Refraction | AP Physics 2 Unit 5 Study Guide

Getting Started

We observe the world through light, but light's path is not always a straight line. When you look at a straw in a glass of water, it appears bent or broken at the water's surface. This chapter investigates the physical system of a light ray crossing the boundary between two different transparent materials, such as air and water. Our core question is: Why does light change direction when it enters a new medium, and how can we predict its new path?

What You Should Be Able to Do

After studying this section, you should be able to:

Explain that refraction is caused by the change in the speed of light as it moves from one medium to another.
Calculate the speed of light in a medium using its index of refraction.
Apply Snell's law to find the angle of refraction for a light ray crossing a boundary.
Describe the conditions required for total internal reflection and calculate the critical angle.

Key Concepts & Mechanisms

System & Preconditions

Our system consists of a light ray, two distinct transparent media (e.g., air and glass), and the flat interface where they meet. To analyze this interaction, we make several idealizations:

We model light as a ray, a straight-line path, which is a good approximation when objects and apertures are much larger than the wavelength of light.
The media are considered isotropic, meaning they have uniform optical properties in all directions.
The boundary between the media is a perfectly smooth, flat plane.

Key Steps / Relations

The Initial Interaction: A light ray traveling in an initial medium (Medium 1) strikes the interface with a second medium (Medium 2). The angle at which it strikes is the angle of incidence ( $θ_{1}$ ), measured between the incoming ray and the normal—an imaginary line drawn perpendicular to the surface at the point of impact.
The Causal Mechanism: Change in Speed: Light is an electromagnetic wave that interacts with the atoms of the medium it travels through. This interaction causes the effective speed of the wave to decrease compared to its speed in a vacuum. The optical density of a medium is characterized by its index of refraction ( $n$ ), a dimensionless quantity defined as the ratio of the speed of light in a vacuum ( $c$ ) to its speed ( $v$ ) in the medium.
$n = \frac{c}{v}$
Here, $c$ is the universal constant $3.00 \times 1 0^{8}$ m/s. Since light slows down in any medium, the index of refraction for any material is always greater than 1 (with $n_{v a c uu m} = 1$ by definition, and $n_{ai r} \approx 1.0003$ ).
The Governing Law: Snell's Law: The change in speed upon crossing the boundary causes the light ray to bend, or refract. The precise relationship between the angles and the indices of refraction is given by Snell's Law. This law conserves a property of the light ray's momentum parallel to the interface.
$n_{1} sin θ_{1} = n_{2} sin θ_{2}$
Here, $n_{1}$ and $θ_{1}$ are the index of refraction and the angle in the initial medium, while $n_{2}$ and $θ_{2}$ are the corresponding values in the second medium. The angle of refraction ( $θ_{2}$ ) is the angle between the exiting ray and the normal.
A Limiting Case: Total Internal Reflection: A special phenomenon can occur if light travels from a medium with a higher index of refraction to one with a lower index ( $n_{1} > n_{2}$ ). According to Snell's law, the ray will bend away from the normal ( $θ_{2} > θ_{1}$ ). As the angle of incidence $θ_{1}$ increases, the angle of refraction $θ_{2}$ will eventually reach its maximum possible value of 90°. The specific angle of incidence that causes this is called the critical angle ( $θ_{c}$ ).
$sin θ_{c} = \frac{n _{2}}{n _{1}} (for n_{1} > n_{2})$
If the angle of incidence exceeds this critical angle ( $θ_{1} > θ_{c}$ ), the light ray does not enter the second medium at all. Instead, it is completely reflected back into the first medium, an effect called total internal reflection.

Outputs & Effects

Change in Direction: The primary effect is refraction, the bending of the light ray's path.
- If $n_{2} > n_{1}$ (e.g., air to water), the ray bends toward the normal ( $θ_{2} < θ_{1}$ ).
- If $n_{2} < n_{1}$ (e.g., water to air), the ray bends away from the normal ( $θ_{2} > θ_{1}$ ).
Change in Speed and Wavelength: The light's speed changes as it enters the new medium. Because the frequency ( $f$ ) of the light wave remains constant, its wavelength ( $λ$ ) must also change, according to the relation $v = f λ$ .
Reflection: At any interface, some light is typically reflected as well as refracted. In the special case of total internal reflection, 100% of the light is reflected.

Regulation & Limits

Snell's law applies to the boundary between two transparent, isotropic media.
All angles ( $θ_{1}, θ_{2}, θ_{c}$ ) must be measured with respect to the normal, not the surface.
Total internal reflection is only possible when a light ray attempts to pass from a medium of higher $n$ to a medium of lower $n$ . It is impossible when going from a lower-n to a higher-n medium.

Key Models & Diagrams

The process of refraction is best understood by linking a physical representation (a ray diagram) to the governing mathematical models.

