A charge `q` is placed at the some distance along the axis of a uniformly charged disc of surface charge density `sigma`. The flux due to the charge `q` through the disc is `phi`. The electric force on charge `q` exerted by the disc is

Options

A. `sigma phi`
B. `(sigma phi)/(4pi)`
C. `(sigma phi)/(2pi)`
D. `(sigma phi)/(3 pi)`

Explanation

To find the electric force on charge \( q \) exerted by the uniformly charged disc with surface charge density \( \sigma \), we can follow these steps:

### Step 1: Understand the Problem
We have a charge \( q \) placed at a distance from a uniformly charged disc with surface charge density \( \sigma \). The electric flux \( \phi \) through the disc due to the charge \( q \) is given.

### Step 2: Relate Electric Flux to Charge
The electric flux \( \phi \) through the disc due to the charge \( q \) can be expressed using Gauss's Law:
\[ \phi = \frac{q}{2 \epsilon_0} (1 - \cos \theta) \]
where \( \theta \) is the half-angle subtended by the disc at the charge \( q \).

### Step 3: Find the Electric Field Due to the Disc
The electric field \( E \) at a distance \( x \) from the center of the disc can be derived from the surface charge density \( \sigma \):
\[ E = \frac{\sigma}{2 \epsilon_0} (1 - \frac{x}{\sqrt{x^2 + R^2}}) \]
where \( R \) is the radius of the disc and \( x \) is the distance from the disc to the charge \( q \).

### Step 4: Calculate the Electric Force on Charge \( q \)
The electric force \( F \) on the charge \( q \) due to the electric field \( E \) from the disc is given by:
\[ F = qE \]
Substituting the expression for \( E \):
\[ F = q \cdot \frac{\sigma}{2 \epsilon_0} \left(1 - \frac{x}{\sqrt{x^2 + R^2}}\right) \]

### Step 5: Simplify the Expression
Now, we can simplify the expression for the electric force:
\[ F = \frac{q \sigma}{2 \epsilon_0} \left(1 - \frac{x}{\sqrt{x^2 + R^2}}\right) \]

### Final Result
Thus, the electric force on charge \( q \) exerted by the disc is:
\[ F = \frac{q \sigma}{2 \epsilon_0} \left(1 - \frac{x}{\sqrt{x^2 + R^2}}\right) \]