Diffraction grating in physics?

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 28°. In both cases the bright fringes are of the same order m. What is the unknown wavelength?

Answer 1

In diffration grating, the constructive interference (brightness) will occur if the difference in their two path lengths is an integral multiple of their wavelength (lamda)

d * sin (theta) = n (lamda)

Theta is the angle of emergence d is the distance between slits (d = 1 / N and N, called the grating constant) and n is the "order number of maxima", (n = + 1, 2, 3, ...)

let grating produce same order (n same) maxima for wavelengths [ lamda-1, and lamda-2] respectively at angles [ theta-1 and theta-2]. First wavelegth is known (420nm) other unknown and angles are [ 26 deg and 28 deg]

we can write
d * sin (theta-1) = n (lamda-1) -----(1)
d * sin (theta-2) = n (lamda-2) ----(2)

divide (1) and (2)

(lamda-2) /(lamda-1) = sin (theta-2) /sin (theta-1)

(lamda-2) /(lamda-1) = sin (28) /sin (26)

(lamda-2) = 420 nm * [sin (28) /sin (26)]

(lamda-2) = 420 nm * [0.4694 /0.4384]

(lamda-2) = 420 nm * [1.0707]

(lamda-2) = 449.69 nm
this is the unknown wavelength for same drating (d-same) and same (n-same) order bright fringes

Answer 2

the grating equation is:

d*sin(θ) = mλ

where d is the spacing, θ is the angle m is the order and λ is the wavelength.

we know:

d*sin(26) = m(420nm)
d*sin(28) = mλ

divide these two, the unknows d and m vanish, and we get:

sin(26)/sin(28) = 420nm/λ

λ = 420nm*sin(28)/sin(26)

λ = 449.79nm