At what angle will the highest-order maximum appear for a wavelength of 412 nm using a grating with 435 lines per mm?

Question

Quizplus · Accepted Answer

We can use the formula for the angles of the maxima for a diffraction grating:

sinθ = mλ/d

where m is the order of the maximum, λ is the wavelength of the light, d is the spacing between the grooves in the grating, and θ is the angle of diffraction.

We can first find the spacing between the grooves:

d = 1/435 mm = 2.30 x 10^-3 mm

We want to find the angle for the highest-order maximum, which corresponds to m = 1 for a diffraction grating. Plugging in the values:

sinθ = (1)(412 nm)/(2.30 x 10^-3 mm)

sinθ = 0.998

Taking the inverse sine, we get:

θ = 81.3°

However, this is the angle for the first-order maximum. To find the angle for the highest-order maximum, we need to subtract the angle for the first-order maximum from 90°:

θ_max = 90° - θ_first

θ_max = 90° - 81.3°

θ_max = 8.7°

Therefore, the angle for the highest-order maximum is 8.7°, which corresponds to choice B.

At What Angle Will the Highest-Order Maximum Appear for a Wavelength