In a vacuum, light always travels at c = 3.00 × 10⁸ m/s. In any material it goes slower, because it interacts with the atoms and electrons it passes. How much slower is measured by the index of refraction:
n = c ÷ v
A bigger n means slower light. Because n compares speeds, it has no units, and it is never less than 1.
Material
Index n
Speed of light (m/s)
Vacuum
1.000
3.00 × 10⁸
Air
1.0003
3.00 × 10⁸
Water
1.33
2.26 × 10⁸
Acrylic
1.49
2.01 × 10⁸
Glass
about 1.50
2.00 × 10⁸
Diamond
2.42
1.24 × 10⁸
Why does a change of speed turn the light? Picture a line of marchers crossing from pavement into mud at an angle. The marchers who reach the mud first slow down, while the others are still moving fast, so the whole line pivots. A wavefront of light does the same thing. The side that enters the slow material first slows first, and the front swings toward the normal. If it goes the other way, from slow into fast, the front swings away.
💡Only the speed and wavelength change in a new material. The frequency stays the same, and the frequency is what sets the color. So light doesn’t change color when it enters water or glass.
BMath
Snell’s Law
How much the light bends depends on the two indices and the angle. Both angles are measured from the normal. This is Snell’s law:
n₁ sin θ₁ = n₂ sin θ₂
Two quick rules give you the direction without any math. If n₂ is greater than n₁, the light bends toward the normal. If n₂ is less than n₁, it bends away. At normal incidence (0°), it doesn’t bend at all.
Example 1: air into glass
Given
Light in air (n = 1.00) hits glass (n = 1.50) at 30° from the normal
Solve
1.00 × sin 30° = 1.50 × sin θ₂
sin θ₂ = 0.500 ÷ 1.50 = 0.3333
θ₂ = sin⁻¹(0.3333) = 19.5°
✓ The light bends toward the normal, from 30° to 19.5°.
Example 2: water into air
Given
Light in water (n = 1.33) hits air (n = 1.00) at 30° from the normal
Solve
1.33 × sin 30° = 1.00 × sin θ₂
sin θ₂ = 1.33 × 0.500 = 0.665
θ₂ = sin⁻¹(0.665) = 41.7°
✓ The light bends away from the normal, from 30° to 41.7°.
Example 3: finding n from data
Given
A beam in air hits a block at θ₁ = 45°, and the angle inside the block is θ₂ = 28.3°
Solve
n₂ = n₁ sin θ₁ ÷ sin θ₂
n₂ = 1.00 × sin 45° ÷ sin 28.3°
n₂ = 0.7071 ÷ 0.4741 = 1.49
✓ The block has an index of 1.49, which is acrylic.
💡The index depends a little on the color of the light. Blue light is slowed a bit more than red, so it bends a bit more. That is dispersion, and it’s why a prism spreads white light into a rainbow.
CExample
Snell Explorer
Start with air above and glass below, and sweep the angle. Then swap them, so the light starts in the glass and heads for the air. Watch for the angle where the refracted ray vanishes.
Light comes down from the top material into the bottom one. Pick the two materials and sweep the angle of incidence. Try water to air, then slide the angle past the critical angle.
1.00 × sin 40° = 1.50 × sin θ₂ sin θ₂ = 0.429, so θ₂ = 25.4°
Material 2 is slower (a higher n), so the light bends toward the normal.
DConcept
The Bent Straw and the Shallow Pool
When light leaves water and goes into your eye, it bends away from the normal at the surface. But your brain assumes light travels in straight lines, so it traces the ray back along the direction it entered your eye. That backward line meets the vertical line through the object at a point higher than the object really is.
For a straight-down view, small angles make sin θ ≈ tan θ, and Snell’s law gives a simple result:
apparent depth ≈ actual depth × (n_observer ÷ n_object’s material)
For an observer in air looking into water, that is apparent depth ≈ actual depth ÷ 1.33.
A deep pool
Given
A pool is 2.0 m deep, and you look straight down into it (n_water = 1.33)
Solve
apparent depth = 2.0 ÷ 1.33
apparent depth = 1.5 m
✓ The pool looks 1.5 m deep, which is a quarter shallower than the real thing.
That is also the straw. The part of the straw under the water looks lifted toward the surface, while the part above the water is seen where it really is. The two don’t line up at the surface, so the straw looks broken. And it explains the coin trick: the water raised the apparent position of the coin up over the rim of the cup.
🔑This answers the unit’s first question. The straw is perfectly straight. Refraction only changes where the light from the underwater half appears to come from.
EMath
Critical Angle and Total Internal Reflection
When light goes from a high-n material to a low-n one, it bends away from the normal. Make the angle of incidence bigger and the refracted ray swings further, until it skims along the boundary at 90°. The angle of incidence that does this is the critical angle. Put θ₂ = 90° into Snell’s law:
θ_c = sin⁻¹(n₂ ÷ n₁) (only when n₁ > n₂)
Past the critical angle, no light can leave. All of it reflects back inside, following the law of reflection. This is total internal reflection, and it has two requirements: the light must be in the higher-n material, and it must strike the boundary at more than θ_c.
Critical angles into air
Given
Glass n = 1.50, water n = 1.33, diamond n = 2.42, each into air (n = 1.00)
Solve
Glass: θ_c = sin⁻¹(1.00 ÷ 1.50) = 41.8°
Water: θ_c = sin⁻¹(1.00 ÷ 1.33) = 48.8°
Diamond: θ_c = sin⁻¹(1.00 ÷ 2.42) = 24.4°
✓ The higher the index, the smaller the critical angle, and the easier it is to trap light.
Using it: prisms, fiber, and sparkle
A glass prism with a 45° face can replace a mirror. Light enters one face straight on, strikes the long face at 45°, and since 45° is more than glass’s 41.8° critical angle, it reflects totally and leaves through the other face, turned by 90°. Binoculars and periscopes use prisms like this because the reflection is perfect, with no silver coating to tarnish.
In a fiber optic cable, light enters a thin glass strand and strikes the wall at an angle bigger than the critical angle every time, so it bounces down the fiber for kilometers. And a diamond’s very small critical angle traps light, which bounces around inside before it finally escapes, which is much of what gives it sparkle.
🔑In Project 6-1-4, a prism can stand in for a mirror in a periscope. Check the arithmetic: a 45° face only works if 45° is greater than the material’s critical angle. For glass and acrylic it is. For water it isn’t, because water’s critical angle is 48.8°.
FWatch Out
Three Common Mix-Ups
⚠️“Light always bends toward the normal when it enters something new.” Only when the new material is slower (a higher n). Going into a faster material, such as glass into air, it bends away from the normal.
⚠️“Total internal reflection can happen going from air into glass.” It can’t. It needs the light to start in the higher-n material. Going from air into glass, light always gets through, bending toward the normal.
⚠️“Light changes color when it goes into water.” It doesn’t. The speed and wavelength change, but the frequency doesn’t, and frequency sets the color. Different colors bend by slightly different amounts, which is a different effect.
GConcept
Quick Check
Four questions. They are not graded. Answer them to find out what still needs work.
Q1Light goes from air into water at 40°. How does the refracted angle compare with 40°?
Q2What is the speed of light in a material with n = 2.0?
Q3Which situation can produce total internal reflection?
Q4A swimming pool is 2.0 m deep. About how deep does it look when you look straight down? (n = 1.33)
Score: 0 of 4
HConcept
Looking Ahead
A prism bends light at flat faces. Curve the faces, and you get something that bends all the rays toward a single point. That is a lens.