Mastering Physics Solutions: A Sparkling Diamond I

A Sparkling Diamond I

Part A = 1.244*10^8 m/s
Part B = 1.224*10^8 m/s
Part C = δ = asin(sin(theta;a) / nred) – asin(sin(theta;a) / nblue)
Part D = δ = 0.287°

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

Calculate vred, the speed of red light in the diamond. To four significant figures, c = 2.998 * 10^8m/s.
Express your answer in meters per second to four significant digits.

The speed of light in a medium is:

c = (2.998 * 10^8) * (1.00 / nmedium)

For red light in diamond:

c = (2.998 * 10^8) * (1.00 / 2.410)
c = 1.244*10^8 m/s

1.244*10^8 m/s

Part B

Calculate vred, the speed of blue light in the diamond. To four significant figures, c = 2.998 * 10^8m/s.
Express your answer in meters per second to four significant digits.

Just solve like in Part A. For blue light in diamond, n = 2.450.

c = (2.998 * 10^8) * (1.00 / 2.450)
c = 1.224*10^8 m/s

1.224*10^8 m/s

Part C

Derive a formula for δ, the angle between the red and blue refracted rays in the diamond.

Start with Snell’s law:

nair * sin(θ1) = ndiamond * sin(θ2)

For red light, the above becomes:

sin(θ1) = nred * sin(θ2, red)
θ2, red = asin(sin(θ1) / nred)

And for blue light:

sin(θ1) = nblue * sin(θ2, blue)
θ2, blue = asin(sin(θ1) / nblue)

Since we want the difference between the angles:

δ = asin(sin(θ1) / nred) – asin(sin(θ1) / nblue)

δ = asin(sin(θ1) / nred) – asin(sin(θ1) / nblue)

Part C

Calculate δ numerically for θa = 45°.

Just use the formula from Part D:

δ = asin(sin(θ1) / nred) – asin(sin(θ1) / nblue)
δ = asin(sin(45°) / 2.410) – asin(sin(45°) / 2.450)
δ = 16.7751° – 17.0619
δ = 0.2868°

round:

0.287°

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4 Responses to Mastering Physics Solutions: A Sparkling Diamond I

  1. Guest says:

    Typo on Part D,
    δ = asin(sin(45°) / 2.450) – asin(sin(45°) / 2.450)
    = 0
    should be:
    δ = asin(sin(45°) / 2.410) – asin(sin(45°) / 2.450)
    correct answer though

  2. Guest says:

    The answer to Part B is incorrect. It should be 1.224*10^8

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