To find the net flux through a Gaussian cube when an infinitely long wire with a uniform linear charge density passes through the cube's diagonal corners, we use

Gauss's Law. The relevant expression is:

Here, is the charge enclosed by the Gaussian surface, and is the permittivity of free space.

Given:
The wire passes through two opposite corners of the cube.

The side length of the cube,

The length of the wire inside the cube is equal to its diagonal, which is .

Thus, the charge enclosed is:
(length of wire inside the cube)
Substitute the given values:

The length of the wire inside the cube is equal to its diagonal, which is .

Thus, the charge enclosed is:
(length of wire inside the cube)
Substitute the given values:

Calculating:

Now apply Gauss's Law for net flux:

Substitute :

Thus, the net flux through the Gaussian cube is