The ratio of the intensity at the cnetre 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 that is one quarter of the fringe width from the center.

### Step-by-step Solution:

1. **Identify the Intensity at the Center of the Bright Fringe:**
- The intensity at the center of a bright fringe is the maximum intensity, denoted as \( I_{\text{max}} \).

2. **Determine the Fringe Width:**
- The fringe width \( \beta \) is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where \( \lambda \) is the wavelength of light, \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.

3. **Calculate the Position for Intensity Measurement:**
- The point we are interested in is one quarter of the fringe width from the center:
\[ y = \frac{\beta}{4} \]

4. **Calculate the Path Difference at Point P:**
- The path difference \( \Delta \) at point \( P \) can be calculated using:
\[ \Delta = y \cdot \frac{d}{D} \]
- Substituting \( y \):
\[ \Delta = \left(\frac{\beta}{4}\right) \cdot \frac{d}{D} \]
- Substitute \( \beta \):
\[ \Delta = \left(\frac{\lambda D}{d \cdot 4}\right) \cdot \frac{d}{D} = \frac{\lambda}{4} \]

5. **Calculate the Phase Difference:**
- The phase difference \( \Delta \phi \) is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta \]
- Substituting \( \Delta \):
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{4} = \frac{\pi}{2} \]

6. **Calculate the Intensity at Point P:**
- The intensity at point \( P \) is given by:
\[ I_P = I_{\text{max}} \cdot \cos^2\left(\frac{\Delta \phi}{2}\right) \]
- Substitute \( \Delta \phi \):
\[ I_P = I_{\text{max}} \cdot \cos^2\left(\frac{\pi}{4}\right) \]
- Since \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):
\[ I_P = I_{\text{max}} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 = I_{\text{max}} \cdot \frac{1}{2} \]

7. **Find the Ratio of Intensities:**
- Now, we can find the ratio of the intensity at the center to the intensity at point \( P \):
\[ \text{Ratio} = \frac{I_{\text{max}}}{I_P} = \frac{I_{\text{max}}}{\frac{I_{\text{max}}}{2}} = 2 \]
- Therefore, the ratio of the intensity at the center of the bright fringe to the intensity at a point one quarter of the fringe width from the center is:
\[ \text{Ratio} = 2:1 \]

### 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 \( 2:1 \).

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