Mastering Physics Solutions: A Sparkling Diamond II

A Sparkling Diamond II

Part A = 24.09°
Part B = 60.97°

Solution Below:

A beam of white light is incident on the surface of a diamond at an angle θa. Since the index of refraction depends on the light’s wavelength, the different colors that comprise white light will spread out as they pass through the diamond. The indices of refraction in diamond are nred = 2.410 for red light and nblue = 2.450 for blue light. The surrounding air has nair = 1.000. Note that the angles in the figure are not to scale.

Part A

Now consider θc, the angle at which the blue refracted ray hits the bottom surface of the diamond. If θc is larger than the critical angle θcrit, the light will not be refracted out into the air, but instead it will be totally internally reflected back into the diamond. Find θcrit.

All you need to do here is incorporate the refractive index:

θcrit = asin(1/n)
θcrit = asin(1/2.450)
θcrit = 24.09°

24.09°

Part B

A diamond is cut such that the angle between its top surface and bottom surface is α. For α = 45°, find the incident angle θa such that the blue light is totally internally reflected off the bottom surface.

Just solve by subtracting the critical angle (Part A) from α:

θd = 45 – 24.09
θd = 20.91

Now use Snell’s law:

sin(θmax) = 2.450 * sin(20.91)
sin(θmax) = 0.8744
θmax = 60.97°

60.97°

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