© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the key information
In a Young’s double slit experiment (YDSE), when light of wavelength 500 \text{ nm} is used, 15 fringes are observed on a certain portion of the screen. When this light is replaced by another light of wavelength \lambda , only 10 fringes are observed on the same portion.
Step 2: Express the number of fringes in terms of wavelength
Let the length of the screen portion where fringes are counted be L . In YDSE, the fringe width \beta is given by
\displaystyle \beta \;=\;\frac{D\,\lambda}{d},
where:
D is the distance between the slits and the screen,
d is the distance between the two slits,
\lambda is the wavelength of the light used.
The number of fringes n in this portion is approximately:
\displaystyle n\;=\;\frac{L}{\beta}\;=\;\frac{L\,d}{D\,\lambda}.
Step 3: Write the relation for two different wavelengths
For the first wavelength \lambda_1 = 500 \text{ nm} :
\displaystyle n_1 = 15 = \frac{L\,d}{D\,\lambda_1}.
For the second wavelength \lambda_2 = \lambda (unknown):
\displaystyle n_2 = 10 = \frac{L\,d}{D\,\lambda_2}.
Step 4: Use the ratio of fringes to find the unknown wavelength
From the two equations, we have:
\displaystyle \frac{n_1}{n_2} \;=\; \frac{\lambda_2}{\lambda_1}.
Substituting n_1 = 15 , n_2 = 10 , and \lambda_1 = 500 \text{ nm} , we get:
\displaystyle \frac{15}{10}\;=\;\frac{\lambda_2}{500},
which simplifies to
\displaystyle \lambda_2 \;=\; \frac{15}{10} \times 500 \;=\; 750 \text{ nm}.
Step 5: State the final answer
Hence, the unknown wavelength \lambda is \boxed{750 \text{ nm}} .
