Let `P(r)=(Q)/(piR^4)r` be the charge density distribution for a solid sphere of radius R and total charge Q. For a point 'p' inside the sphere at distance `r_1` from the centre of the sphere, the magnitude of electric field is:

Options

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

Answer

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

### Step 1: Understand the Charge Density Distribution
The charge density \( P(r) \) is given by:
\[ P(r) = \frac{Q}{\pi R^4} r \]
where \( Q \) is the total charge and \( R \) is the radius of the sphere.

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

### Step 3: Calculate the Differential Charge \( dQ \)
The differential charge \( dQ \) in the thin shell can be expressed as:
\[ dQ = P(r) \cdot dV = P(r) \cdot dA \cdot dr = \left(\frac{Q}{\pi R^4} r\right) \cdot (4\pi r^2) \cdot dr \]
Substituting for \( P(r) \):
\[ dQ = \frac{Q}{\pi R^4} r \cdot 4\pi r^2 \cdot dr = \frac{4Q}{R^4} r^3 dr \]

### Step 4: Calculate the Electric Field Contribution \( dE \)
The electric field \( dE \) at point \( P \) due to the charge \( dQ \) located at a distance \( r \) from the center is given by Coulomb's law:
\[ dE = \frac{k \cdot dQ}{r^2} \]
where \( k = \frac{1}{4\pi \epsilon_0} \). Substituting \( dQ \):
\[ dE = \frac{k \cdot \frac{4Q}{R^4} r^3 dr}{r^2} = \frac{4kQ}{R^4} r dr \]

### Step 5: Integrate to Find the Total Electric Field \( E \)
To find the total electric field \( E \) at point \( P \), we need to integrate \( dE \) from \( r = 0 \) to \( r = r_1 \):
\[ E = \int_0^{r_1} dE = \int_0^{r_1} \frac{4kQ}{R^4} r \, dr \]
Calculating the integral:
\[ E = \frac{4kQ}{R^4} \left[ \frac{r^2}{2} \right]_0^{r_1} = \frac{4kQ}{R^4} \cdot \frac{r_1^2}{2} = \frac{2kQ r_1^2}{R^4} \]

### Step 6: Substitute \( k \)
Substituting \( k = \frac{1}{4\pi \epsilon_0} \):
\[ E = \frac{2 \cdot \frac{1}{4\pi \epsilon_0} Q r_1^2}{R^4} = \frac{Q r_1^2}{2\pi \epsilon_0 R^4} \]

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