In a region of space having a spherically symmetric distribution of charge, the electric flux enclosed by a concentric spherical Gaussian surface, varies with radius r as
`phi=(phi_0r^3)/R^3` for `r le R`
`phi=phi_0` for r > R
where R and `phi_0` are constants.
The volume charge density in the region is given as

Options

A. `(3phi_0epsilon_0r^2)/(4piepsilon_0R^5)` for `r le R , 0` for r gt R
B. `(3phi_0epsilon_0)/(4piR^3)` for `r le R`, 0 for r gt R
C. `(phi_0 epsilon_0)/(4piR^2r)` for `r le R`, 0 for r gt R
D. `(3phi_0epsilon_0)/(piR^3)` for `r le R` , 0 for r gt R

Answer

To find the volume charge density in the region of space with a spherically symmetric distribution of charge, we can use Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (ε₀).

### Step-by-Step Solution:

1. **Understand the Electric Flux**:
The electric flux (φ) is given by:
\[ \phi = \frac{\phi_0 r^3}{R^3} \quad \text{for } r \leq R \]
\[ \phi = \phi_0 \quad \text{for } r > R \]

2. **Apply Gauss's Law**:
According to Gauss's law:
\[ \phi = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \(Q_{\text{enc}}\) is the charge enclosed by the Gaussian surface.

3. **Relate Charge Enclosed to Volume Charge Density**:
The charge enclosed can be expressed in terms of volume charge density (ρ) as:
\[ Q_{\text{enc}} = \rho \cdot V \]
where \(V\) is the volume of the sphere of radius \(r\):
\[ V = \frac{4}{3} \pi r^3 \]

4. **Substituting Charge Enclosed into Gauss's Law**:
For \(r \leq R\), we can substitute \(Q_{\text{enc}}\) into Gauss's law:
\[ \frac{\phi_0 r^3}{R^3} = \frac{\rho \cdot \frac{4}{3} \pi r^3}{\epsilon_0} \]

5. **Canceling \(r^3\)**:
We can cancel \(r^3\) from both sides:
\[ \frac{\phi_0}{R^3} = \frac{4 \pi \rho}{3 \epsilon_0} \]

6. **Solving for Volume Charge Density (ρ)**:
Rearranging the equation to solve for ρ gives:
\[ \rho = \frac{3 \epsilon_0 \phi_0}{4 \pi R^3} \]

### Final Answer:
The volume charge density in the region is given by:
\[ \rho = \frac{3 \epsilon_0 \phi_0}{4 \pi R^3} \]