Refraction of Light Notes

Learning Target(s)

• Define refraction as the change in direction of light when it passes from one medium to another.
• Explain how the index of refraction determines the bending of light at the boundary between two mediums.
• Apply Snell’s Law to calculate the angle of refraction for light passing through different substances.
• Describe total internal reflection and identify conditions under which it occurs.
• Explain dispersion of light and how different wavelengths refract at different angles.
• Analyze real-world applications of refraction, such as mirages, optical fibers, and rainbows.

Introduction

• What happens to light when it enters a new medium?

New Vocabulary

• index of refraction
• critical angle
• total internal refraction
• dispersion

Review Vocabulary

• Refraction: the change in direction of waves at the boundary between two different mediums.

Light and Boundaries

• When light encounters a transparent or translucent medium, some light is reflected from the surface, and some is transmitted through the medium.
• When light crosses a boundary between two mediums, it bends. This phenomenon is called refraction.

Snell’s Law of Refraction

• The angle of incidence $\theta_1$ is the angle at which the light ray strikes the surface. It is measured from the normal to the surface.
• The angle of refraction $\theta_2$ is the angle at which the transmitted light leaves the surface. It is also measured with respect to the normal.
• The index of refraction $(n)$ determines the angle of refraction between two mediums.

Snell’s Law

• Snell’s law of refraction relates the indexes of refraction, the angle of incidence, and the angle of refraction for any two substances.
• Snell's Law of Refraction: $n<em>1\sin(\theta</em>1) = n<em>2\sin(\theta</em>2)$

Snell's Law Example Problem

• Problem: A ray of light is incident upon a layer of flint glass $(n = 1.62)$ at an angle of $19.0^\circ$. What is the angle of refraction?
  • Known:
    • $n_1 = 1.00$
    • $n_2 = 1.62$
    • $\theta_1 = 19.0^\circ$
  • Unknown: $\theta_2 = ?$
  • Solution: Use Snell’s law to find the angle of refraction.
• Evaluate the answer: Since $n<em>1 < n</em>2$, then $\theta<em>2$ should be smaller than $\theta</em>1$. This agrees with our answer.

The Meaning of the Index of Refraction

• By examining refraction using the wave model of light and using some geometry, we find that $\frac{\sin(\theta<em>1)}{\sin(\theta</em>2)} = \frac{v<em>1}{v</em>2}$
• By combining this with Snell’s law, we see that $\frac{n<em>2}{n</em>1} = \frac{v<em>1}{v</em>2}$

Index of Refraction

• If $n<em>2 = 1.00$ (a vacuum), then $v</em>2 = c$ and we see that the index of refraction for a medium is related to the speed of light in the medium.
• Index of Refraction: $n = \frac{c}{v}$

Total Internal Reflection

• When $n<em>1 > n</em>2$, the angle of refraction is greater than the angle of incidence.
• At a certain angle of incidence, known as the critical angle $\theta_c$, the refracted light ray lies along the boundary of two mediums.

Critical Angle

• Total internal refraction occurs when light traveling from a region of higher $n$ to a region of lower $n$ strikes the boundary at an angle greater than the critical angle such that all light reflects back into the region of higher $n$.

Optical Fiber

• The light traveling through a transparent optical fiber always hits the internal boundary of the optical fiber at an angle greater than the critical angle, so all of the light is reflected and none of the light is transmitted through the boundary.

Mirages

• The speed of light in a substance (and thus the index of refraction) depends on temperature.
• Mirages can form when light rays are refracted as they travel through air at different temperatures.
• For example, a thermal layering of hot air above the road causes light traveling toward the road to gradually bend upward. This makes the light appear to be coming from a reflection in a pool.
• In the case of a mirage, the Huygens’ wavelets closer to the ground travel faster than those higher up, causing the wavefronts to gradually turn upward.

Dispersion of Light

• The speed of light and the index of refraction vary for different wavelengths of light in the same liquid or solid medium.
• White light separates into a spectrum of colors when it passes through a glass prism. This phenomenon is called dispersion.
• Notice that violet is refracted more than red. This occurs because the speed of violet light through glass is less than the speed of red light through glass.
• This results in glass having a slightly higher index of refraction for violet light than it has for red light.

Rainbows

• A rainbow is a spectrum formed when sunlight is dispersed by water droplets in the atmosphere. Sunlight that falls on a water droplet is refracted and dispersed.
• At the back surface of the droplet, some of the light undergoes internal reflection.
• On the way out of the droplet, the light is again refracted and dispersed.

Quiz Questions

• Which determines the angle of refraction between two mediums? Index of Refraction
• Which is the correct formula for the index of refraction? $n = \frac{c}{v}$
• Which is the certain angle of incidence when the refracted light ray lies along the boundary of two mediums? Critical Angle
• Which occurs when light traveling from a region of higher n to a region of lower n strikes the boundary at an angle greater than the critical angle such that all light reflects back into the region of higher n? Total Internal Reflection
• Which occurs when white light separates into a spectrum of colors when it passes through a glass prism? Dispersion