Problem Statement: Charge is distributed uniformly in some space. The net flux passing through the surface of an imaginary cube of side a in the space is phi. The net flux passing through the surface of an imaginary sphere of radius a in the space will be? Explain in detail.

Solution:

Step 1: Understanding Flux
Flux is defined as the amount of electric field passing through a given area. It is given by the formula:
Flux = E . A . cos(theta)

Where,
E = Electric field
A = Area
theta = Angle between the electric field and the normal to the area

Step 2: Finding the Flux through the Cube
Given, the charge is distributed uniformly in space and the net flux passing through the surface of an imaginary cube of side a is phi.

We know that the electric field due to a uniformly charged cube at any point inside or outside the cube is given by:
E = (1/4πε) * Q / r^2

Where,
ε = Permittivity of free space
Q = Total charge of the cube
r = Distance of the point from the center of the cube

The flux through each face of the cube can be calculated using the formula mentioned in step 1. Since the cube is symmetric, the flux through each face will be the same.

Therefore, total flux through the cube will be:
Flux = 6 * (E * A)
Flux = 6 * (E * a^2)
Flux = 6 * [(1/4πε) * Q / a^2 * a^2]
Flux = 6 * [(1/4πε) * Q]

Hence, the net flux passing through the surface of the imaginary cube is 6 times the flux through each face, which is equal to 6 times [(1/4πε) * Q].

Step 3: Finding the Flux through the Sphere
To find the flux passing through the surface of an imaginary sphere of radius a, we can use Gauss's law, which states that the net flux passing through a closed surface is equal to the total charge enclosed by the surface divided by the permittivity of free space.

Therefore, the net flux passing through the surface of the imaginary sphere will be:
Flux = Q / ε

Where, Q is the total charge enclosed by the sphere, which can be calculated using the charge density of the space and the volume of the sphere.

Q = (charge density) * (volume of sphere)
Q = (charge density) * [(4/3)πa^3]

Substituting the value of Q in the equation for flux, we get:
Flux = (charge density) * [(4/3)πa^3] / ε

Since the charge is distributed uniformly in space, the charge density is constant throughout the space. Therefore, we can write:
Flux = (total charge in space) * [(4/3)πa^3] / ε

Step 4: Comparing Flux through the Cube and Sphere
We can compare the flux passing through the imaginary cube and sphere by dividing the flux through the sphere by the flux through the cube.

Flux ratio = Flux through sphere / Flux through cube
Flux ratio = [(total charge in space)