Question

Calculate the first and second order angles for light of wavelength 400. nm and 700. nm of the grating contains 1.00 x 104 lines/cm.

Answer 1

Answer:

$23.58^{\circ}$ and $53.13^{\circ}$

$44.43^{\circ}$, second order does not exist

Explanation:

n = Number of lines grating = $1\times10^4\ \text{Lines/cm}$

$\lambda$ = Wavelength

m = Order

Distance between slits is given by

$d=\dfrac{1}{n}\\\Rightarrow d=\dfrac{1}{1\times 10^4}\\\Rightarrow d=10^{-6}\ \text{m}$

$\lambda=400\ \text{nm}$

m = 1

We have the relation

$d\sin\theta=m\lambda\\\Rightarrow \theta=\sin^{-1}\dfrac{m\lambda}{d}\\\Rightarrow \theta=\sin^{-1}\dfrac{1\times 400\times 10^{-9}}{10^{-6}}\\\Rightarrow \theta=23.58^{\circ}$

m = 2

$\theta=\sin^{-1}\dfrac{2\times 400\times 10^{-9}}{10^{-6}}\\\Rightarrow \theta=53.13^{\circ}$

The first and second order angles for light of wavelength 400 nm are $23.58^{\circ}$ and $53.13^{\circ}$.

$\lambda=700\ \text{nm}$

m = 1

$\theta=\sin^{-1}\dfrac{1\times 700\times 10^{-9}}{10^{-6}}\\\Rightarrow \theta=44.43^{\circ}$

m = 2

$\theta=\sin^{-1}\dfrac{2\times 700\times 10^{-9}}{10^{-6}}$

Here $\dfrac{2\times 700\times 10^{-9}}{10^{-6}}=1.4>1$ so there is no second order angle for this case.

The first order angle for light of wavelength 700 nm are $44.43^{\circ}$.

Second order angle does not exist.