In the Young's double slit experiment, the intensities at two points `P_(1)` and `P_(2)` on the screen are respectively `I_(1)` and `I_(2)` If `P_(1)` is located at the centre of a bright fringe and `P_(2)` is located at a distance equal to a quarter of fringe width from `P_(1)` then `I_(1)/I_(2)` is

Options

A. 2
B. 3
C. 4
D. None of these

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the setup
In the Young's double slit experiment, we have two points \( P_1 \) and \( P_2 \) on the screen. \( P_1 \) is at the center of a bright fringe, and \( P_2 \) is at a distance equal to a quarter of the fringe width from \( P_1 \).

### Step 2: Define 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.

### Step 3: Calculate the distance for \( P_2 \)
Since \( P_2 \) is located at a distance equal to a quarter of the fringe width from \( P_1 \), we can express this distance as:
\[ y = \frac{\beta}{4} = \frac{\lambda D}{4d} \]

### Step 4: Calculate the path difference
The path difference \( \Delta x \) for the points \( P_1 \) and \( P_2 \) can be expressed as:
\[ \Delta x = \frac{\lambda y}{\beta} = \frac{\lambda \cdot \frac{\beta}{4}}{\beta} = \frac{\lambda}{4} \]

### Step 5: Calculate the phase difference
The phase difference \( \phi \) corresponding to the path difference \( \Delta x \) is given by:
\[ \phi = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{4} = \frac{\pi}{2} \]

### Step 6: Calculate the intensity at \( P_2 \)
The intensity at point \( P_2 \) can be calculated using the formula:
\[ I_2 = I_1 \cos^2\left(\frac{\phi}{2}\right) \]
Substituting \( \phi = \frac{\pi}{2} \):
\[ I_2 = I_1 \cos^2\left(\frac{\pi}{4}\right) \]
Since \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):
\[ I_2 = I_1 \left(\frac{1}{\sqrt{2}}\right)^2 = I_1 \cdot \frac{1}{2} \]

### Step 7: Find the ratio \( \frac{I_1}{I_2} \)
Now we can find the ratio of the intensities:
\[ \frac{I_1}{I_2} = \frac{I_1}{\frac{I_1}{2}} = 2 \]

### Conclusion
Thus, the ratio \( \frac{I_1}{I_2} \) is:
\[ \frac{I_1}{I_2} = 2 \]

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