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. `(2pi_(q)theta)/(in_(0))`
B. `(q)/(2in_(0)) (1-cos theta)`
C. `(q(2theta))/(4pi in_(0))`
D. `(q theta)/(2pi in_(0))`

Explanation

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

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a spherical shell of radius \( R \) with a point charge \( q \) at its center. We need to find the electric flux through a cap of the shell that subtends a half-angle \( \theta \) at the center.

2. **Electric Field Due to Point Charge**:
- The electric field \( E \) due to a point charge \( q \) at a distance \( r \) from the charge is given by:
\[ E = \frac{kq}{r^2} \]
- Here, \( k \) is Coulomb's constant, and for our case, \( r = R \) (the radius of the shell).

3. **Determine the Area Element**:
- Consider a small area element \( dA \) on the cap of the sphere. The area of a small ring at an angle \( \phi \) from the vertical axis can be expressed as:
\[ dA = R^2 \sin \phi \, d\phi \, d\theta \]
- The total area of the cap can be integrated from \( \phi = 0 \) to \( \phi = \theta \).

4. **Calculate the Electric Flux**:
- The electric flux \( \Phi \) through the cap is given by:
\[ \Phi = \int E \cdot dA \]
- Since \( E \) is constant over the cap and directed outward, we can simplify this to:
\[ \Phi = E \int dA \]
- Substitute \( E = \frac{kq}{R^2} \):
\[ \Phi = \frac{kq}{R^2} \int_0^\theta R^2 \sin \phi \, d\phi \, d\theta \]

5. **Integrate the Area**:
- The integral of \( \sin \phi \) from \( 0 \) to \( \theta \) is:
\[ \int_0^\theta \sin \phi \, d\phi = -\cos \phi \bigg|_0^\theta = 1 - \cos \theta \]
- Therefore, the total flux becomes:
\[ \Phi = \frac{kq}{R^2} R^2 (1 - \cos \theta) = kq (1 - \cos \theta) \]

6. **Final Expression**:
- Using \( k = \frac{1}{4\pi \epsilon_0} \), we can write the flux as:
\[ \Phi = \frac{q}{4\pi \epsilon_0} (1 - \cos \theta) \]

### Conclusion:
The electric flux through the cap of the spherical shell is:
\[ \Phi = \frac{q}{4\pi \epsilon_0} (1 - \cos \theta) \]