Question

Yellow light of wavelength 590 nm passes through a diffraction grating and makes an interference pattern on a screen 80 cm away. The first bright fringes are 1.9 cm from the central maximum. How many lines per mm does this grating have?
A. 20
B. 40
C. 80
D. 200

Answer 1

Answer:B

Explanation:

Given

Wavelength of light $\lambda =590\ nm$

Screen distance $L=80\ cm$

First fringe is at a distance $y_1=1.9\ cm$

No of lines per mm is given by N

$N=\frac{1}{d}$

where d=slit width

From N-slits Experiment

$\sin \theta _m=\frac{m\lambda }{d}$

$d=\frac{m\lambda }{\sin \theta _m}-----1$

Position of bright fringe is given by

$y=\tan \theta _m\cdot L$

$\tan \theta _m=\frac{y}{L}$

$\theta _m=\tan^{-1}(\frac{y}{L})$

Put the value of $\theta _m$  in eq. 1

$d=\frac{m\lambda }{\sin (\tan^{-1}(\frac{y}{L}))}$

Therefore $N=d^{-1}$

$N=\frac{\sin (\tan^{-1}(\frac{y}{L}))}{m\lambda }$

for $m=1$

$N=\frac{\sin (\tan^{-1}(\frac{1.9\times 10^{-2}}{0.8}))}{1\times 590\times 10^{-9}}$

$N=40243\ line/m$

$N=40\ line/mm$