Question 220: Waves Optics

Question

Sodium light of wavelengths 650 nm and 655 nm is used to study diffraction at a single slit of aperture 0.5 mm. The distance between the slit and the screen is 2.0 m. The separation between the positions of the first maxima of diffraction pattern obtained in the two cases is ___________ $$	imes$$ 10$$-$$5 m.

Accepted Answer

Condition for diffraction maximum is $a \sin 	heta=(2 n+1) \frac{\pi}{2}$
For first Maxima, $n=1, a \sin 	heta=\frac{3 \lambda}{2}$
Since $	heta$ is very less, so
$$
\begin{aligned}
& \sin 	heta \approx 	an 	heta \\
& 	herefore a 	an 	heta=\frac{3 \lambda}{2}
\end{aligned}
$$
$$
\begin{aligned}
a\left(\frac{y}{D}ight)  =\frac{3 \lambda}{2} \\
\Rightarrow y  =\frac{3 \lambda \mathrm{D}}{2 a}
\end{aligned}
$$
For the two cases given
$$
\begin{aligned}
y_2-y_1 & =\frac{3 \mathrm{D}}{2 a}\left(\lambda_2-\lambda_1ight) \\
& =\frac{3}{2} 	imes \frac{2}{0 \cdot 5 	imes 10^{-3}}(655-650) 	imes 10^{-9} \\
& =6 	imes 10^3 	imes 5 	imes 10^{-9} \\
& =3 	imes 10^{-5} \mathrm{~m}
\end{aligned}
$$

Waves Optics

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