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

Explanation

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

### Step 1: Understand the Geometry
The spherical shell has a radius \( R \) and the point charge \( q \) is at the center. The cap of the shell is defined by a half-angle \( \theta \).

### Step 2: Calculate the Total Surface Area of the Sphere
The total surface area \( A \) of a sphere is given by:
\[ A = 4\pi R^2 \]

### Step 3: Determine the Area of the Cap
The area of the cap can be calculated using the formula for the area of a spherical cap:
\[ A_{\text{cap}} = 2\pi R^2 (1 - \cos \theta) \]
This formula arises from integrating the area of infinitesimal circular strips that make up the cap.

### Step 4: Calculate the Electric Field at the Surface
The electric field \( E \) due to a point charge \( q \) at a distance \( R \) from the charge (on the surface of the shell) is given by:
\[ E = \frac{q}{4\pi \epsilon_0 R^2} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 5: Calculate the Electric Flux through the Cap
The electric flux \( \Phi \) through the cap is given by the product of the electric field and the area of the cap:
\[ \Phi = E \cdot A_{\text{cap}} = \left(\frac{q}{4\pi \epsilon_0 R^2}\right) \cdot \left(2\pi R^2 (1 - \cos \theta)\right) \]

### Step 6: Simplify the Expression
Substituting the expressions for \( E \) and \( A_{\text{cap}} \) into the flux equation:
\[ \Phi = \frac{q}{4\pi \epsilon_0 R^2} \cdot 2\pi R^2 (1 - \cos \theta) \]
This simplifies to:
\[ \Phi = \frac{q}{2\epsilon_0} (1 - \cos \theta) \]

### Final Answer
Thus, the electric flux through the cap of the shell of half-angle \( \theta \) is:
\[ \Phi = \frac{q}{2\epsilon_0} (1 - \cos \theta) \]