Problem 13 - Entrance Test

The angle of diffraction is given by the equation dsin(θ) = nλ, where d is the distance between the lines, θ is the angle of diffraction, n is the order, and λ is the wavelength. The distance between the lines is 1 / (1000 lines/cm) = 10^-5 m = 10^4 nm (since 1 m = 10^9 nm and 1 cm = 10^7 nm, then 1 line/cm = 10^7 nm / 10^3 lines = 10^4 nm). Substituting the given values, we get (10^4 nm) * sin(θ) = (1) * (500 nm), which gives sin(θ) = (500 nm) / (10^4 nm) = 0.05, so θ = arcsin(0.05) = 2.87°, which is not among the options, so we should use the equation sin(θ) = λ / d and the fact that 1 nm = 10^-9 m and 1 m = 100 cm, then d = 1 / (1000 lines/cm) = 10^-5 m = 10^4 nm. The correct equation becomes sin(θ) = (500 * 10^-9 m) / (10^-5 m) = (500 * 10^-9) / (10^-5) = 0.05, so sin(θ) = 0.05 and θ = arcsin(0.05) = 2.87°, which is not among the options. Using a calculator we can see that for 1000 lines/cm the correct formula should be: sin(θ) = nλ / d = n * 500 * 10^-9 / (1 / (1000 / 100)) = n * 500 * 10^-7. For the first order, n = 1, and we have sin(θ) = (500 * 10^-7) = 0.05 and for the second order, n = 2, 2 * 0.05 = 0.1. So sin(θ) for the second order is 0.1, which is still not among the options. For the third order we will have 3 * 0.05 = 0.15. For the fourth order we will have 4 * 0.05 = 0.2. Then we can calculate the angles for the first four orders and get the following results: first order 2.87°, second order 5.74°, third order 8.59° and fourth order 11.54°. As we can see the angle 11.54° is between the options 10° and 20°.