Question

What is the highest order spectrum which may be seen with monochromatic light of wavelength $600{\text{ }}nm$ by means of a diffraction grating with $5000{\text{ }}\dfrac{{lines}}{{cm}}$?

Answer 1

For the given question, we have to find the relation between order of spectrum, wavelength and diffraction grafting to find the answer. Then by substituting all the values we will find the order of diffraction. As it is mentioned that the order is maximum then the maximum value of sine angle is to be used.

Complete step by step answer:
It is stated that a monochromatic light of wavelength $600{\text{ }}nm$ falls on a surface with diffraction grating $5000{\text{ }}\dfrac{{lines}}{{cm}}$. We have to find the highest order of spectrum that could be formed by this monochromatic light due to diffraction.
We know that the equation of grafting is given as,
$n\lambda = d\sin \theta - - - - \left( 1 \right)$
The variables are defined as,
$n =$ the order of diffraction due to the monochromatic light.
$\lambda =$ wavelength of the light that is used.
$d =$ width of the slit
Since, the diffraction grating is given as $5000{\text{ }}\dfrac{{lines}}{{cm}}$.
The width of the slit$d = \dfrac{1}{{5000}}{\text{ }}cm = \dfrac{1}{{5000 \times 100}}{\text{ }}m$
The wavelength of the monochromatic light is given as, $\lambda = 600{\text{ }}nm = 600 \times {10^{ - 7}}{\text{ }}m$
For the highest order the value of $\sin \theta$ must be maximum.
The maximum value of $\sin \theta = 1$,
Substituting all the values in equation $\left( 1 \right)$ we get,
$n \times 600 \times {10^{ - 9}} = \dfrac{1}{{5000 \times 100}} \times 1$
From cross multiplication we get,
$n = \dfrac{1}{{5000 \times 100 \times 600 \times {{10}^{ - 9}}}} = \dfrac{{100}}{{30}} = 3.33$
So, we can write $n \approx 3$
The highest order spectrum which may be seen with monochromatic light of wavelength $600{\text{ }}nm$ by means of a diffraction grating with $5000{\text{ }}\dfrac{{lines}}{{cm}}$ is $3$.

Note: It must be noted that the order is maximum then the maximum value of sine angle is to be used. Diffraction is defined as the slight bending of light when it passes through the edge of an object. To analyse diffraction monochromatic light is to be used.