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
a.2 b.3 c.4 d.None of these

Options

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

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the positions of points P1 and P2
- Point P1 is located at the center of a bright fringe, which corresponds to a position where the path difference is zero.
- Point P2 is located at a distance equal to a quarter of the fringe width (β) from P1.

### Step 2: Define the fringe width (β)
- The fringe width (β) in the Young's double slit experiment is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where:
- λ is the wavelength of light,
- D is the distance from the slits to the screen,
- d is the distance between the slits.

### Step 3: Calculate the position of P2
- Since P2 is at a distance of \( \frac{\beta}{4} \) from P1, we can express this as:
\[ y_{P2} = \frac{\beta}{4} = \frac{\lambda D}{4d} \]

### Step 4: Calculate the path difference (Δx)
- The path difference (Δx) for point P2 can be calculated as:
\[ \Delta x = \frac{y_{P2} \cdot d}{D} = \frac{\left(\frac{\lambda D}{4d}\right) \cdot d}{D} = \frac{\lambda}{4} \]

### Step 5: Calculate the phase difference (Δφ)
- The phase difference (Δφ) corresponding to the path difference can be calculated using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Substituting Δx:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{4} = \frac{\pi}{2} \]

### Step 6: Calculate the intensities (I1 and I2)
- The intensity at point P1 (I1) is at the center of a bright fringe, so:
\[ I_1 = I_0 \]
where \( I_0 \) is the maximum intensity.
- The intensity at point P2 (I2) can be calculated using:
\[ I_2 = I_0 \cos^2\left(\frac{\Delta \phi}{2}\right) \]
Substituting Δφ:
\[ I_2 = I_0 \cos^2\left(\frac{\pi}{4}\right) = I_0 \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{2} \]

### Step 7: Calculate the ratio of intensities (I1/I2)
- Now we can find the ratio:
\[ \frac{I_1}{I_2} = \frac{I_0}{\frac{I_0}{2}} = 2 \]

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