Electric Flux

• The total flux through the cuboid is:
• As the cuboid has 6 faces, the flux through each face is:

Step 1: Analyze the Position of Charges Relative to the Sphere

A sphere of radius centered at the origin includes the charge located at since . The charge at lies outside the sphere since .

Step 2: Apply Gauss’s Law

According to Gauss’s law, the electric flux through a closed surface depends only on the net charge enclosed by the surface:

Since only the charge is inside the sphere, the enclosed charge .

Step 3: Calculate the Electric Flux

So, the correct answer is:

To determine the total electric flux through all the faces of the cube due to the line charge, we apply Gauss's Law, which states:

where is the electric flux, is the total charge enclosed, and is the permittivity of free space.

The line charge has a linear charge density and a length , thus the total charge is:

Substituting into Gauss's Law gives:

Given that the charge is symmetrically placed in the cube, this flux is evenly distributed through all six faces of the cube. The flux through one face of the cube (assuming symmetry) is one-sixth of the total flux:

However, for each face to have equal charge distribution and considering the geometry provided, the total flux through all six faces must match the charge distribution correctly. Thus, reevaluating using geometric considerations for the specific setup:

The correct total electric flux through all faces based on options provided is:

This value accounts for equidistant distribution and symmetry relative to the specific placement within the cube.