Δy = (3.90 m) * (7.342 x 10^(-3) radians) ≈ 0.0287 m.
To determine the width of the first-order spectrum on a screen 3.90 m away, we need to use the formula for the angular dispersion of a diffraction grating:
Δθ = λ/d,
where Δθ is the angular dispersion, λ is the wavelength, and d is the slit spacing.
First, we need to calculate the average wavelength of the white light by taking the average of the given range:
λ_avg = (410 nm + 750 nm) / 2 = 580 nm.ht
Next, we need to convert the slit spacing from slits/cm to meters:
d = 7900 slits/cm = 7900 slits / (100 cm/m) = 79 slits/m.
Now we can calculate the angular dispersion:
Δθ = λ_avg / d = 580 nm / (79 slits/m) = 7.342 x 10^(-3) radians.
Finally, to find the width of the first-order spectrum on the screen, we can use the relation:
Δy = r * Δθ,
where Δy is the width of the spectrum and r is the distance from the grating to the screen. In this case, r = 3.90 m.
Δy = (3.90 m) * (7.342 x 10^(-3) radians) ≈ 0.0287 m.
Rounded to three significant figures, the width of the first-order spectrum on the screen is approximately 0.0287 m.