Question

Light emitted by element X passes through a diffraction grating that has 1200 slits/mm/mm. The interference pattern is observed on a screen 79.0 cmcm behind the grating. First-order maxima are observed at distances of 58.2 cmcm , 65.2 cmcm , and 92.5 cmcm from the central maximum.

What are the wavelengths of light emitted by element X?

Answer 1

The wavelengths are

`\lambda_1  =  614\ nm`

`\lambda_2  =  687\ nm`

`\lambda_3 = 975\ nm`

Step-by-step explanation:

From the question we are told that

The number of slits per mm is N = 1200

The distance of the screen is
`D = 79.0 \ cm = 0.79 \ m`

The first distance where First-order maxima is observed is
`y_1 = 58.2 cm = 0.582 \ m`

The second distance where First-order maxima is observed is
`y_2 = 65.2 \ cm = 0.652 \ m`

The second distance where First-order maxima is observed is
`y_2 = 92.5 \ cm = 0.925 \ m`

Generally the distance of separation between the slits is mathematically represented as

`d = (1)/(1200)= 8.33 *10^(-4) \ mm = 8.33 *10^(-7) \ m`

Considering the first distance where First-order maxima is observed

Generally the the first distance where First-order maxima is observed is mathematically represented as

`y_1 = (\lambda_1 * D)/(d)`

=>
`\lambda_1 = (0.582 * 8.33 *10^(-7) )/(0.79)`

=>
`\lambda_1  =  6.14 *10^(-7) m`

=>
`\lambda_1  =  614\ nm`

Considering the second distance where First-order maxima is observed

Generally the the second distance where First-order maxima is observed is mathematically represented as

`y_2 = (\lambda_2 * D)/(d)`

=>
`\lambda_2 = (0.652 * 8.33 *10^(-7) )/(0.79)`

=>
`\lambda_2  =  6.87 *10^(-7) m`

=>
`\lambda_2  =  687\ nm`

Considering the third distance where First-order maxima is observed

Generally the the third distance where First-order maxima is observed is mathematically represented as

`y_3 = (\lambda_3 * D)/(d)`

=>
`\lambda_3 = (0.925 * 8.33 *10^(-7) )/(0.79)`

=>
`\lambda_3  =  9.75 *10^(-7) m`

=>
`\lambda_3 =  975\ nm`