Question

Light of wavelength 650 nm is normally incident on the rear of a grating. The first bright fringe (other than the central one) is at an angle of 5o with respect to the normal.

A. Find the number of slits per centimeter in the grating.
B. Two rays of light of wavelength 650 nm and 420 nm are normally incident on a different grating. If the grating has 5000 slits/cm, what is the angular seperation of of the two light rays' second order maximum?

Answer 1

A

`N = 1340.86 \ slits / cm`

B

`\theta = 15.7^o`

Step-by-step explanation:

From the question we are told that

The wavelength is
`\lambda = 650 \ nm = 650 *10^(-9) \ m`

The angle of first bright fringe is
`\theta = 5^o`

The order of the fringe considered is n =1

Generally the condition for constructive interference is

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

=>
`d = (1 * 650 *10^(-9 ))/( sin(5))`

=>
`d = 7.458 *10^(-6) \ m`

Converting to cm

`d = 7.458 *10^(-6) \ m = 7.458 *10^(-6) * 100 = 0.0007458 \ cm`

Generally the number of grating pre centimeter is mathematically represented as

`N = (1)/(d)`

=>
`N = (1)/(0.0007458)`

=>
`N = 1340.86 \ slits / cm`

Considering question B

From the question we are told that

The first wavelength is
`\lambda_1 = 650 \ nm = 650 *10^(-9) \ m`

The second wavelength is
`\lambda_2 = 429 \ m = 420 *10^(-9 ) \ m`

The order of the fringe is
`n = 2`

The grating is
`N = 5000 \ slits / cm`

Generally the slit width is mathematically represented as

`d = (1)/(N )`

=>
`d = (1)/( 5000 )`

=>
`d = 0.0002 \ c m = 2.0 *10^(-6) \ m`

Generally the condition for constructive interference for the first ray is mathematically represented as

`d sin(\theta_1) = n * \lambda_1`

=>
`\theta_1 = sin^(-1) [( 2 * \lambda )/(d)]`

=>
`\theta_1 = sin^(-1) [( 2 * 650 *10^(-9) )/( 2*10^(-6))]`

=>
`\theta_1 = 40.5 ^o`

Generally the condition for constructive interference for the second ray is mathematically represented as

`d sin(\theta_2) = n * \lambda_2`

=>
`\theta_2 = sin^(-1) [( 2 * \lambda_1 )/(d)]`

=>
`\theta_2 = sin^(-1) [( 2 * 420 *10^(-9) )/( 2*10^(-6))]`

=>
`\theta_2 = 24.8 ^o`

Generally the angular separation is mathematically represented as

`\theta = \theta_1 - \theta_1`

=>
`\theta = 42.5^o - 24.8^o`

=>
`\theta = 15.7^o`