33-5 Reflection and Refraction#

Prompts

Sketch a light ray reflecting from an interface. Identify the incident ray, reflected ray, normal, angle of incidence, and angle of reflection. State the law of reflection.
Sketch refraction at an interface. What is Snell’s law? Define the index of refraction.
When light goes from medium 1 to medium 2: if \(n_2 > n_1\), which way does the ray bend? What if \(n_2 < n_1\)? What if \(n_2 = n_1\)?
What is chromatic dispersion? For red and blue light entering glass from air, which bends more? Which has the larger angle of refraction?
How are the primary and secondary rainbows formed? Why are they circular arcs? At what angles from the antisolar point do they appear?

Lecture Notes#

Overview#

Geometrical optics: light represented as straight-line rays (valid when wavelength \(\ll\) feature sizes).
At a boundary between two transparent media: reflected ray and refracted ray; both lie in the plane of incidence (containing incident ray and normal).
Reflection: angle of reflection = angle of incidence.
Refraction: Snell’s law relates angles via indices of refraction; bending occurs only at the interface.
Chromatic dispersion: index depends on wavelength; white light spreads into colors (e.g., prism, rainbow).

Reflection#

Law of reflection: The reflected ray lies in the plane of incidence and makes an angle with the normal equal to the angle of incidence \(\theta_1\):

(243)#\[ \theta_1' = \theta_1 \]

The incident ray, reflected ray, and normal all lie in one plane.

Poll: Mirror deflects beam—find angle

A mirror deflects a horizontal laser beam by 60°. What is the angle \(\phi\) (as defined in the figure)?

[FIGURE: Horizontal incident ray, mirror at angle, reflected ray deflected 60° from horizontal; label φ as the mirror angle or the angle between rays]

(A) 20°
(B) 30°
(C) 40°
(D) 45°

Poll: Two mirrors—beam returns parallel to incident

Two mirrors meet at angle \(\theta\). A laser beam is incident on one mirror. For the reflected beam to always be parallel to the incident beam (regardless of incident direction), what must \(\theta\) be?

(A) 45°
(B) 60°
(C) 90°
(D) 120°

Poll: Two mirrors at 80°—angle of reflected beam

Same setup: two mirrors at angle \(\theta = 80°\). What is the angle \(\phi\) of the reflected beam (relative to the incident beam)?

(A) 15°
(B) 20°
(C) 30°
(D) 45°

Refraction and Snell’s law#

Refraction is the change in direction when light crosses an interface. Bending occurs only at the interface; inside a uniform medium, light travels in straight lines.

Index of refraction \(n\): \(n = c/v\), where \(v\) is the speed of light in the medium and \(c\) is the speed in vacuum. Vacuum: \(n = 1\); air: \(n \approx 1\); typical glass: \(n \approx 1.5\); water: \(n \approx 1.33\).

Snell’s law:

(244)#\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]

where \(\theta_1\) is the angle of incidence (in medium 1) and \(\theta_2\) is the angle of refraction (in medium 2), both measured from the normal.

Direction of bending:

\(n_2\) vs \(n_1\)	Bending
\(n_2 = n_1\)	No bending; ray continues straight
\(n_2 > n_1\)	Ray bends toward the normal (\(\theta_2 < \theta_1\))
\(n_2 < n_1\)	Ray bends away from the normal (\(\theta_2 > \theta_1\))

Important

Refraction occurs only at an interface between two media—not in the interior of a uniform material.

Poll: Spear at fish—where to aim?

A fish swims below the water surface at point P. Where should a fisherman throw a spear to hit it?

[FIGURE: Fish at P in water; fisherman above; apparent image of fish (higher); actual ray path from fish to eye]

(A) Toward where he sees the fish
(B) Above where he sees the fish
(C) Below where he sees the fish

Poll: Laser at fish—where to point?

Same setup. The fisherman points a laser to hit the fish. Where should he aim?

(A) Toward where he sees the fish
(B) Above where he sees the fish
(C) Below where he sees the fish

Poll: Fish sees fisherman—where?

A fisherman stands above the water. A fish at P looks up at the fisherman’s eye. Where does the fish see the eye?

(A) Exactly where it really is
(B) Above where it really is
(C) Below where it really is

Poll: Fix the incorrect refraction sketch

The sketch shows light incident from air onto a blue glass plate. The drawing is incorrect. What would you suggest to fix it?

(A) Make \(\theta_2\) smaller
(B) State that \(n_2 < n_1\) (as drawn)
(C) Curve the ray in the lower medium
(D) Measure all angles from the perpendicular
(E) None of these

Poll: Light exits glass—which direction?

Light enters a glass plate, travels through it, and exits. Which direction does the ray take when it exits?

[FIGURE: Ray entering glass (bends toward normal), traveling through, exiting. Options as arrows.]

(A) Same as incident
(B) Parallel to incident but offset
(C) Bends away from normal (as drawn)
(D) Bends toward normal

Poll: Reaching for object in stream—which sketch?

Friends miss a gold object at the bottom of a stream on their first attempt. Which sketch correctly shows why they were reaching in the wrong place?

[FIGURE: Multiple sketch options. Correct one shows apparent image above actual object; they reached toward apparent position (too high); should reach lower.]

(A) Sketch showing rays and apparent vs actual position
(B) Sketch showing they reached above the object
(C) Sketch showing they reached at the apparent position
(D) Sketch showing they reached below the object

Example: Pool depth from shadow

A 4.0 m wide swimming pool is filled to the top. The bottom becomes completely shaded when the sun is 20° above the horizon. How deep is the pool?

Solution: The shadow extends from the far edge. For the near edge to be at the shadow boundary, the ray from the sun at 20° grazes the near edge and hits the far bottom corner. So \(\tan 20° = 4.0/d\) \(\Rightarrow\) \(d = 4.0/\tan 20° \approx 11\) m.

Chromatic dispersion#

The index of refraction \(n\) depends on wavelength (except in vacuum). Shorter wavelengths (blue) typically have larger \(n\) than longer wavelengths (red).

Chromatic dispersion: When white light (many wavelengths) refracts at an interface, different colors bend by different amounts—the light spreads into a spectrum.

Direction	Blue vs red bending
Air → glass	Blue bends more (smaller \(\theta_2\))
Glass → air	Blue bends more (larger \(\theta_2\))

A prism enhances the separation by refracting at two surfaces.

Rainbows#

Primary rainbow: Sunlight enters a raindrop, reflects once from the inner surface, and refracts out. Colors appear at 42° from the antisolar point \(A\) (the point directly opposite the Sun).

Secondary rainbow: Two reflections inside the drop; colors at 52° from \(A\). Order of colors is reversed compared to the primary; secondary is wider and dimmer.

Circular arcs: Drops at 42° (or 52°) from \(A\) in any direction contribute to the rainbow. The locus of such directions is a cone—hence a circular arc. The top of a rainbow is never more than 42° above the horizon.

Summary#

Reflection: \(\theta_1' = \theta_1\); both rays in plane of incidence.
Refraction: Snell’s law \(n_1\sin\theta_1 = n_2\sin\theta_2\); bending only at interface.
Bending: \(n_2 > n_1\) → toward normal; \(n_2 < n_1\) → away from normal.
Chromatic dispersion: \(n\) depends on \(\lambda\); blue bends more than red.
Rainbows: Primary at 42° from antisolar point (one reflection); secondary at 52° (two reflections, reversed colors).