Two wavelengths \( \lambda_1 \) and \( \lambda_2 \) are used in Young's double slit experiment. \( \lambda_1 = 450 \, \text{nm} \) and \( \lambda_2 = 650 \, \text{nm} \). The minimum order of fringe produced by \( \lambda_2 \), which overlaps with the fringe produced by \( \lambda_1 \), is \( n \). The value of \( n \) is _____.
Correct Answer: 9
Approach Solution - 1
Condition for Overlapping Fringes:
In Young’s double slit experiment, the condition for overlapping fringes for different wavelengths is given by:
\[ n_2 \lambda_2 = n_1 \lambda_1 \]
where \( n_1 \) and \( n_2 \) are the fringe orders for wavelengths \( \lambda_1 \) and \( \lambda_2 \), respectively.
Determine the Ratio of Wavelengths:
Given:
\[ \lambda_1 = 450 \, \text{nm}, \quad \lambda_2 = 650 \, \text{nm} \]
The ratio of the wavelengths is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{450}{650} = \frac{9}{13} \]
Find the Minimum Order of Overlapping Fringes:
For the fringes to overlap, \( n_2 \lambda_2 = n_1 \lambda_1 \).
Let \( n_1 = 13 \) and \( n_2 = 9 \) (the smallest integers that satisfy the ratio):
\[ n_2 = 9 \]
Conclusion:
The minimum order of fringe produced by \( \lambda_2 \) which overlaps with the fringe produced by \( \lambda_1 \) is \( n = 9 \).
Approach Solution -2
Two wavelengths are used in Young’s double-slit experiment:
\[ \lambda_1 = 450 \, \text{nm}, \quad \lambda_2 = 650 \, \text{nm}. \] We are asked to find the minimum order of fringe \( n \) for \( \lambda_2 \) that coincides with a fringe of \( \lambda_1 \).
Step 2: Condition for overlapping of fringes.
For the fringes of two wavelengths to coincide (overlap), the path difference must be the same for both wavelengths.
Hence, the condition for coincidence is: \[ n_1 \lambda_1 = n_2 \lambda_2 \] where \( n_1 \) and \( n_2 \) are fringe orders for \( \lambda_1 \) and \( \lambda_2 \) respectively.
Step 3: Finding the minimum integer values.
We want the minimum integer \( n_2 \) such that the ratio \[ \frac{n_1}{n_2} = \frac{\lambda_2}{\lambda_1} = \frac{650}{450} = \frac{13}{9}. \] So: \[ n_1 : n_2 = 13 : 9. \] The smallest integer values satisfying this ratio are \( n_1 = 13 \) and \( n_2 = 9 \).
Step 4: Final Answer.
\[ \boxed{n = 9} \]
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