Let there be a spherical symmetric charge density varying as `p(r )=p_(0)(r )/(R )` upto r = R and `rho(r )=0` for `r gt R`, where r is the distance from the origin. The electric field at on a distance `r(r lt R)` from the origin is given by -

Options

A. `(rho_(0)r^(2))/(4epsilon_(0)R)`
B. `(rho_(0)r)/(4epsilon_(0)R)`
C. `(rho_(0)r^(4))/(epsilon_(0)R)`
D. `(rho_(0)r^(2))/(epsilon_(0)R)`

Answer

To solve the problem, we need to find the electric field at a distance \( r \) from the origin, where \( r < R \) and the charge density is given by \( \rho(r) = \frac{\rho_0 r}{R} \).

### Step-by-Step Solution:

1. **Understanding Charge Density**:
The charge density is given as:
\[ \rho(r) = \frac{\rho_0 r}{R} \quad \text{for } r \leq R \]
and \( \rho(r) = 0 \) for \( r > R \).

2. **Finding the Volume of a Shell**:
Consider a spherical shell of radius \( r \) and thickness \( dr \). The volume \( dV \) of this shell can be calculated as:
\[ dV = 4\pi r^2 dr \]

3. **Calculating Charge in the Shell**:
The charge \( dq \) in this shell can be expressed as:
\[ dq = \rho(r) \cdot dV = \left(\frac{\rho_0 r}{R}\right) \cdot (4\pi r^2 dr) = \frac{4\pi \rho_0 r^3}{R} dr \]

4. **Total Charge Enclosed**:
To find the total charge \( Q \) enclosed within a radius \( r \), we integrate \( dq \) from \( 0 \) to \( r \):
\[ Q = \int_0^r dq = \int_0^r \frac{4\pi \rho_0 r'^3}{R} dr' \]
where \( r' \) is a dummy variable for integration. The integral becomes:
\[ Q = \frac{4\pi \rho_0}{R} \int_0^r r'^3 dr' = \frac{4\pi \rho_0}{R} \left[\frac{r'^4}{4}\right]_0^r = \frac{4\pi \rho_0}{R} \cdot \frac{r^4}{4} = \frac{\pi \rho_0 r^4}{R} \]

5. **Applying Gauss's Law**:
According to Gauss's law, the electric field \( E \) at a distance \( r \) from the center is given by:
\[ \Phi_E = \frac{Q}{\epsilon_0} \]
The electric flux \( \Phi_E \) through a spherical surface of radius \( r \) is:
\[ \Phi_E = E \cdot 4\pi r^2 \]
Setting these equal gives:
\[ E \cdot 4\pi r^2 = \frac{Q}{\epsilon_0} = \frac{\pi \rho_0 r^4}{R \epsilon_0} \]

6. **Solving for Electric Field \( E \)**:
Rearranging the equation to solve for \( E \):
\[ E \cdot 4\pi r^2 = \frac{\pi \rho_0 r^4}{R \epsilon_0} \]
\[ E = \frac{\rho_0 r^4}{4 R \epsilon_0 r^2} = \frac{\rho_0 r^2}{4 R \epsilon_0} \]

### Final Result:
The electric field at a distance \( r \) from the origin (for \( r < R \)) is given by:
\[ E = \frac{\rho_0 r^2}{4 R \epsilon_0} \]