Question

A grating has 2000 slits/cm/cm. How many full spectral orders can be seen (400 to 700 nmnm) when it is illuminated by white light

Answer 1

With the use of the formula SinФ = nλ / d, there are 16 spectral orders which can be seen when it is illuminated by white light.

Given that a grating has 2000 slits/cm. That is,

d = 0.01 / 2000

d = 5 x $10^{-6}$ m

The wavelength λ = (700 - 400) nm

λ = 300 x $10^{-9}$ m

To calculate how many full spectral orders that can be seen (400 to 700 nm) when it is illuminated by white light, we will use the below formula

SinФ = nλ / d

Φ = $Sin^{-1}$ (nλ / d)

When n = 1

Φ = $Sin^{-1}$(300 x $10^{-9}$/ 5 x $10^{-6}$ )

Φ = 3.4 degrees

when n = 2

Φ = $Sin^{-1}$(2 x 300 x $10^{-9}$/ 5 x $10^{-6}$ )

Ф = 6.9 degrees

When n = 3

Ф = $Sin^{-1}$(3 x 300 x $10^{-9}$/ 5 x $10^{-6}$ )

When n = 16

Ф = $Sin^{-1}$(16 x 300 x $10^{-9}$/ 5 x $10^{-6}$ )

Ф = $Sin^{-1}$(0.96)

Ф = 73.7 degrees

when n = 17

Ф = $Sin^{-1}$(17 x 300 x $10^{-9}$/ 5 x $10^{-6}$ )

Ф = $Sin^{-1}$(1.05)

Ф = Error ( that is, it does not exist)

Therefore, there are 16 spectral orders which can be seen when it is illuminated by white light.

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