A diffraction grating has 450 lines per mm. What is the highest order m that contains the entire visible spectrum from 400 nm to 700 nm?

Highest order = 6 orders

Explanation:

The formula for maxima of diffraction gratings is given by;

dsin θ = mλ

Where;

λ is the wavelength of the light

d is the distance between slits

θ is the angle between the central axis and the corresponding point

m is the order of the interference

Now, for maximum value of the sine function, let's set sin θ = 1

Thus;

d = mλ

To find d, we are told that the grating has 450 lines per mm = 450 x 10³ lines per m

Thus;

d = 1/(450 × 10³)

d = 2.22 × 10^(-6) m

At λ = 700nm = 700 × 10^(-9)m, we have;

m = d/λ = [2.22 × 10^(-6)]/[700 × 10^(-9)]

m = 3.17 ≈ 3

At,λ = 400nm = 400 × 10^(-9)m, we have;

m = d/λ = [2.22 × 10^(-6)]/[400 × 10^(-9)] = 5.55 ≈ 6

Since the fourth maxima occurs further than 3.17 and the seventh maxima occurs further than 5.55,we can say that number of maxima on each side of the central maximum = 6 - 3 = 3.

Thus,there are 6 orders

nah.   i'm just sleepy.