When red light illuminates a grating with 5000 lines per centimeter, its forth maximum (m=4) is at 47.2°. What is the wavelength of this light in nanometers?
Question
When red light illuminates a grating with 5000 lines per centimeter, its forth maximum (m=4) is at 47.2°. What is the wavelength of this light in nanometers?
Solution
To solve this problem, we can use the grating equation:
dsin(θ) = mλ
where:
- d is the distance between the slits (or lines) in the grating,
- θ is the angle of the mth maximum,
- m is the order of the maximum, and
- λ is the wavelength of the light.
First, we need to find the distance between the lines in the grating. We know there are 5000 lines per centimeter, so:
d = 1 cm / 5000 lines = 210^-4 cm = 210^-6 m
Next, we can plug the known values into the grating equation and solve for λ:
210^-6 m * sin(47.2°) = 4λ
λ = (2*10^-6 m * sin(47.2°)) / 4
After calculating the above expression, we find that λ is approximately 6.5*10^-7 m, or 650 nm. So, the wavelength of the red light is approximately 650 nanometers.
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