A monochromatic light from He-Ne laser λ=6328Ao is incident normally on a diffraction grating having 6000 lines/cm. Find angles at which first & second order maxima would be observed
Best answer
Answer: First order at \(\theta_1 = 22.3^\circ\), second order at \(\theta_2 = 49.4^\circ\)
Grating equation: \((a+b)\sin\theta = n\lambda\).
\((a+b) = \dfrac{1}{6000}\ \text{cm} = 1.667 \times 10^{-4}\) cm, \(\lambda = 6328\ \text{Å} = 6.328 \times 10^{-5}\) cm.
First order (n = 1): \(\sin\theta_1 = \dfrac{6.328 \times 10^{-5}}{1.667 \times 10^{-4}} = 0.3797 \Rightarrow \theta_1 = 22.3^\circ\)
Second order (n = 2): \(\sin\theta_2 = 2 \times 0.3797 = 0.7594 \Rightarrow \theta_2 = 49.4^\circ\)
Answer: First order at about \(22.3^\circ\), second order at about \(49.4^\circ\)
Grating element:
\[d = \frac{1}{6000}\ \mathrm{cm} = 1.667\times10^{-6}\ \mathrm{m}\]
Grating equation at normal incidence: \(d\sin\theta = n\lambda\), with \(\lambda = 6328\ \text{Å} = 6.328\times10^{-7}\ \mathrm{m}\).
First order (\(n = 1\)):
\[\sin\theta_1 = \frac{6.328\times10^{-7}}{1.667\times10^{-6}} = 0.3797\]
\[\theta_1 \approx 22.3^\circ\]
Second order (\(n = 2\)):
\[\sin\theta_2 = 2\times0.3797 = 0.7594\]
\[\theta_2 \approx 49.4^\circ\]
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