13.3 Refraction

Refraction is the bending of light when it crosses between two media because light changes speed. You quantify the slowdown with the index of refraction (n = c/v), predict the bending angle with Snell's law (n1 sin θ1 = n2 sin θ2), and find when light gets trapped using the critical angle for total internal reflection.

Why This Matters for the AP Physics 2 Exam

Refraction is a core part of Geometric Optics, which carries real weight on the exam. You will use the index of refraction and Snell's law to predict how a light ray bends at a boundary, connect that bending to a change in light speed, and reason about total internal reflection. These same ideas set up the next topic on lenses, where refraction is what forms images.

Optics also fits the experimental side of the course. You might be asked to plan a procedure, collect angle data at a boundary, and linearize it to find an index of refraction from a best-fit line. Being comfortable with Snell's law as both an equation and a graphable relationship helps on data-analysis and free-response reasoning.

Key Takeaways

• Refraction happens because light travels at different speeds in different media; the direction change occurs only at the boundary.
• The index of refraction is n = c/v, so a larger n means light moves slower in that medium.
• Snell's law (n1 sin θ1 = n2 sin θ2) relates the angles and indices; angles are always measured from the normal.
• Light bends toward the normal when entering a higher-index medium and away from the normal when entering a lower-index medium.
• At normal incidence (θ1 = 0), light passes straight through with no bending.
• Total internal reflection only happens going from higher index to lower index, and only beyond the critical angle θ_critical = sin⁻¹(n2/n1).

Refraction of Light Between Media

Why light changes direction

When light passes from one transparent medium into another, it usually changes direction at the boundary. That bending is refraction, and it happens because light travels at different speeds in different materials.

• The amount of bending depends on the difference in indices of refraction and the angle at which light hits the boundary.
• Entering a higher-index medium (like air to water) bends the ray toward the normal line.
• Entering a lower-index medium (like water to air) bends the ray away from the normal line.
• The normal is the line perpendicular to the surface at the point where the ray hits.

Speed of light in media

Light moves fastest in a vacuum and slows down when it travels through matter. That speed change is the cause of refraction.

• In a vacuum, light travels at about 3 × 10⁸ m/s, denoted c.
• In air, light travels only slightly slower than in vacuum.
• In water, light travels at about 75% of its vacuum speed.
• In glass, light travels at roughly 67% of its vacuum speed.
• In diamond, light slows to about 41% of its vacuum speed.

The more a material slows light, the more strongly the ray bends when entering or exiting at an angle.

Index of refraction

The index of refraction measures how much a material slows light compared to vacuum.

$n=\frac{c}{v}$

Where:

• n is the index of refraction (dimensionless)
• c is the speed of light in vacuum (3 × 10⁸ m/s)
• v is the speed of light in the medium

Common indices of refraction:

• Vacuum: exactly 1.0
• Air: approximately 1.0003 (often rounded to 1.00)
• Water: about 1.33
• Glass: typically 1.5 to 1.6
• Diamond: about 2.42

A higher index means light slows more in that medium and bends more sharply at an angled boundary.

Snell's law

Snell's law gives the math relationship between the angle of incidence and the angle of refraction when light crosses between media with different indices.

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

Where:

• n₁ is the index of the first medium
• θ₁ is the angle of incidence (measured from the normal)
• n₂ is the index of the second medium
• θ₂ is the angle of refraction (measured from the normal)

What this tells you:

• Going from lower index to higher index (air to water), the ray bends toward the normal, so θ₂ < θ₁.
• Going from higher index to lower index (water to air), the ray bends away from the normal, so θ₂ > θ₁.
• If light hits the boundary along the normal (θ₁ = 0°), it passes straight through with no bending (θ₂ = 0°).
• The bigger the difference between the indices, the more pronounced the bend.

Total internal reflection

Total internal reflection is a special case that happens when light tries to move from a higher-index medium into a lower-index medium at a large enough angle.

• It only occurs going from a higher index to a lower index (like water to air).
• It happens when the angle of incidence is larger than a specific value called the critical angle.
• At the critical angle, the refracted ray would travel exactly along the boundary, at 90° to the normal.
• The critical angle is found with:

$\theta_{\text{critical}} = \sin^{-1}\left(\frac{n_2}{n_1}\right)$

Here n₁ is the index of the medium the light starts in, and n₂ is the index of the medium it would enter.

