To solve the problem, we need to find the positions of the zero-order and tenth-order maxima when the wavelength is changed from \( \lambda_1 = 6000 \, \text{Å} \) to \( \lambda_2 = 5000 \, \text{Å} \).
### Step-by-Step Solution:
1. **Identify Given Data:**
- Wavelength 1: \( \lambda_1 = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} \)
- Wavelength 2: \( \lambda_2 = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} \)
- Position of zero-order maximum \( Y_0 = 12.34 \, \text{mm} = 0.01234 \, \text{m} \)
- Position of tenth-order maximum \( Y_{10} = 14.73 \, \text{mm} = 0.01473 \, \text{m} \)
2. **Calculate the Fringe Width for the First Wavelength:**
- The fringe width \( \beta \) is given by:
\[
\beta = \frac{\lambda D}{d}
\]
- The difference in position between the tenth and zero-order maxima is:
\[
Y_{10} - Y_0 = 14.73 \, \text{mm} - 12.34 \, \text{mm} = 2.39 \, \text{mm} = 0.00239 \, \text{m}
\]
- Since the distance between the tenth and zero-order maxima is equal to \( 10 \beta_1 \):
\[
10 \beta_1 = 0.00239 \, \text{m}
\]
- Therefore, the fringe width \( \beta_1 \) is:
\[
\beta_1 = \frac{0.00239}{10} = 0.000239 \, \text{m} = 0.239 \, \text{mm}
\]
3. **Relate Fringe Widths for Different Wavelengths:**
- Since the fringe width is directly proportional to the wavelength, we can write:
\[
\frac{\beta_1}{\beta_2} = \frac{\lambda_1}{\lambda_2}
\]
- Rearranging gives:
\[
\beta_2 = \beta_1 \cdot \frac{\lambda_2}{\lambda_1}
\]
- Substituting the values:
\[
\beta_2 = 0.239 \, \text{mm} \cdot \frac{5000 \, \text{Å}}{6000 \, \text{Å}} = 0.239 \, \text{mm} \cdot \frac{5000}{6000}
\]
- Calculate \( \beta_2 \):
\[
\beta_2 = 0.239 \, \text{mm} \cdot \frac{5}{6} = 0.199 \, \text{mm}
\]
4. **Calculate New Positions of Maxima:**
- The position of the zero-order maximum \( Y_0 \) remains unchanged:
\[
Y_0 = 12.34 \, \text{mm}
\]
- The position of the tenth-order maximum \( Y_{10} \) with the new fringe width is:
\[
Y_{10} = Y_0 + 10 \beta_2 = 12.34 \, \text{mm} + 10 \cdot 0.199 \, \text{mm}
\]
- Calculate \( Y_{10} \):
\[
Y_{10} = 12.34 \, \text{mm} + 1.99 \, \text{mm} = 14.33 \, \text{mm}
\]
### Final Results:
- Position of zero-order maximum \( Y_0 = 12.34 \, \text{mm} \)
- Position of tenth-order maximum \( Y_{10} = 14.33 \, \text{mm} \)
