Light Reflection and Refraction

26.2 Light as an Electromagnetic Wave

As established in Chapter 24, electromagnetic waves consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. Light is transverse: the field oscillations are perpendicular to the direction of travel. This transverse character is what makes polarization possible (Chapter 28).

Because light is an electromagnetic wave, it obeys all the general wave relationships

field oscillates 750 trillion times per second.

Unlike sound, light requires no medium. It propagates freely through the vacuum of

meter is now based on this exact value). When light enters a transparent medium, it slows down. The ratio c/v gives the index of refraction n of that medium (Section 26.9).

observing that the orbital period of Io (a moon of Jupiter) appeared shorter when Earth was approaching Jupiter and longer when receding. He inferred this was due to the changing time for light to cross the varying Earth–Jupiter distance, and estimated c

2.2 × 10⁸ m/s — within 25% of the correct value, using only a telescope and a

pendulum clock.

Physics: Foundations of Physical Science

26.3 Propagation of Light in Vacuum and Media

In a uniform, isotropic medium (one with the same properties in all directions), light travels in straight lines. This is the basis of the ray model of light: we represent a beam of light as a geometric ray — a directed line segment pointing in the direction of propagation. Ray optics (also called geometric optics) is an excellent approximation when the wavelength is much smaller than any obstacles or apertures the light encounters, which is true for most everyday optical situations.

In vacuum, the ray model is exact: light always travels in a straight line at speed c. In a homogeneous medium (uniform composition and density), light also travels in straight

changes only when the medium changes.

In a non-uniform medium (such as the atmosphere with temperature gradients, or glass with stress patterns), the index of refraction varies from point to point. Rays follow curved paths through such media, continuously bending toward regions of higher n (slower speed). This is the physics behind mirages, the twinkling of stars, and the atmospheric bending that keeps radio signals from escaping into space.

500 nm), geometric optics works for objects larger than about 0.1 mm. For smaller features, wave optics (Chapter 28) is required.

26.4 Reflection of Light

Reflection is the redirection of light back into the medium from which it came when it strikes a surface or interface. All surfaces reflect some light; the fraction reflected depends on the optical properties of both media and the angle of incidence.

Specular reflection occurs from smooth, polished surfaces (mirrors, still water, polished metal). Parallel incoming rays reflect as parallel outgoing rays, producing a clear, undistorted image. The surface appears shiny.

Diffuse reflection occurs from rough surfaces (matte paper, walls, skin, unpolished wood). Parallel incoming rays scatter in many directions because each small patch of the surface faces a slightly different direction. No coherent image is formed. The surface appears dull or matte. Most surfaces in everyday life produce diffuse reflection, which is why we can see them from many viewing angles without them appearing as mirrors.

Partial reflection occurs at the interface of two transparent media (air–glass, water–air).

Some light is reflected back, and some is transmitted into the second medium. The Physics: Foundations of Physical Science fraction reflected depends on the index of refraction difference and the angle of incidence. For normal incidence at an air–glass interface (n_glass 1.5), about 4% of the light is reflected and 96% transmitted. This partial reflection is why windows act as both transparent surfaces and weak mirrors.

26.5 The Law of Reflection

The law of reflection governs the direction of specular reflection. It states that the angle of reflection equals the angle of incidence, both measured from the normal to the reflecting surface at the point of incidence.

The normal is an imaginary line perpendicular to the surface at the point where the ray strikes. It is always measured from the normal, not from the surface. A common mistake is to measure angles from the surface; this gives the complement of the correct angle.

The incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane, called the plane of incidence. The reflected ray is on the opposite side of the normal from the incident ray, in the same plane. This coplanarity is part of the law of reflection.

Example — Reflection angle A ray strikes a mirror at 35° to the surface (i.e., 90°

between incident and reflected rays is 55° + 55° = 110°.

Always measure reflection and refraction angles from the normal, never from the surface. If a problem gives the angle from the surface, subtract from 90° to get the angle from the normal before applying any formula.

26.6 Plane Mirrors and Image Formation

A plane mirror is a perfectly flat, specularly reflecting surface. When an object is placed in front of a plane mirror, reflected rays diverge from the mirror in a way that appears to an observer as if they are coming from a point behind the mirror. This apparent source location is the image.

