The ratio of the intensity at the centre of a bright fringe to the intensity at a point one quarter of the fringe width from the centre is

Options

A. 1
B. `1/2`
C. `3/4`
D. `1/4`

Explanation

To solve the problem, we need to find the ratio of the intensity at the center of a bright fringe to the intensity at a point one quarter of the fringe width from the center.

### Step-by-Step Solution:

1. **Understanding the Fringe Width**:
The fringe width (w) is the distance between two consecutive bright or dark fringes in an interference pattern. The center of the pattern corresponds to the central maximum.

2. **Intensity at the Center**:
At the center of the bright fringe (central maximum), the intensity is maximum. We denote this intensity as \( I_{\text{max}} \).

3. **Finding the Position of Interest**:
We need to find the intensity at a point that is one quarter of the fringe width from the center. This position can be expressed as:
\[ x = \frac{w}{4} \]

4. **Intensity Formula in Interference**:
The intensity at a point in an interference pattern can be expressed as:
\[ I = I_{\text{max}} \cos^2\left(\frac{\pi x}{w}\right) \]
where \( I_{\text{max}} \) is the maximum intensity, and \( x \) is the distance from the center.

5. **Calculating Intensity at \( x = \frac{w}{4} \)**:
Substitute \( x = \frac{w}{4} \) into the intensity formula:
\[ I\left(\frac{w}{4}\right) = I_{\text{max}} \cos^2\left(\frac{\pi \left(\frac{w}{4}\right)}{w}\right) \]
Simplifying this gives:
\[ I\left(\frac{w}{4}\right) = I_{\text{max}} \cos^2\left(\frac{\pi}{4}\right) \]
Since \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):
\[ I\left(\frac{w}{4}\right) = I_{\text{max}} \left(\frac{1}{\sqrt{2}}\right)^2 = I_{\text{max}} \cdot \frac{1}{2} \]

6. **Finding the Ratio of Intensities**:
Now we need to find the ratio of the intensity at the center to the intensity at \( \frac{w}{4} \):
\[ \text{Ratio} = \frac{I_{\text{max}}}{I\left(\frac{w}{4}\right)} = \frac{I_{\text{max}}}{\frac{I_{\text{max}}}{2}} = 2 \]

### Final Answer:
The ratio of the intensity at the center of a bright fringe to the intensity at a point one quarter of the fringe width from the center is **2**.