Representation (Ray Diagram)	Governing Equations	Predicted Observables
A diagram showing a boundary between Medium 1 ( $n_{1}$ ) and Medium 2 ( $n_{2}$ ). An incident ray strikes the boundary at angle $θ_{1}$ to the normal. A refracted ray leaves the boundary at angle $θ_{2}$ to the normal.	Index of Refraction: $n = \frac{c}{v}$ Snell's Law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ Critical Angle: $sin θ_{c} = \frac{n _{2}}{n _{1}}$ (only if $n_{1} > n_{2}$ )	- The speed of light in each medium.- The value of the refracted angle, $θ_{2}$ .- Whether the ray bends toward or away from the normal.- Whether total internal reflection is possible and, if so, the value of the critical angle.

Key Components & Evidence

Refraction: The change in direction of a light ray as it passes from one medium to another. Evidence: A "bent" straw in a glass of water.
Index of Refraction ( $n$ ): A dimensionless property of a material that quantifies how much it slows down light. $n_{ai r} \approx 1.00$ , $n_{w a t er} \approx 1.33$ , $n_{g l a ss} \approx 1.50$ .
Speed of light in a vacuum ( $c$ ): A fundamental constant, approximately $3.00 \times 1 0^{8}$ m/s. This is the maximum speed in the universe.
Speed of light in a medium ( $v$ ): The speed of light within a material, measured in m/s. It is always less than or equal to $c$ .
Angle of Incidence ( $θ_{1}$ ): The angle between the incoming light ray and the normal to the surface, measured in degrees or radians.
Angle of Refraction ( $θ_{2}$ ): The angle between the transmitted light ray and the normal to the surface, measured in degrees or radians.
Snell's Law: The fundamental law of refraction, $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ , which relates the angles and indices of refraction.
Normal: An imaginary line constructed perpendicular to the boundary surface at the point where the light ray hits. It is a crucial reference line for measuring angles.
Total Internal Reflection (TIR): The complete reflection of light at the boundary of two media, which occurs when the angle of incidence is greater than the critical angle. Evidence: The operating principle of fiber optic cables.
Critical Angle ( $θ_{c}$ ): The minimum angle of incidence for which total internal reflection occurs.

Skill Snapshots

Causation

The interaction of a light wave with the atoms of a transparent medium causes the wave's effective speed to decrease.
A change in the speed of light as it crosses a boundary at an angle causes the light ray to change its direction of travel.
For a ray moving from a higher-n to a lower-n medium, increasing the angle of incidence causes the angle of refraction to increase until it reaches 90°, at which point total internal reflection begins.

Comparison

A light ray entering a medium with a higher index of refraction bends toward the normal, whereas a ray entering a medium with a lower index of refraction bends away from the normal.
The index of refraction for a vacuum is defined as exactly 1, whereas the index for any physical medium (like water or glass) is always greater than 1.
Refraction describes light passing through an interface and changing direction, whereas reflection describes light bouncing off an interface.

Change Over Time

This framework can be applied by considering how the refracted ray changes as the incident ray's angle is varied.

Baseline State: A light ray travels from water ( $n_{1} = 1.33$ ) into air ( $n_{2} = 1.00$ ) at an angle of incidence $θ_{1} = 3 0^{\circ}$ .
Change 1: As the angle of incidence $θ_{1}$ is increased from 30°, the angle of refraction $θ_{2}$ also increases, following Snell's law, and the refracted ray bends further away from the normal.
Change 2: As $θ_{1}$ reaches the critical angle, $θ_{c} = sin^{- 1} (1.00/1.33) \approx 48. 8^{\circ}$ , the angle of refraction becomes $θ_{2} = 9 0^{\circ}$ , and the ray travels along the surface. For any $θ_{1} > 48. 8^{\circ}$ , the ray undergoes total internal reflection.
Continuity: Throughout this process of changing angles, the frequency of the light wave remains constant.

Common Misconceptions & Clarifications

Misconception: The angles in Snell's law are measured from the surface of the material.
- Clarification: The angles of incidence and refraction ( $θ_{1}$ and $θ_{2}$ ) are always measured with respect to the normal, a line perpendicular to the surface.
Misconception: Light always bends when it enters a new material.
- Clarification: If a light ray strikes the boundary along the normal (an angle of incidence of $0^{\circ}$ ), it will change speed but will not deviate from its path (the angle of refraction will also be $0^{\circ}$ ).
Misconception: Total internal reflection can happen whenever light moves from one medium to another.
- Clarification: Total internal reflection can only occur when light is traveling from a medium with a higher index of refraction to a medium with a lower index of refraction (e.g., from glass to air, but not from air to glass).
Misconception: The index of refraction has units of speed, like m/s.
- Clarification: The index of refraction is a ratio of two speeds ( $n = c / v$ ), so the units cancel out. It is a dimensionless quantity.

One-Paragraph Summary

Refraction is the bending of light that occurs when it crosses the boundary between two media with different optical properties. This phenomenon is caused by the change in the speed of light as it enters the new medium, a property quantified by the dimensionless index of refraction, $n = c / v$ . The relationship between the angle of incidence, the angle of refraction, and the indices of the two media is precisely described by Snell's Law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ . A critical consequence of this law is total internal reflection, which can occur when light travels from a denser to a less dense medium ( $n_{1} > n_{2}$ ) at an angle greater than the critical angle. These principles are essential for explaining the function of lenses, prisms, and fiber optic technology.

Refraction - AP Physics 2: Algebra-Based Study Guide