With a monochromatic light, the fringe-width obtained in a Young’s double slit experiment is 0.133 cm. The whole set- up is immersed in water of refractive index 1.33, then the new fringe-width is

Options

A. Does not change
B. 1.0 mm
C. 2.0 mm
D. 2.69 mm

Answer

To solve the problem of finding the new fringe width when the Young's double slit experiment setup is immersed in water, we can follow these steps:

### Step 1: Understand the formula for fringe width
The fringe width (β) in a Young's double slit experiment is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where:
- \( \lambda \) = wavelength of the light
- \( D \) = distance from the slits to the screen
- \( d \) = distance between the slits

### Step 2: Determine the effect of immersion in water
When the setup is immersed in water, the wavelength of the light changes. The new wavelength (\( \lambda' \)) in a medium with refractive index \( \mu \) is given by:
\[ \lambda' = \frac{\lambda}{\mu} \]
In this case, the refractive index of water is given as \( \mu = 1.33 \).

### Step 3: Express the new fringe width
The new fringe width (\( \beta' \)) when the setup is in water can be expressed as:
\[ \beta' = \frac{\lambda' D}{d} = \frac{\left(\frac{\lambda}{\mu}\right) D}{d} = \frac{\beta}{\mu} \]
This shows that the new fringe width is the original fringe width divided by the refractive index.

### Step 4: Substitute the known values
We know from the problem that the original fringe width \( \beta = 0.133 \, \text{cm} \) and the refractive index \( \mu = 1.33 \). Thus, we can substitute these values into the equation:
\[ \beta' = \frac{0.133 \, \text{cm}}{1.33} \]

### Step 5: Calculate the new fringe width
Now, performing the calculation:
\[ \beta' = \frac{0.133}{1.33} \approx 0.1 \, \text{cm} \]

### Step 6: Convert to mm if necessary
Since the answer may need to be in millimeters, we convert \( 0.1 \, \text{cm} \) to mm:
\[ 0.1 \, \text{cm} = 1 \, \text{mm} \]

### Final Answer
Thus, the new fringe width when the setup is immersed in water is:
\[ \beta' = 1 \, \text{mm} \]