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

Options

A. `phi`
B. `3/(4pi) phi`
C. `(2pi)/3 phi`
D. `(4pi)/3 phi`

Answer

To solve the problem, we will use Gauss's Law, which states that the electric flux through a closed surface is proportional to the charge enclosed by that surface.

### Step-by-step Solution:

1. **Understand the Problem**: We have a uniform charge distribution in space. We know the electric flux through a cube of side `a` is given as `φ`. We need to find the electric flux through a sphere of radius `a`.

2. **Calculate Charge Enclosed in the Cube**:
- The volume of the cube, \( V_{\text{cube}} \), is given by:
\[ V_{\text{cube}} = a^3 \]
- Let the volume charge density be \( \rho \). The total charge \( Q_{\text{cube}} \) enclosed in the cube is:
\[ Q_{\text{cube}} = \rho \cdot V_{\text{cube}} = \rho \cdot a^3 \]

3. **Apply Gauss's Law for the Cube**:
- According to Gauss's Law:
\[ \Phi_{\text{cube}} = \frac{Q_{\text{cube}}}{\epsilon_0} = \frac{\rho \cdot a^3}{\epsilon_0} \]
- We know from the problem statement that \( \Phi_{\text{cube}} = \phi \). Therefore:
\[ \phi = \frac{\rho \cdot a^3}{\epsilon_0} \]

4. **Calculate Charge Enclosed in the Sphere**:
- The volume of the sphere, \( V_{\text{sphere}} \), is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi a^3 \]
- The total charge \( Q_{\text{sphere}} \) enclosed in the sphere is:
\[ Q_{\text{sphere}} = \rho \cdot V_{\text{sphere}} = \rho \cdot \frac{4}{3} \pi a^3 \]

5. **Apply Gauss's Law for the Sphere**:
- According to Gauss's Law:
\[ \Phi_{\text{sphere}} = \frac{Q_{\text{sphere}}}{\epsilon_0} = \frac{\rho \cdot \frac{4}{3} \pi a^3}{\epsilon_0} \]

6. **Relate the Flux through the Sphere to the Flux through the Cube**:
- We already know that \( \phi = \frac{\rho \cdot a^3}{\epsilon_0} \). Therefore, we can express \( \Phi_{\text{sphere}} \) in terms of \( \phi \):
\[ \Phi_{\text{sphere}} = \frac{4}{3} \pi \cdot \frac{\rho \cdot a^3}{\epsilon_0} = \frac{4}{3} \pi \cdot \phi \]

7. **Final Answer**:
- Thus, the net flux passing through the surface of the imaginary sphere of radius `a` is:
\[ \Phi_{\text{sphere}} = \frac{4}{3} \pi \phi \]