A non-conducting spherical shell of radius R surrounds a point charge q (q at center). The electric flux through a cap of the shell of half angle `theta` is:

Options

A. (a)`(2piq theta)/(in_(0))`
B. (b)`(q)/(2in_(0))(1-cos theta)`
C. (c)`(q(2theta))/(4pi in_(0))`
D. (d)`(q theta)/(2pi in_(0))`

Explanation

To find the electric flux through a cap of a non-conducting spherical shell surrounding a point charge \( q \) at its center, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Electric Field**:
The electric field \( E \) due to a point charge \( q \) at a distance \( r \) from the charge is given by:
\[ E = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2} \]
Here, \( \epsilon_0 \) is the permittivity of free space.

2. **Determine the Area of the Cap**:
The area \( A \) of the spherical cap with half-angle \( \theta \) on a sphere of radius \( R \) can be calculated using the formula:
\[ A = 2\pi R^2 (1 - \cos \theta) \]
This formula arises from integrating the area element over the cap.

3. **Calculate the Electric Flux**:
The electric flux \( \Phi_E \) through the cap is given by:
\[ \Phi_E = E \cdot A \]
Substituting the expressions for \( E \) and \( A \):
\[ \Phi_E = \left(\frac{1}{4\pi \epsilon_0} \frac{q}{R^2}\right) \cdot \left(2\pi R^2 (1 - \cos \theta)\right) \]

4. **Simplify the Expression**:
Now, we can simplify the expression:
\[ \Phi_E = \frac{q}{4\pi \epsilon_0} \cdot 2\pi (1 - \cos \theta) \]
\[ \Phi_E = \frac{q}{2 \epsilon_0} (1 - \cos \theta) \]

5. **Final Result**:
Thus, the electric flux through the cap of the shell is:
\[ \Phi_E = \frac{q}{2 \epsilon_0} (1 - \cos \theta) \]