Question

Visible light passes through a diffraction grating that has 900 slits per centimeter, and the interference pattern is observed on a screen that is 2.30m from the grating.In the first-order spectrum, maxima for two different wavelengths are separated on the screen by 2.98 mm. What is the difference between these wavelengths?

Answer 1

`\Delta \lambda=14.3\ nm`

Step-by-step explanation:

It is given that,

The number of lines per unit length, N = 900 slits per cm

Distance between the formed pattern and the grating, l = 2.3 m

n the first-order spectrum, maxima for two different wavelengths are separated on the screen by 2.98 mm,
`\Delta Y=2.98\ mm = 0.00298\ m`

Let d is the slit width of the grating,

`d=(1)/(N)`

`d=(1)/(900\ cm)`

`d=1.11* 10^(-5)\ m`

For the first wavelength, the position of maxima is given by :

`y_1=(L\lambda_1)/(d)`

For the other wavelength, the position of maxima is given by :

`y_2=(L\lambda_2)/(d)`

So,

`\Delta \lambda=(\Delta y d)/(l)`

`\Delta \lambda=(0.00298* 1.11* 10^(-5))/(2.3)`

`\Delta \lambda=1.43* 10^(-8)\ m`

or

`\Delta \lambda=14.3\ nm`

So, the difference between these wavelengths is 14.3 nm. Hence, this is the required solution.