What is the wavelength of light?

Question

If the incident angle that light hits a grating is 48 o and the diffraction angle is 20 o, calculate the wavelength of light at n = 2. For the given grating d = 689.7 nm/groove and the grating has 1450 grooves/mm.

Answer 1

λ ≈ 374 nm

The grating equation with a non-normal incident beam

d(sin α + sin β) = nλ

where α is the incident angle and β the diffraction angle, both measured from the grating normal. The simpler d sin θ = nλ you may have seen is just this equation with α = 0 — it only applies when light arrives perpendicular to the grating, which is not the case here.

Substituting

sin 48° = 0.7431
sin 20° = 0.3420
Sum = 1.0851

λ = d(sin α + sin β) / n = (689.7 nm)(1.0851) / 2 = 748.4 / 2 = 374 nm

First, verify the two grating numbers agree

You were given both d and the groove density, and they should be reciprocals:

d = 1 mm / 1450 grooves = 6.897 × 10⁻⁴ mm = 689.7 nm ✓

They match, so the second value is a consistency check rather than extra data. Confirming that takes ten seconds and tells you nothing was mistyped in the problem.

A note on the sign convention

Whether the two sine terms are added or subtracted depends on which side of the grating normal the diffracted beam leaves. Same side → add; opposite sides → subtract. With subtraction you would get:

λ = (689.7)(0.7431 − 0.3420) / 2 = 138 nm

374 nm is almost certainly what is intended — it sits at the near-UV/violet edge of the spectrum, an ordinary result for a visible-range spectrometer, whereas 138 nm is vacuum ultraviolet, which ordinary gratings and air optics cannot handle at all. If your setup diagram shows the beams on opposite sides, use the subtraction form; otherwise go with 374 nm.

Sanity check on the order: for n = 2 the wavelength must be under d/2 × the sine sum — and 374 nm is comfortably below the 690 nm groove spacing, as it must be. A first-order (n = 1) measurement of the same geometry would correspond to 748 nm, in the near infrared.