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 need to determine the net electric flux passing through the surface of an imaginary sphere of radius \( a \) when the net flux through an imaginary cube of side \( a \) is given as \( \phi \).

### Step-by-Step Solution:

1. **Understanding Electric Flux**:
Electric flux (\( \Phi \)) through a closed surface is defined by Gauss's Law:
\[ \Phi = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( Q_{\text{enc}} \) is the total charge enclosed within the surface and \( \epsilon_0 \) is the permittivity of free space.

2. **Flux through the Cube**:
Given that the net flux through the surface of the cube of side \( a \) is \( \phi \), we can express this as:
\[ \phi = \frac{Q_{\text{enc, cube}}}{\epsilon_0} \]
This implies that the charge enclosed by the cube is:
\[ Q_{\text{enc, cube}} = \phi \cdot \epsilon_0 \]

3. **Charge Density**:
If we assume that the charge is uniformly distributed in space, we can define the charge density \( \rho \) as:
\[ \rho = \frac{Q_{\text{enc, cube}}}{V_{\text{cube}}} \]
where \( V_{\text{cube}} = a^3 \) is the volume of the cube. Thus,
\[ \rho = \frac{Q_{\text{enc, cube}}}{a^3} = \frac{\phi \cdot \epsilon_0}{a^3} \]

4. **Flux through the Sphere**:
Now, we need to find the flux through a sphere of radius \( a \). The volume of the sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi a^3 \]
The charge enclosed by the sphere can be calculated as:
\[ Q_{\text{enc, sphere}} = \rho \cdot V_{\text{sphere}} = \left(\frac{\phi \cdot \epsilon_0}{a^3}\right) \cdot \left(\frac{4}{3} \pi a^3\right) \]
Simplifying this gives:
\[ Q_{\text{enc, sphere}} = \frac{4}{3} \pi \phi \cdot \epsilon_0 \]

5. **Calculating the Flux through the Sphere**:
Now we can find the flux through the sphere using Gauss's Law:
\[ \Phi_{\text{sphere}} = \frac{Q_{\text{enc, sphere}}}{\epsilon_0} = \frac{\frac{4}{3} \pi \phi \cdot \epsilon_0}{\epsilon_0} = \frac{4}{3} \pi \phi \]

Thus, the net flux passing through the surface of the imaginary sphere of radius \( a \) is:
\[ \Phi_{\text{sphere}} = \frac{4}{3} \pi \phi \]

### Final Answer:
The net flux passing through the surface of the imaginary sphere of radius \( a \) is \( \frac{4}{3} \pi \phi \).