Laser light of wavelength 1260 nm incident on a pair of slits produces an interference pattern in which the fringes are seprated by 8.1mm .A second laser light produces an interference pattern in which the fringes are seprated by 7.2 mm.Calculate the wavelength of the second light

Answer

To solve the problem, we need to use the relationship between the fringe width (β), the wavelength (λ), and the constants involved in the interference pattern. The formula that relates these quantities is:

\[ \beta = \frac{\lambda D}{d} \]

Where:
- β is the fringe width (distance between adjacent bright or dark fringes),
- λ is the wavelength of the light,
- D is the distance from the slits to the screen,
- d is the distance between the slits.

Since D and d remain constant for both lasers, we can establish a proportional relationship between the fringe widths and the wavelengths of the two lasers:

\[ \frac{\beta_1}{\lambda_1} = \frac{\beta_2}{\lambda_2} \]

### Step-by-step solution:

1. **Identify the known values:**
- Wavelength of the first laser light (λ₁) = 1260 nm
- Fringe width of the first laser light (β₁) = 8.1 mm
- Fringe width of the second laser light (β₂) = 7.2 mm
- Wavelength of the second laser light (λ₂) = ?

2. **Set up the proportional relationship:**
\[ \frac{\beta_1}{\lambda_1} = \frac{\beta_2}{\lambda_2} \]

3. **Rearrange the equation to solve for λ₂:**
\[ \lambda_2 = \frac{\beta_2 \cdot \lambda_1}{\beta_1} \]

4. **Substitute the known values into the equation:**
- Convert the fringe widths from mm to nm for consistency:
- β₁ = 8.1 mm = 8100 nm
- β₂ = 7.2 mm = 7200 nm
\[ \lambda_2 = \frac{7200 \, \text{nm} \times 1260 \, \text{nm}}{8100 \, \text{nm}} \]

5. **Calculate λ₂:**
\[ \lambda_2 = \frac{7200 \times 1260}{8100} \]
\[ \lambda_2 = \frac{9072000}{8100} \approx 1120 \, \text{nm} \]

6. **Final answer:**
The wavelength of the second laser light (λ₂) is approximately **1120 nm**.