Question

A laser beam of wavelength 670 nm shines through a diffraction grating that has 750 lines/mm. Sketch the pattern that appears on a screen 1.0 m behind the grating, noting distances on your drawing and explaining where these numbers come from.

Answer 1

The question asks to sketch a diffraction pattern for a laser beam passing through a diffraction grating with specific characteristics. To sketch the pattern, one must calculate the diffraction angles using the grating equation and then determine the positions on a screen using trigonometry.

Step-by-step explanation:

The student's question is related to the diffraction patterns produced by a laser beam passing through a diffraction grating, and the subsequent analysis of those patterns on a screen. The diffraction grating has a density of 750 lines/mm, which implies that the distance between adjacent grating lines (grating spacing) is 1/mm divided by the number of lines per mm.

The distance between the grating and screen is given as 1.0 m. The diffraction angle, θ, for each order n can be calculated using the equation d sin(θ) = nλ, where d is the grating spacing and λ is the wavelength of the light.

For a grating with 750 lines/mm, the spacing d is 1/(750 mm-1), or 1.33×10-6 m. By inserting the wavelength of 670 nm (6.70×10-7 m) into the diffraction equation for various orders of n, the diffraction angles can be calculated.

To sketch the pattern, one needs to calculate the angles first and then use trigonometric relationships to find the corresponding positions on the screen. The positions x on the screen for each order n can be found using the tangent function: x = L tan(θ), where L is the distance from the grating to the screen (1.0 m).

Answer 2

only first order exist with vertical distance of y = 0.58 m from central order

Step-by-step explanation:

Given:

- The wavelength of laser beam λ = 670 nm

- Diffraction grating N = 750 lines/mm

- Distance of screen from grating x = 1.0 m

Find:

Sketch the pattern that appears on a screen 1.0 m behind the grating, noting distances on your drawing and explaining where these numbers come from.

Solution:

- We will use the results derived from Young's experiment that relates the position of the fringes relative to central order depending on the wavelength of the incident light as follows:

sin(Q) = m* λ*N m = 0, 1, 2, 3, ....

- We will find the first order of bright fringe @ m = 1:

Q_1 = sin^-1 ( 670*750*10^-6)

Q_1 = 0.53 radians

- The vertical distance from the central order is:

y_1 = x*tan(Q_1)

y_1 = 1.0*tan(0.53)

y_1 = 0.58 m

- Re-do the calculations for m = 2:

Q_2 = sin^-1 (2* 670*750*10^-3)

Q_2 = sin^-1 ( > 1)

Q_2 = does not exist

- Hence the second order is too far apart and does on exist in our case. So we have a pair of bright fringe above ans below the central order displaced @ 0.58 m. (See attachment for sketch)