Problem:
Two equal point charges Q are placed at two opposite corners of a square, while two equal point charges Q are placed at the other two opposite corners. The value of Q is 2√2 μC. What must be the value of q so that the resultant force on Q is zero?

Solution:

Step 1: Analyzing the Forces
To find the value of q that will result in zero net force on Q, we need to analyze the forces acting on it.

Force due to Q charges:
The two charges Q at the opposite corners will exert a repulsive force on Q. Since the charges are equal, the magnitude of this force can be calculated using Coulomb's law:

F = k * (Q1 * Q2) / r^2

Where:
- F is the force between the charges
- k is the Coulomb's constant (9 * 10^9 N m^2/C^2)
- Q1 and Q2 are the magnitudes of the charges
- r is the distance between the charges

In this case, Q1 = Q2 = Q and the distance between the charges is the length of one side of the square.

Force due to q charges:
The two charges q at the other corners will also exert a repulsive force on Q. Similarly, the magnitude of this force can be calculated using Coulomb's law:

F = k * (q1 * q2) / r^2

Where:
- F is the force between the charges
- k is the Coulomb's constant (9 * 10^9 N m^2/C^2)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges

In this case, q1 = q2 = q and the distance between the charges is the length of one side of the square.

Step 2: Setting up the Equations
Since the forces due to the charges Q and q are equal in magnitude and opposite in direction, we can set up the following equations:

FQ = Fq

k * (Q * Q) / r^2 = k * (q * q) / r^2

Simplifying the equation:

Q^2 = q^2

Taking the square root of both sides:

Q = q

Step 3: Substituting Values and Solving
Given that Q = 2√2 μC, we can substitute this value into the equation:

2√2 μC = q

Squaring both sides:

8 μC^2 = q^2

Taking the square root of both sides:

q = 2√2 μC

Therefore, the value of q that will result in zero net force on Q is 2√2 μC.