Gauss Law

The potential at the center due to a half ring is given by:

For two concentric half rings:

Thus, the correct answer is .

Step 1: Visualize the Cube and Electric Field
The electric field is in the x-direction and its magnitude varies with x. The cube has side length .

Step 2: Calculate the Electric Flux
The electric flux through a surface is given by: Since the electric field is only in the x-direction, the only flux is through the face of the cube at (the face ABCD in the given figure). The electric field at this face is , and the area vector is . Therefore, the net flux through the cube is:

Step 3: Apply Gauss’s Law
Gauss’s law states that the net electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space ( ): where is the enclosed charge.

Step 4: Calculate the Enclosed Charge
Combining the flux calculation and Gauss’s law, we have: Substitute the given values, , , and :

Step 5: Find the Value of Q
The enclosed charge is given as . We have found . Therefore, .

Conclusion: The value of is 288.

1. **Using Gauss’s Law:**
When a charge is placed at the center of one face of a cube, it can be visualized that the charge contributes equally to two adjacent cubes.
2. **Flux Calculation:**
According to Gauss’s law, the total flux due to charge in a closed surface is given by:
Since the charge is shared equally between two adjacent cubes, the flux through each cube is:

Answer: