the electric field inside a sphere which carries a charge density proportional to the distance from the origin `p=alpha` r (`alpha` is a constant) is :

Options

A. `propto` `r^(3)/(4epsilon_(0))`
B. `propto` `r^(2)/(4epsilon_(0))`
C. `propto` `r^(3)/(3epsilon_(0))`
D. none of these

Answer

To find the electric field inside a sphere with a charge density that is proportional to the distance from the origin, we can follow these steps:

### Step 1: Define the Charge Density
The charge density \( p \) is given as:
\[ p = \alpha r \]
where \( \alpha \) is a constant and \( r \) is the distance from the origin.

### Step 2: Calculate the Total Charge Enclosed
To find the electric field at a distance \( r \) from the center of the sphere, we first need to calculate the total charge enclosed within a radius \( r \).

The volume element in spherical coordinates is:
\[ dV = r'^2 \sin \theta \, dr' \, d\theta \, d\phi \]

The total charge \( Q \) enclosed within a radius \( r \) is given by:
\[ Q = \int_0^r p \, dV \]
Substituting \( p = \alpha r' \):
\[ Q = \int_0^r \alpha r' \, (r'^2 \sin \theta \, dr' \, d\theta \, d\phi) \]
Integrating over the angles \( \theta \) and \( \phi \):
\[ Q = \alpha \int_0^r r'^3 \, dr' \int_0^\pi \sin \theta \, d\theta \int_0^{2\pi} d\phi \]
The angular integrals yield:
\[ \int_0^\pi \sin \theta \, d\theta = 2 \quad \text{and} \quad \int_0^{2\pi} d\phi = 2\pi \]
Thus,
\[ Q = \alpha \cdot 2\pi \cdot 2 \int_0^r r'^3 \, dr' = 4\pi \alpha \left[ \frac{r'^4}{4} \right]_0^r = \pi \alpha r^4 \]

### Step 3: Apply Gauss's Law
According to Gauss's Law:
\[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( \Phi_E \) is the electric flux through a Gaussian surface of radius \( r \):
\[ \Phi_E = E \cdot 4\pi r^2 \]
Setting the two expressions for electric flux equal gives:
\[ E \cdot 4\pi r^2 = \frac{\pi \alpha r^4}{\epsilon_0} \]

### Step 4: Solve for the Electric Field \( E \)
Rearranging the equation to solve for \( E \):
\[ E = \frac{\pi \alpha r^4}{4\pi \epsilon_0 r^2} = \frac{\alpha r^2}{4 \epsilon_0} \]

### Final Result
Thus, the electric field inside the sphere at a distance \( r \) from the center is:
\[ E = \frac{\alpha r^2}{4 \epsilon_0} \]

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