The fringe width at a distance of 50 cm from the slits in young's experiment for light of wavelength `6000 Å` is `0.048cm`. The fringe width at the samedistance for `lambda = 5000 Å` will be

Options

A. `0.04 cm`
B. `0.4 cm`
C. `0.14 cm`
D. `0.45 cm`

Explanation

To solve the problem, we need to find the fringe width for a wavelength of \( \lambda = 5000 \, \text{Å} \) given that the fringe width for \( \lambda = 6000 \, \text{Å} \) is \( 0.048 \, \text{cm} \) at a distance of \( 50 \, \text{cm} \) from the slits.

### Step-by-Step Solution:

1. **Understand the relationship between fringe width and wavelength**:
The fringe width \( \beta \) in Young's double-slit experiment 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. Since \( D \) and \( d \) are constants for this experiment, we can conclude that the fringe width \( \beta \) is directly proportional to the wavelength \( \lambda \).

2. **Set up the proportionality**:
Since \( \beta \) is directly proportional to \( \lambda \), we can write:
\[ \frac{\beta_1}{\beta_2} = \frac{\lambda_1}{\lambda_2} \]
where:
- \( \beta_1 = 0.048 \, \text{cm} \) (fringe width for \( \lambda_1 = 6000 \, \text{Å} \))
- \( \lambda_1 = 6000 \, \text{Å} \)
- \( \lambda_2 = 5000 \, \text{Å} \)

3. **Substituting values**:
We want to find \( \beta_2 \):
\[ \beta_2 = \beta_1 \cdot \frac{\lambda_2}{\lambda_1} \]
Substituting the known values:
\[ \beta_2 = 0.048 \, \text{cm} \cdot \frac{5000 \, \text{Å}}{6000 \, \text{Å}} \]

4. **Calculating \( \beta_2 \)**:
\[ \beta_2 = 0.048 \, \text{cm} \cdot \frac{5000}{6000} \]
Simplifying the fraction:
\[ \beta_2 = 0.048 \, \text{cm} \cdot \frac{5}{6} \]
Now, calculate \( \beta_2 \):
\[ \beta_2 = 0.048 \cdot 0.8333 \approx 0.04 \, \text{cm} \]

5. **Final Answer**:
Therefore, the fringe width at the same distance for \( \lambda = 5000 \, \text{Å} \) is:
\[ \beta_2 \approx 0.04 \, \text{cm} \]