The diffraction grating contains 100 strokes per 1 mm. The distance from the grid to the screen is 5 m. The grid is illuminated with white light (wavelengths from 400 nm to 780 nm). What is the width of the spectrum of the first order?

To solve this problem, we need to use the formula for the diffraction grating:

sin(θ) = mλ/d

where:

• θ is the angle of diffraction,
• m is the order of the spectrum (in this case, m=1),
• λ is the wavelength of light, and
• d is the distance between the strokes on the grating.

First, we need to convert the distance between the strokes from mm to nm. Since there are 100 strokes per 1 mm, the distance between the strokes is 1 mm / 100 = 0.01 mm. Converting this to nm gives us 0.01 mm * 1,000,000 nm/mm = 10,000 nm.

Next, we need to find the angles of diffraction for the minimum and maximum wavelengths of white light (400 nm and 780 nm, respectively). Using the formula above, we get:

sin(θ_min) = (1)(400 nm) / (10,000 nm) = 0.04 sin(θ_max) = (1)(780 nm) / (10,000 nm) = 0.078

To find the angles themselves, we take the inverse sine of these values:

θ_min = arcsin(0.04) = 2.29 degrees θ_max = arcsin(0.078) = 4.48 degrees

The width of the spectrum on the screen is the difference between these two angles, multiplied by the distance from the grating to the screen:

Width = (θ_max - θ_min) * Distance Width = (4.48 degrees - 2.29 degrees) * 5 m Width = 2.19 degrees * 5 m

To convert this from degrees to meters, we use the fact that 1 degree = 0.0174533 radians:

Width = 2.19 degrees * 0.0174533 rad/degree * 5 m = 0.191 m

So, the width of the first-order spectrum on the screen is approximately 0.191 meters, or 19.1 cm.