The image formed by a plane mirror has four key properties that follow directly from the law of reflection applied geometrically. First, the image is virtual: the reflected rays Physics: Foundations of Physical Science diverge and do not actually converge at the image location; the image exists only where rays appear to originate when extended backward. Second, the image is upright (not inverted). Third, the image is the same size as the object (magnification = 1).

Fourth, the image distance behind the mirror equals the object distance in front of it: d_image = d_object.

Plane mirrors produce left–right reversal, more precisely described as a front–back reversal (the image’s depth axis is reversed relative to the object’s). When you raise your right hand, the image appears to raise its left hand, because the axis pointing toward the mirror is reversed while the lateral axes are unchanged.

mirrors at 90°, three images appear. For mirrors facing each other (parallel), an infinite series of images is produced, each dimmer than the last as the intensity is reduced by repeated partial reflection.

A virtual image cannot be projected onto a screen because the light rays do not actually pass through the image location. A real image, formed by converging rays, can be projected (as in a movie projector or camera). The mirror in a bathroom forms a virtual image.

26.7 Refraction of Light

Refraction is the change in direction of a ray of light as it crosses the boundary between two media with different speeds of light. When light moves from a medium in which it travels fast into one where it travels slower, the ray bends toward the normal.

When it moves into a faster medium, the ray bends away from the normal.

Refraction occurs because different parts of the wave front reach the boundary at

new medium first slows down (or speeds up) while the rest of the front continues at the original speed. This differential speed change rotates the wave front, changing the direction of propagation. (This is the wave explanation of Snell’s Law.) Everyday examples of refraction: a spoon appears bent in a glass of water (the underwater portion is displaced toward the surface in your perception because rays from it bend as they leave the water); a pool appears shallower than it is (light from the bottom bends away from the normal as it enters air, making the bottom appear higher); the apparent position of the setting Sun is slightly above the horizon even after the Sun has geometrically set (atmospheric refraction bends the rays over the horizon).

Refraction is simultaneously transmission: when light is refracted, it is transmitted into Physics: Foundations of Physical Science the second medium. The refracted ray and reflected ray coexist at any real interface; their relative intensities depend on the materials and the angle.

26.8 Why Light Bends Between Media

The physical reason for refraction is the change in wave speed. When a wave enters a medium where it travels more slowly, each part of the wave front that crosses the boundary first is slowed before the rest of the front reaches the boundary. The wave front pivots around the point of contact, changing the direction of propagation.

An analogy: imagine a column of soldiers marching in a line across a field, hitting a muddy region at an angle. The soldiers who reach the mud first slow down while the others continue at full speed. The formation rotates — the direction of the column’s advance changes. This is exactly what happens to a wave front entering a slower medium.

hf) is conserved as it crosses the interface; the frequency, set by the source, is a fixed

wavelength. The color of light perceived by the eye corresponds to frequency (not wavelength), which is why the color does not change even though the wavelength does when light enters glass.

At any boundary: frequency is unchanged (set by source). Speed changes

normal incidence).

26.9 Index of Refraction

The index of refraction (symbol n) of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in the medium:

1.00 (usually approximated as

exactly 1 for most calculations). Water has n 1.33, meaning light travels at 343/1.33

2.26 × 10⁸ m/s in water. Dense flint glass has n up to about 1.9.

Physics: Foundations of Physical Science Medium Index of Refraction (n) Speed of Light (m/s) Vacuum

1.000 (exact)

2.998 × 10⁸

Air (STP) 1.000293

2.997 × 10⁸

Water (20°C) 1.333

2.249 × 10⁸

Ice 1.309

2.291 × 10⁸

Crown glass 1.52

1.97 × 10⁸

Dense flint glass 1.89

1.59 × 10⁸

Diamond 2.42

1.24 × 10⁸

Sapphire 1.77

1.69 × 10⁸

Table 26.2

The index of refraction varies slightly with wavelength — a property called dispersion (Section 26.12). Materials with higher indices refract light more strongly and have more dispersion. Diamond’s high index (2.42) and high dispersion are responsible for both its brilliant sparkle and its fire (the rainbow colors it scatters).

26.10 Snell's Law

Snell's Law quantifies the direction of the refracted ray. It was established empirically by Willebrord Snell in 1621 and derived theoretically by Huygens from wave principles. It states:

ray bends toward the normal. If n₂ < n₁ (entering a less dense medium), the ray bends away from the normal.

