The ratio of the intensity at the centre of a bright fringe to the intensity at a point one-quarter of the distance between two fringe from the centre is

Options

A. (a) `2`
B. (b) `1//2`
C. (c) `4`
D. (d) `16`

Explanation

To solve the problem of finding the ratio of the intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center, we can follow these steps:

### Step 1: Understand the Intensity Formula
The intensity \( I \) at any point on the screen in a double-slit interference pattern is given by the formula:
\[ I = 4 I_0 \cos^2\left(\frac{\phi}{2}\right) \]
where \( I_0 \) is the intensity from one slit and \( \phi \) is the phase difference between the two waves arriving at that point.

### Step 2: Calculate Intensity at the Center of a Bright Fringe
At the center of a bright fringe, the phase difference \( \phi \) is 0 (constructive interference). Thus, we can substitute \( \phi = 0 \) into the intensity formula:
\[ I_{\text{center}} = 4 I_0 \cos^2\left(\frac{0}{2}\right) = 4 I_0 \cos^2(0) = 4 I_0 \cdot 1 = 4 I_0 \]

### Step 3: Determine the Phase Difference at One-Quarter Distance
The distance between two consecutive bright fringes corresponds to a phase difference of \( 2\pi \). Therefore, one-quarter of this distance corresponds to a phase difference of:
\[ \phi = \frac{2\pi}{4} = \frac{\pi}{2} \]

### Step 4: Calculate Intensity at One-Quarter Distance
Now we can substitute \( \phi = \frac{\pi}{2} \) into the intensity formula:
\[ I_{\frac{1}{4}} = 4 I_0 \cos^2\left(\frac{\pi/2}{2}\right) = 4 I_0 \cos^2\left(\frac{\pi}{4}\right) = 4 I_0 \left(\frac{1}{\sqrt{2}}\right)^2 = 4 I_0 \cdot \frac{1}{2} = 2 I_0 \]

### Step 5: Calculate the Ratio of Intensities
Now we can find the ratio of the intensity at the center of the bright fringe to the intensity at one-quarter distance:
\[ \text{Ratio} = \frac{I_{\text{center}}}{I_{\frac{1}{4}}} = \frac{4 I_0}{2 I_0} = 2 \]

### Conclusion
The ratio of the intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center is:
\[ \boxed{2} \]