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 \( n = 9 \).
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32