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

`23.58^(\circ)` and
`53.13^(\circ)`

`44.43^(\circ)`, second order does not exist

Step-by-step explanation:

n = Number of lines grating =
`1*10^4\ \text{Lines/cm}`

`\lambda` = Wavelength

m = Order

Distance between slits is given by

`d=(1)/(n)\\\Rightarrow d=(1)/(1* 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)(m\lambda)/(d)\\\Rightarrow \theta=\sin^(-1)(1* 400* 10^(-9))/(10^(-6))\\\Rightarrow \theta=23.58^(\circ)`

m = 2

`\theta=\sin^(-1)(2* 400* 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)(1* 700* 10^(-9))/(10^(-6))\\\Rightarrow \theta=44.43^(\circ)`

m = 2

`\theta=\sin^(-1)(2* 700* 10^(-9))/(10^(-6))`

Here
`(2* 700* 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.