Step 1: Identify: n₁, θ₁, n₂. Draw the normal at the point of incidence.

Physics: Foundations of Physical Science Step 4: Check:

> θ₁.

Example — Snell's Law calculation Light in air (θ₁ = 40°) enters water (n = 1.33).

ray bends toward the normal as expected (water is denser: n₂ > n₁).

Example — Snell's Law reversed Light in glass (n = 1.52, θ₁ = 25°) exits into air.

ray bends away from the normal as expected.

26.11 Total Internal Reflection

When light travels from a denser medium (higher n) into a less dense medium (lower

refracted ray would need to travel at 90° from the normal — along the surface. For any

reflected back into the denser medium. This is total internal reflection (TIR).

(n₁ > n₂) For glass (n = 1.52) to air (n = 1.00): θ_c = arcsin(1.00/1.52) = arcsin(0.658) = 41.1°.

Any ray inside the glass striking the glass–air surface at more than 41.1° from the normal is totally reflected.

very small critical angle means that light entering the gem from almost any direction is trapped inside and undergoes many internal reflections before finally escaping from one of the carefully angled facets. This is why diamonds sparkle so brilliantly: the facets are cut at angles specifically designed to direct internally reflected light upward toward the observer.

Optical fibers exploit TIR to guide light around curves without loss. A glass or plastic fiber with a high-index core and a lower-index cladding confines light within the core by TIR at every surface the ray reaches. As long as the bend radius of the fiber is not too sharp, virtually 100% of the light is guided along the fiber. This enables high-speed internet connections (light pulses carrying data), medical endoscopes (imaging inside the body), and laser delivery for surgery.

Physics: Foundations of Physical Science

John Tyndall demonstrated light guiding by TIR in 1870, directing a beam of light through a curved stream of water and showing that the light followed the curve. This demonstration — light appearing to bend around a corner — was the foundational experiment of fiber optic technology, over a century before fiber optics became commercially practical.

26.12 Dispersion and Prisms

Dispersion is the variation of the speed of light (and therefore the index of refraction) with wavelength within a medium. Most transparent materials have higher indices of refraction for shorter wavelengths (violet) than for longer wavelengths (red). This means violet light bends more than red light when refracted.

When white light (a mixture of all visible wavelengths) passes through a prism, each wavelength refracts by a slightly different angle. After two refractions (entering and exiting the prism), the wavelengths are spread out into a spectrum. Violet light emerges at the largest angle from the original direction; red light at the smallest. The result is the familiar rainbow of colors produced by a glass prism.

Natural rainbows arise from the same physics. Sunlight enters raindrops, refracts at the entry surface, reflects off the back of the drop (TIR or partial reflection), and refracts again on exit. Each wavelength exits at a slightly different angle. Red light exits at about 42° from the anti-solar point (the point directly opposite the Sun from the observer); violet at about 40°. The observer sees a circular arc with red on the outside and violet on the inside.

Dispersion is a practical challenge in optical systems: a simple lens refracts different colors to slightly different focal points,

colored fringing around images (chromatic aberration). Telescope and camera lens designers use combinations of glasses with different dispersion characteristics (achromatic doublets) to correct this, bringing at least two wavelengths to the same focus.

26.13 Atmospheric Refraction and Mirages

The atmosphere is not uniform: air density (and therefore index of refraction) decreases with altitude. Rays from astronomical objects outside the atmosphere are continuously

atmospheric refraction makes celestial objects appear higher above the horizon than they actually are.

The effect is largest near the horizon, where the air path is longest. The Sun and Moon Physics: Foundations of Physical Science appear about 0.5° higher than their actual positions, which equals almost exactly one solar or lunar diameter. This means that when you see the bottom of the Sun touching the horizon, the entire Sun has actually set geometrically — you are seeing it above its true position due to atmospheric refraction.

Mirages are optical illusions caused by atmospheric refraction in non-standard temperature gradients. In a hot desert (or over a hot road), the air near the surface is much hotter and less dense than the air above, creating a gradient in which the index of refraction decreases toward the ground. Rays from the sky travel downward and then curve back upward due to this gradient, creating TIR-like behavior at the hot surface layer. The observer sees an image of the sky below the horizon — which the brain interprets as a pool of water reflecting the sky.

