Explanation:

To find the value of Q, we need to use the principle of superposition of electric forces. The electric force on a charge is the vector sum of the forces due to all other charges in the system.

Step 1: Analyze the forces acting on the charge Q due to the other three charges.

Let the three charges at the corners of the square be q1, q2, and q3. Without loss of generality, assume that q1 is located at the origin, q2 is located at (L,0), and q3 is located at (0,L). Let Q be located at (L,L).

The electric force on Q due to q1 is given by Coulomb's law as F1 = kqQ/L^2. The direction of this force is along the line joining q1 and Q.

Similarly, the electric force on Q due to q2 is given by F2 = kqQ/(2L^2) and is directed along the line joining q2 and Q. The electric force on Q due to q3 is given by F3 = kqQ/(2L^2) and is directed along the line joining q3 and Q.

Step 2: Use vector addition to find the net force on Q.

The net force on Q is the vector sum of the forces F1, F2, and F3. To find this vector sum, we can use the parallelogram law of vector addition. The diagonal of the parallelogram gives the direction and magnitude of the net force.

It is easy to see that the diagonal of the parallelogram is zero if Q is located at the midpoint of the diagonal of the square. This is because the forces F1, F2, and F3 are symmetrically placed with respect to this point and cancel out.

Step 3: Calculate the value of Q.

Let the side of the square be L. Using Coulomb's law, we can equate the magnitude of the forces F1, F2, and F3 to get:

kqQ/L^2 = kqQ/(2L^2) + kqQ/(2L^2)

Solving for Q, we get:

Q = (2L^2)/(3)

Therefore, the charge Q that needs to be placed at the empty corner of the square is (2L^2)/(3).