To solve the problem, we will use the relationship between the fringe width (β), the wavelength (λ), and the distance between the slits (d) and the screen (D). The formula we will use is:
\[
\beta = \frac{\lambda D}{d}
\]
Since D and d are constant for both cases, we can establish a proportional relationship between the fringe widths and the wavelengths of the two laser lights.
### Step-by-Step Solution:
1. **Identify the Given Data:**
- Wavelength of the first laser light, \( \lambda_1 = 315 \, \text{nm} \)
- Fringe width for the first laser light, \( \beta_1 = 8.1 \, \text{mm} \)
- Fringe width for the second laser light, \( \beta_2 = 7.2 \, \text{mm} \)
- Wavelength of the second laser light, \( \lambda_2 \) (to be calculated)
2. **Establish the Proportionality:**
Using the relationship between fringe width and wavelength, we can write:
\[
\frac{\beta_1}{\lambda_1} = \frac{\beta_2}{\lambda_2}
\]
3. **Rearranging the Equation:**
Rearranging the equation to solve for \( \lambda_2 \):
\[
\lambda_2 = \frac{\beta_2 \cdot \lambda_1}{\beta_1}
\]
4. **Substituting the Values:**
Now substitute the known values into the equation:
\[
\lambda_2 = \frac{7.2 \, \text{mm} \cdot 315 \, \text{nm}}{8.1 \, \text{mm}}
\]
5. **Calculating \( \lambda_2 \):**
First, convert all units to nanometers for consistency:
- \( 7.2 \, \text{mm} = 7200 \, \text{µm} = 7200000 \, \text{nm} \)
- \( 8.1 \, \text{mm} = 8100 \, \text{µm} = 8100000 \, \text{nm} \)
Now plug in the values:
\[
\lambda_2 = \frac{7200000 \, \text{nm} \cdot 315 \, \text{nm}}{8100000 \, \text{nm}}
\]
6. **Perform the Calculation:**
\[
\lambda_2 = \frac{2268000000 \, \text{nm}^2}{8100000 \, \text{nm}} = 280 \, \text{nm}
\]
7. **Final Result:**
The wavelength of the second laser light is:
\[
\lambda_2 = 280 \, \text{nm}
\]
