Q \cdot q|}{(d/2)^2} = k \frac{4|Q \cdot q|}{d^2}
\]

Forces on one of the charges \( Q \) (let's take the left charge):
- The charge \( q \) at the midpoint exerts a force on \( Q \) in one direction.
- The other charge \( Q \) exerts a repulsive force on this charge from the opposite direction.

For equilibrium, the force between the two \( Q \)'s must balance the force due to \( q \) on \( Q \).

1. Force between the two \( Q \)'s:

\[
F_{QQ} = k \frac{Q^2}{d^2}
\]

2. Force between \( Q \) and \( q \) (distance \( d/2 \)):

\[
F_{Qq} = k \frac{4|Q \cdot q|}{d^2}
\]

Condition for equilibrium:
The force between the two \( Q \)'s must equal the force between \( Q \) and \( q \):

\[
k \frac{Q^2}{d^2} = k \frac{4|Q \cdot q|}{d^2}
\]

Canceling out the common terms:

\[
Q^2 = 4|Q \cdot q|
\]

\[
q = \frac{Q}{4}
\]

Conclusion:
The magnitude of the third charge \( q \) that should be placed at the midpoint for equilibrium is:

\[
q = \frac{- Q}{4}
\]