Question

What is the angular separation of the two second-order spectral lines having wavelengths 417 nm and 388 nm using a diffraction grating having 456 lines/mm

Answer 1

The value is
`\theta = 1.63^o`

Step-by-step explanation:

From the question we are told that

The wavelength of the first spectral line is
`\lambda _1 = 417 nm = 417*10^(-9) \ m`

The wavelength of the second spectral line is
`\lambda _2 = 388 nm = 388 *10^(-9) \ m`

The diffraction grating is
`k =456 lines/mm = 456000 \ lines / m`

Generally the condition for constructive interference is

`dsin (\theta ) = n * \lambda`

Here n is the order of maxima which is n = 2 given that order of the spectral lines is second order

d is this the distance of slit separation which is evaluated as

`d = (1)/(k)`

=>
`d = (1)/(456000)`

=>
`d = 2.19298 *10^(-6) \ m`

Considering the first spectral lines with
`\lambda _1 = 417 nm = 417*10^(-9) \ m`

`\theta_1 = sin ^(-1) [( 2 * 417 * 10^(-9))/(2.19298 *10^(-6)) ]`

=>
`\theta_1 = 22.35^o`

Considering the first spectral lines with
`\lambda _2 = 388 nm = 388 *10^(-9) \ m`

`\theta_2 = sin ^(-1) [( 2 * 388 * 10^(-9))/(2.19298 *10^(-6)) ]`

=>
`\theta_2 = 20.72^o`

Generally the angular separation is

`\theta = \theta_1 - \theta_2`

=>
`\theta = 22.35 - 20.72`

=>
`\theta = 1.63^o`