Question

A diffraction grating with exactly 1000 slits per centimeter is illuminated by a He-Ne laser of wavelength \(633 \mathrm{nm} .\) a) What is the highest order of diffraction that could be observed with this grating? b) What would be the highest order if there were exactly 10,000 slits per centimeter?

Short answer

Answer: The highest orders of diffraction are (a) m = 15 and (b) m = 1.

Step-by-step solution

Find the distance between the slits

To find the distance between the slits for a grating with 1000 slits per centimeter, we can simply take the reciprocal of the slits per centimeter value and convert it to meters: d = 1 / 1000 * 0.01 m = 0.00001 m

Solve for the highest order of diffraction

To find the highest order of diffraction, we must find the largest value of m for which the grating equation holds when θ = 90 degrees. When θ = 90 degrees, sin(θ) = 1: 0.00001 * 1 = m * 633 * 10^{-9} m = 0.00001 / 633 * 10^{-9} m = 15.797 Since the diffraction order, m, must be an integer, the maximum order of diffraction in this case is m = 15. #a) 10,000 slits per centimeter#

Find the distance between the slits

To find the distance between the slits for a grating with 10,000 slits per centimeter, we can simply take the reciprocal of the slits per centimeter value and convert it to meters: d = 1 / 10,000 * 0.01 m = 0.000001 m

Solve for the highest order of diffraction

To find the highest order of diffraction, we must find the largest value of m for which the grating equation holds when θ = 90 degrees. When θ = 90 degrees, sin(θ) = 1: 0.000001 * 1 = m * 633 * 10^{-9} m = 0.000001 / 633 * 10^{-9} m = 1.5797 Since the diffraction order, m, must be an integer, the maximum order of diffraction in this case is m = 1.