An inferior mirage (the most common type, seen on hot roads) shows an image below the actual object. A superior mirage occurs when cold air lies below warmer air (a temperature inversion): rays curve downward, bringing images of objects over the horizon into view. Mirages of ships and coastlines at extreme distances have been observed in the Arctic and over large cold lakes.

26.14 Common Mistakes in Geometric Optics

Measuring angles from the surface instead of the normal. The law of reflection and Snell’s Law both require angles measured from the normal. If given the angle from the surface, subtract from 90° before using any formula.

Forgetting that Snell's Law is symmetric. You can apply it in either direction: n₁

the larger n always has the smaller angle.

Applying TIR to the wrong direction. TIR only occurs when light tries to move from a higher-n medium into a lower-n medium. Light going from air into glass cannot undergo TIR at the air–glass boundary.

frequency (unchanged); wavelength in the medium is shorter than in vacuum.

applies only approximately. In precise calculations, n must be specified for the wavelength in question. This is why Snell's Law gives slightly different angles for red and violet light.

at visible wavelengths. Light slows down, never speeds up, when entering a material medium.

Physics: Foundations of Physical Science

When applying Snell's Law, always identify which medium is 1 (incident) and which is 2 (refracted) before substituting. A common error is to swap n₁ and n₂ while keeping the correct angles, producing an inverted (and physically wrong) result.

Chapter Summary

= c/n in a material medium, where n is the index of refraction.

Visible light spans 380–700 nm in wavelength. Color corresponds to frequency (unchanged at boundaries); wavelength changes with speed: λ = v/f = λ₀/n.

in all directions from rough surfaces.

behind the mirror.

Refraction: direction change at a speed boundary. Snell's Law: n₁ sinθ₁ = n₂ sinθ₂.

The ray bends toward the normal when entering a denser medium.

2.42) bends light more than water (n = 1.33).

TIR occurs when light in a denser medium hits a boundary at more than the critical angle: θ_c = arcsin(n₂/n₁). Exploited in optical fibers and gem cutting.

different angles. Responsible for prism spectra and rainbows.

Atmospheric refraction: non-uniform air bends rays, making objects near the horizon appear higher than their true positions. Mirages arise from extreme temperature-gradient refraction.

Key Terms

electromagnetic radiation: Self-sustaining oscillation of electric and magnetic fields propagating through space; light is a form of electromagnetic radiation.

constant of nature; the maximum speed in the universe. visible spectrum: The range of electromagnetic wavelengths detectable by the human Physics: Foundations of Physical Science eye: approximately 380–700 nm. ray model (geometric optics): Representation of light as straight directed lines (rays) perpendicular to wave fronts; valid when wavelength ≪obstacle size. specular reflection: Mirror-like reflection from a smooth surface; parallel incoming rays produce parallel outgoing rays; θ_r = θ_i. diffuse reflection: Scattering of light in many directions from a rough surface; no coherent image formed. normal: A line perpendicular to a surface at the point of incidence; reference line for measuring angles of incidence, reflection, and refraction.

measured from the normal. virtual image: An image from which reflected or refracted rays appear to diverge but through which no light actually passes; cannot be projected onto a screen.

with different wave speeds; governed by Snell’s Law.

in a medium; always ≥1.

indices at a refracting boundary. total internal reflection (TIR): Complete reflection of light back into a denser medium when the angle of incidence exceeds the critical angle θ_c = arcsin(n₂/n₁).

TIR occurs. optical fiber: A thin, flexible glass or plastic fiber that guides light along its length by total internal reflection at the core–cladding boundary. dispersion: The variation of index of refraction with wavelength; causes different colors to refract at different angles and separates white light into a spectrum. chromatic aberration: The failure of a simple lens to focus different wavelengths to the same point due to dispersion. mirage: An optical illusion caused by atmospheric refraction in a temperature gradient, producing images of objects in incorrect positions.

Concept Review Questions

Physics: Foundations of Physical Science Answer in complete sentences. No calculations required unless specified.

1. Why does a swimming pool appear shallower than it actually is? Draw a ray

diagram to support your explanation.

2. Explain why the color of light does not change when it crosses a glass surface,

even though the wavelength does. What property of the light corresponds to color?

3. A smooth lake surface acts as a mirror in calm weather but not in choppy

weather. Explain the physics difference between these two cases.

