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?

To solve this problem, we can use the grating equation which is given by:

dsin(θ) = mλ

where:

• d is the distance between the lines in the grating
• θ is the angle of the maximum
• m is the order of the maximum
• λ is the wavelength of the light

First, we need to convert the number of lines per centimeter to the distance between the lines. Since there are 5000 lines per centimeter, the distance between the lines (d) is 1/5000 = 0.0002 cm. But we need this distance in meters, so d = 0.0002 * 10^-2 = 2 * 10^-6 m.

Next, we need to convert the angle from degrees to radians. θ = 47.2° = 47.2 * π/180 = 0.823 rad.

Now we can substitute these values into the grating equation and solve for λ:

2 * 10^-6 m * sin(0.823 rad) = 4 * λ

Solving for λ gives:

λ = (2 * 10^-6 m * sin(0.823 rad)) / 4 = 4.14 * 10^-7 m

Finally, we need to convert this wavelength from meters to nanometers. Since 1 m = 10^9 nm, we have:

λ = 4.14 * 10^-7 m * 10^9 nm/m = 414 nm

So, the wavelength of the red light is approximately 414 nm.