When the angle of incidence is beyond the critical angle:

• No light passes into the second medium.
• All the light reflects back into the first medium.
• The reflection follows the law of reflection (angle of incidence equals angle of reflection).

Total internal reflection is the idea behind several real applications, such as fiber optic communications, where light signals travel down a glass or plastic core surrounded by lower-index cladding, the sparkle of cut diamonds, and reflecting prisms in binoculars and periscopes. Treat these as applications of the concept, not separate AP requirements.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Always measure θ from the normal, not from the surface. Mixing this up is the fastest way to get a wrong Snell's law answer.
• Decide the direction of bending before computing. If n2 > n1, expect θ2 to be smaller; if n2 < n1, expect θ2 to be larger. Use that as a sanity check.
• For critical angle problems, confirm the light is going from the higher-index medium to the lower-index one. If it is not, there is no critical angle.
• Keep units consistent and remember n is dimensionless because c and v cancel.

Free Response

• When asked to explain, connect the bending to the change in speed, not just to "the rule." Stating that v changes at the boundary earns the reasoning.
• If a problem gives speed instead of n, find n = c/v first, then apply Snell's law.

Experimental Reasoning

• A common lab uses measured angle pairs at a boundary. Plot sin θ1 versus sin θ2 to get a straight line whose slope relates to the ratio of indices, then use the best-fit line to find an unknown index.
• Be ready to identify sources of error, such as imprecise angle measurement or a non-flat boundary.

Practice Problem 1: Index of Refraction

Solution

Use the relationship between vacuum speed and medium speed:

$n = \frac{c}{v}$

Where:

• n is the index of refraction
• c is the speed of light in vacuum (3.00 × 10⁸ m/s)
• v is the speed of light in the medium (1.94 × 10⁸ m/s)

Substituting:

$n = \frac{3.00 \times 10^8 \text{ m/s}}{1.94 \times 10^8 \text{ m/s}} = 1.55$

The index of refraction is 1.55, which is typical of certain types of glass.

Practice Problem 2: Snell's Law

Solution

Use Snell's law:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

Where:

• n₁ = 1.00 (air)
• θ₁ = 42° (angle of incidence)
• n₂ = 1.33 (water)
• θ₂ = ? (angle of refraction)

Rearranging for θ₂:

$\sin \theta_2 = \frac{n_1 \sin \theta_1}{n_2} = \frac{1.00 \times \sin 42°}{1.33}$

$\sin \theta_2 = \frac{0.669}{1.33} = 0.503$

$\theta_2 = \sin^{-1}(0.503) = 30.2°$

The ray refracts at 30.2° to the normal in the water. The angle decreased as light entered the higher-index medium, which is exactly what you should expect.

Practice Problem 3: Total Internal Reflection

Solution

Use the critical angle formula:

$\theta_{\text{critical}} = \sin^{-1}\left(\frac{n_2}{n_1}\right)$

Where:

• n₁ = 2.42 (diamond)
• n₂ = 1.00 (air)

Substituting:

$\theta_{\text{critical}} = \sin^{-1}\left(\frac{1.00}{2.42}\right) = \sin^{-1}(0.413) = 24.4°$

The critical angle at the diamond-air boundary is 24.4°. Any ray inside the diamond that hits this boundary at more than 24.4° from the normal undergoes total internal reflection. This small critical angle is part of why cut diamonds appear so brilliant: much of the light is trapped by repeated total internal reflection before it exits.

Common Misconceptions

• Refraction does not slow light "more and more" as it travels through a medium. The speed change is set by the medium, and the direction change happens only at the boundary.
• Angles in Snell's law are measured from the normal, not from the surface. A ray "close to the surface" actually has a large angle of incidence.
• A higher index of refraction means light moves slower, not faster. Bigger n equals smaller v.
• Total internal reflection cannot happen going from a lower-index medium into a higher-index one. It requires starting in the higher-index medium.
• The critical angle is not where total internal reflection starts to allow some light through. At and below the critical angle some light still refracts out; only beyond the critical angle is all light reflected.
• Light bending toward the normal does not mean it speeds up. Bending toward the normal happens when light enters a slower, higher-index medium.

Related AP Physics 2 Guides

• 13.1 Reflection
• 13.2 Images Formed by Mirrors
• 13.4 Images Formed by Lenses