A Young's Double slit experiment is performed in air and in water. Which of the following relationship is true regarding fringe width `(beta)` ?

Options

A. `beta_("AIR") gtbeta_("WATER")`
B. `beta_("WATER")gtbeta_("AIR") `
C. `beta_("AIR") =beta_("WATER")`
D. `beta_("WATER")=0`

Explanation

To solve the problem regarding the fringe width (β) in Young's Double Slit Experiment performed in air and in water, we need to understand how the fringe width is affected by the medium in which the experiment is conducted.

### Step-by-Step Solution:

1. **Understanding Fringe Width Formula**:
The fringe width (β) in Young's Double Slit Experiment is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where:
- \( \lambda \) = wavelength of light
- \( D \) = distance from the slits to the screen
- \( d \) = distance between the slits

2. **Effect of Medium on Wavelength**:
When light travels from one medium to another, its wavelength changes according to the refractive index (μ) of the medium. The relationship is given by:
\[ \lambda' = \frac{\lambda}{\mu} \]
where:
- \( \lambda' \) = wavelength in the new medium
- \( \mu \) = refractive index of the medium

3. **Refractive Indices**:
- The refractive index of air (μ_air) is approximately 1.
- The refractive index of water (μ_water) is approximately 1.33.

4. **Calculating Fringe Width in Different Media**:
- In air:
\[ \beta_{air} = \frac{\lambda D}{d} \]
- In water:
\[ \beta_{water} = \frac{\lambda'}{d} D = \frac{\lambda}{\mu_{water}} \cdot \frac{D}{d} = \frac{\lambda D}{1.33 d} \]

5. **Comparing Fringe Widths**:
To compare the fringe widths:
\[ \beta_{water} = \frac{\beta_{air}}{1.33} \]
This shows that the fringe width in water is less than that in air because the factor of 1.33 (the refractive index of water) reduces the fringe width.

6. **Conclusion**:
Since the fringe width in air is greater than that in water, we conclude that:
\[ \beta_{air} > \beta_{water} \]
Therefore, the correct relationship is:
**Option 1: Fringe width of air is greater than fringe width of water.**