Let `rho(r)=(Qr)/(piR^(4))` be the charge density distribution for a soild sphere of radius R and total charge Q. For a point P inside the sphere at a distance `r_(1)` from the centre of the sphere, the magnitude of electric field is

Options

A. `Q/(4piepsilon_(0)r_(1)^(2))`
B. `(Qr_(1)^(2))/(4piepsilon_(0)R^(4))`
C. `(Qr_(1)^(2))/(3piepsilon_(0)R^(4))`
D. zero

Explanation

To find the magnitude of the electric field at a point P inside a solid sphere with a given charge density distribution, we will follow these steps:

### Step 1: Understand the Charge Density
The charge density is given by:
\[ \rho(r) = \frac{Qr}{\pi R^4} \]
This indicates that the charge density varies linearly with the distance \( r \) from the center of the sphere.

### Step 2: Consider a Differential Element
To find the electric field at point P, which is at a distance \( r_1 \) from the center, we consider a thin spherical shell of radius \( r \) and thickness \( dr \). The surface area of this shell is:
\[ dA = 4\pi r^2 \]

### Step 3: Calculate the Charge in the Differential Shell
The charge \( dq \) in the differential shell can be expressed as:
\[ dq = \rho(r) \cdot dV = \rho(r) \cdot (dA \cdot dr) = \frac{Qr}{\pi R^4} \cdot (4\pi r^2 \cdot dr) = \frac{4Qr^3}{R^4} dr \]

### Step 4: Apply Gauss's Law
According to Gauss's Law, the electric field \( dE \) due to the charge \( dq \) at point P is given by:
\[ dE = \frac{1}{4\pi \epsilon_0} \cdot \frac{dq}{r_1^2} \]
Substituting \( dq \):
\[ dE = \frac{1}{4\pi \epsilon_0} \cdot \frac{4Qr^3}{R^4 r_1^2} dr \]

### Step 5: Integrate to Find Total Electric Field
To find the total electric field \( E \) at point P, we integrate \( dE \) from \( r = 0 \) to \( r = r_1 \):
\[ E = \int_0^{r_1} dE = \int_0^{r_1} \frac{1}{4\pi \epsilon_0} \cdot \frac{4Qr^3}{R^4 r_1^2} dr \]
This simplifies to:
\[ E = \frac{Q}{\epsilon_0 R^4 r_1^2} \int_0^{r_1} r^3 dr \]
Calculating the integral:
\[ \int_0^{r_1} r^3 dr = \left[ \frac{r^4}{4} \right]_0^{r_1} = \frac{r_1^4}{4} \]
Thus, substituting back:
\[ E = \frac{Q}{\epsilon_0 R^4 r_1^2} \cdot \frac{r_1^4}{4} = \frac{Q r_1^2}{4 \epsilon_0 R^4} \]

### Final Result
The magnitude of the electric field at point P inside the sphere is:
\[ E = \frac{Q r_1^2}{4 \pi \epsilon_0 R^4} \]