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Step-by-Step Solution
Step 1: Identify the Relevant Formula
The position of the nth bright fringe in a Young’s double-slit experiment is given by
y = n \frac{D \lambda}{d} ,
where
y is the linear distance of the bright fringe from the central maximum,
D is the distance between the slits and the screen,
d is the separation between the two slits,
\lambda is the wavelength of the light used,
n is the order of the bright fringe (here, for the first bright fringe, n = 1 ).
Step 2: Express the Wavelengths in Terms of Given Data
For the first red fringe, its distance from the center is 3.5 mm (that is 3.5 \times 10^{-3} m). Hence,
y_r = \frac{D \, \lambda_r}{d} \quad \Longrightarrow \quad \lambda_r = \frac{y_r \, d}{D}.
For the first violet fringe, its distance from the center is 2.0 mm ( 2.0 \times 10^{-3} m). Hence,
y_v = \frac{D \, \lambda_v}{d} \quad \Longrightarrow \quad \lambda_v = \frac{y_v \, d}{D}.
Step 3: Substitute the Values
Given data:
y_r = 3.5 \times 10^{-3}\,\text{m}
y_v = 2.0 \times 10^{-3}\,\text{m}
d = 0.3 \times 10^{-3}\,\text{m}
D = 1.5\,\text{m}
Therefore,
\lambda_r = \frac{3.5 \times 10^{-3} \times 0.3 \times 10^{-3}}{1.5}, \quad
\lambda_v = \frac{2.0 \times 10^{-3} \times 0.3 \times 10^{-3}}{1.5}.
Step 4: Compute the Difference in Wavelengths
From the above expressions, the difference in wavelengths is:
\lambda_r - \lambda_v
= \left(\;3.5 \times 10^{-3} - 2.0 \times 10^{-3}\right) \times \frac{0.3 \times 10^{-3}}{1.5}
= 1.5 \times 10^{-3} \times \frac{0.3 \times 10^{-3}}{1.5}.
Simplifying,
\lambda_r - \lambda_v = 1.5 \times 10^{-3} \times \frac{0.3 \times 10^{-3}}{1.5}
= 1.5 \times 10^{-3} \times 0.2 \times 10^{-3}
= 3.0 \times 10^{-7}\,\text{m}.
Expressing this in nanometers (since 1\,\text{m} = 10^9\,\text{nm} ), we get
3.0 \times 10^{-7}\,\text{m} = 3.0 \times 10^{-7} \times 10^{9}\,\text{nm} = 300\,\text{nm}.
Step 5: State the Final Answer
The difference in wavelengths of the red and violet light is
300 nm.
