#### Prisms and Diffraction Gratings

A particular beam of light looks white but rather than being made up of all wavelengths in the visible spectrum it actually contains only three wavelengths, 450 nm, 550 nm, and 650 nm.

If the beam is incident on a prism made from the same glass as the prism we used in class last week, which of the three options correctly shows the way the prism disperses the light?

- Option 1
- Option 2
- Option 3
- None of the above

In most materials the index of refraction decreases as wavelength increases, so the longest wavelength (red at 650 nm) is bent the least and the shortest wavelength (blue at 450 nm) is bent the most. Option 2 is correct.

The prism is replaced by a diffraction grating. Remembering that the condition for constructive interference is

d sinθ = mλ, which option correctly shows what the light does after passing through the grating.

- Option 1
- Option 2
- Option 3
- None of the above

For the grating the longest wavelength (red at 650 nm) has the largest angle and the shortest wavelength (blue at 450 nm) has the smallest angle. Option 2 is correct.

This diffraction grating has a grating spacing of d = 2000 nm. Recall that the wavelengths in the beam of light are 450 nm, 550 nm, and 650 nm. Which option correctly shows all the observed orders?

- Option 1
- Option 2
- Option 3
- Option 4

Here we have d sinθ = mλ, and the largest sinθ can be is 1. We can set up a table to see at what angle the different orders will appear for the different colors.

Order | Angle for blue | Angle for green | Angle for red |

m=1 | sinθ = 450/2000 | sinθ = 550/2000 | sinθ = 650/2000 |

m=2 | sinθ = 900/2000 | sinθ = 1100/2000 | sinθ = 1300/2000 |

m=3 | sinθ = 1350/2000 | sinθ = 1650/2000 | sinθ = 1950/2000 |

m=4 | sinθ = 1800/2000 | sinθ = 2200/2000 | sinθ = 2600/2000 |

The fourth-order green and red lines are not observed since it doesn't make sense to have sinθ > 1, and the fourth-order blue line is at a smaller angle than the third-order red. Option 1 shows the correct result.