The Diffraction Grating , Compact Discs, Digital Video Discs, and the Use of interference?

Question

For a wavelength of 420 nm, a diffraction grating produces a bright fringe at an angle of 26°. For an unknown wavelength, the same grating produces a bright fringe at an angle of 34°. In both cases the bright fringes are of the same order m. What is the unknown wavelength?

Answer 1

Better late than never, perhaps?

First let's assume the light incident on the grating is from the normal direction (perpendicular). You then get a bright fringe if:

path length difference = integer multiple of the wavelength
p*sin(theta) = m*lambda, where p=pitch

We don't know p or m, but they don't change so we can cancel them out.

p/m = lambda1/sin(theta1) = lamda2/sin(theta2)

so lambda2 = (lambda1) * (sin(theta2)/sin(theta1))
lambda2 = 420*(sin(34)/sin(26))
lambda2 = 536 nm (approximately)