In Young's double slit experiment , light waves of `lambda = 5.4 xx 10^(2)` nm and `lambda = 6.85 xx 10^(1)` nm are used in turn , keeping the same geometry of the set up . Calculate the ratio of the fringe widths in the two cases .

Options

A. 1.3
B. 4.3
C. 7.9
D. 9.5

Explanation

To solve the problem of finding the ratio of the fringe widths in Young's double slit experiment for two different wavelengths, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Wavelengths:**
- Let \( \lambda_1 = 5.4 \times 10^2 \) nm
- Let \( \lambda_2 = 6.85 \times 10^1 \) nm

2. **Convert the Wavelengths to a Common Unit:**
- \( \lambda_1 = 5.4 \times 10^2 \) nm = 540 nm
- \( \lambda_2 = 6.85 \times 10^1 \) nm = 68.5 nm

3. **Understand the Formula for Fringe Width:**
- The fringe width \( \beta \) in Young's double slit experiment is given by the formula:
\[ \beta = \frac{D \lambda}{d} \]
- Where \( D \) is the distance from the slits to the screen, \( d \) is the distance between the slits, and \( \lambda \) is the wavelength of light used.

4. **Establish the Ratio of Fringe Widths:**
- Since the geometry of the setup remains unchanged, the ratio of the fringe widths \( \beta_1 \) and \( \beta_2 \) can be expressed as:
\[ \frac{\beta_1}{\beta_2} = \frac{\lambda_1}{\lambda_2} \]

5. **Substitute the Values:**
- Substitute the values of \( \lambda_1 \) and \( \lambda_2 \) into the ratio:
\[ \frac{\beta_1}{\beta_2} = \frac{540 \, \text{nm}}{68.5 \, \text{nm}} \]

6. **Calculate the Ratio:**
- Perform the division:
\[ \frac{\beta_1}{\beta_2} = \frac{540}{68.5} \approx 7.87 \]

7. **Final Result:**
- The ratio of the fringe widths is approximately:
\[ \frac{\beta_1}{\beta_2} \approx 7.87 \]