4. Explain why optical fibers can guide light around curves. What would happen if

the fiber were bent too sharply?

5. Why do rainbows always appear as arcs centered on the anti-solar point? Why is

red on the outside and violet on the inside?

6. Total internal reflection can only occur from inside the denser medium. Explain

why TIR cannot happen when light goes from air into glass.

7. A diamond is cut with many small flat facets at specific angles. Explain using TIR

why this maximizes the brilliance of the stone.

8. Explain what a mirage is. Is the mirage an image of water on the road, or an

image of the sky? How does the physics of refraction in a temperature gradient produce it?

Reflection Problems

9. A ray of light strikes a flat mirror at 28° from the surface. (a) What is the angle

of incidence? (b) What is the angle of reflection? (c) What is the angle between the incident and reflected rays?

10. An object stands 0.40 m in front of a plane mirror. Where is the image? Is it real

or virtual? Is it upright or inverted?

11. Two plane mirrors are arranged at 60° to each other. How many images of an

12. A ray travels horizontally and strikes a mirror tilted 25° from the vertical. Find the

direction of the reflected ray relative to the horizontal.

13. A person 1.70 m tall stands 2.0 m from a plane mirror. What is the minimum

mirror length needed so the person can see their full reflection? Does the distance from the mirror matter? Explain.

Physics: Foundations of Physical Science

Refraction Problems

14. Light in air strikes water (n = 1.33) at 50° from the normal. Find the angle of

refraction in the water.

15. Light inside glass (n = 1.52) hits the glass–air surface at 20° from the normal.

Find the angle of refraction in air.

16. A ray of light enters a flat glass slab (n = 1.60) from air at 45° from the normal.

(a) Find the angle inside the glass. (b) Find the angle as it exits the opposite parallel face back into air. Explain why the exit angle equals the original entry angle.

17. What is the speed of light in diamond (n = 2.42)?
18. Light travels from water (n = 1.33) into ice (n = 1.31). A ray hits the boundary at

30°. Find the refracted angle. Does the ray bend toward or away from the normal?

Snell's Law Exercises

19. Find the critical angle for the glass–air interface (n_glass = 1.52, n_air = 1.00).
20. Find the critical angle at a glass–water interface (n_glass = 1.52, n_water = 1.33).

Is TIR possible at this interface? Under what conditions?

21. A diamond ring is submerged in water. Find the critical angle for diamond–water

angle (24.4°). What implication does this have for the diamond’s appearance underwater?

22. White light enters a crown glass prism (n_red = 1.513, n_violet = 1.532) at 30°.

Find the refracted angles for red and violet light. What is the angular separation (dispersion angle) between them inside the glass?

Applied Optics Problems

Physics: Foundations of Physical Science

23. A fish at depth 1.20 m in water looks straight up. What is the radius of the

circular patch of the surface through which it can see the entire above-water

24. An optical fiber has a core with n = 1.62 and cladding with n = 1.50. Find the

critical angle. What is the maximum angle (from the fiber axis) at which a ray can enter the fiber end face and still undergo TIR? (This is the acceptance angle:

n_clad²).)

25. The Sun is actually 0.53° below the geometric horizon when an observer sees it

touching the horizon due to atmospheric refraction. How long (in minutes) has the Sun actually set before it appears to touch the horizon? (Earth rotates 360°

26. A flat glass window (n = 1.52, thickness 4.0 mm) has a ray striking at 40°. Find

the perpendicular displacement of the ray as it exits the far side parallel to the

Before moving on to Chapter 27, confirm you can do each of the following.

I can… See Section State the speed of light in vacuum and explain what the index of refraction measures 26.1–26.2 Explain when the ray model is valid and when wave optics is needed 26.3 Distinguish specular from diffuse reflection and explain partial reflection 26.4 Apply the law of reflection and measure angles correctly from the normal 26.5 Describe the four image properties of a plane mirror 26.6

argument 26.7–26.8 Calculate the speed of light in a medium given its index of 26.9 Physics: Foundations of Physical Science refraction Apply Snell's Law to find angles of refraction; identify toward/away from normal 26.10 Derive and apply the critical angle formula for TIR 26.11 Explain dispersion and how it produces rainbows and chromatic aberration 26.12

26.13 Identify and avoid the most common errors in geometric optics problems 26.14