A diffraction grating gives a second-order maximum at as angle of 31° for violet light (λ = 4.0 × 102 nm). If the diffraction grating is 1.0 cm in width, how many lines are on this diffraction grating?
We can use the formula for the position of a maximum in a diffraction grating:
d sinθ = mλ
where d is the distance between the lines on the grating, θ is the angle of the maximum, m is the order of the maximum, and λ is the wavelength of the light.
For a second-order maximum at an angle of 31° for violet light with a wavelength of 4.0 × 10^-7 m:
d sin 31° = 2(4.0 × 10^-7 m)
d = (2 × 4.0 × 10^-7 m) / sin 31°
d = 7.78 × 10^-7 m
The width of the grating is 1.0 cm, or 0.01 m. The number of lines on the grating is equal to the length of the grating divided by the distance between the lines:
number of lines = width / d
number of lines = 0.01 m / 7.78 × 10^-7 m
number of lines = 12,844
Therefore, there are approximately 12,844 lines on this diffraction grating.