Problem: Four charge particles each having charge q are placed at the corner of a square of side length d. A fifth charge Q is placed at the mid point of one of its sides. The net force on it is?

Solution:

To find the net force on the fifth charge Q, we need to calculate the force exerted by each of the four charges at the corners of the square. We can use Coulomb's law to calculate the force between two charges:

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

Where F is the force between the two charges, k is Coulomb's constant, q1 and q2 are the charges of the two particles, and r is the distance between them.

We can use vector addition to find the net force on the fifth charge Q due to the four corner charges. Let's assume that each corner charge has a charge of q and is located at a corner of the square of side length d. The fifth charge Q has a charge of Q and is located at the midpoint of one of the sides of the square.

We can break down the force vectors into their x and y components, and then add them up to find the net force on the fifth charge Q. The force due to each corner charge will have both an x and y component.

Calculations:

Let the distance between each corner charge and the fifth charge Q be r. Then the distance from the midpoint of the side to each corner charge is (d/2) * sqrt(2).

The force due to each corner charge on the fifth charge Q is given by Coulomb's law:

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

The x-component of the force due to each corner charge is given by:

Fx = F * cos(theta)

where theta is the angle between the force vector and the x-axis. Similarly, the y-component of the force due to each corner charge is given by:

Fy = F * sin(theta)

We can use trigonometry to find the x and y components of the force vector for each corner charge, and then add them up to find the net force on the fifth charge Q.

Conclusion:

In summary, the net force on the fifth charge Q due to the four corner charges can be found by calculating the force due to each corner charge using Coulomb's law, breaking down each force vector into its x and y components, and then adding up the x and y components to find the net force vector.

The net force on the fifth charge will be zero since the net force on any two charges along the diagonal gets cancel with each other.