AP Physics 2 · Topic 13.1

Topic 13.1: Reflection

Unit 13: Geometric Optics12-15% of the multiple-choice section

The law of reflection says the angle between the incident ray and the normal equals the angle between the reflected ray and the normal. The normal is the line perpendicular to the surface, so measure from that line, never from the mirror face. Smooth surfaces reflect uniformly, rough ones scatter.

AP Physics: Unit 13 (topics 13.1 Reflection). AP Physics 2 Unit 13, Topic 13.1. Two learning objectives. 13.1.A asks students to describe light as a ray, supported by 13.1.A.1 (a light ray is a straight line perpendicular to the wavefront of a light wave, pointing in the direction of travel) with sub-statements 13.1.A.1.i (rays determine behavior in geometric optics, where the wave nature can be neglected), 13.1.A.1.ii (rays are not sufficient to understand the spreading of light; in interference and diffraction the wave nature is important) and 13.1.A.1.iii (a laser is a common source of a single coherent, monochromatic beam that can be modeled as a ray; its wave nature is considered in Unit 14), plus 13.1.A.2 (ray diagrams depict the path of light before and after an interaction with matter). 13.1.B asks students to describe the reflection of light from a surface, supported by 13.1.B.1 (light incident on a surface can be reflected), 13.1.B.2 (the law of reflection, relevant equation theta_i = theta_r, with the angles measured between each ray and the normal, the line perpendicular to the surface), 13.1.B.3 (diffuse reflection from a rough surface, because the surface normal varies over the illuminated area) and 13.1.B.4 (specular reflection from a smooth surface, because the surface normal has an approximately constant direction over the area the light strikes). The topic prints no boundary statement; the only boundary statement in Unit 13 sits under Topic 13.2 and limits the study of mirrors to plane mirrors, convex spherical mirrors, and concave spherical mirrors. Suggested skills are 1.A, 2.B, 2.C and 3.B, listed identically on the topic page and in the Unit at a Glance table. Unit 13 is weighted at 12 to 15 percent of the multiple-choice section across a suggested 8 to 12 class periods.

What Topic 13.1 requires

Topic 13.1 carries two learning objectives and prints no boundary statement.

13.1.A, describe light as a ray. Essential knowledge 13.1.A.1 defines a light ray as a straight line that is perpendicular to the wavefront of a light wave and points in the direction of travel of the wave. Three sub-statements qualify it. 13.1.A.1.i says light rays can be used to determine the behavior of light in geometric optics, where the wave nature of light can be neglected. 13.1.A.1.ii says rays are not sufficient to understand the spreading of light, and that in interference and diffraction the wave nature of the light is important. 13.1.A.1.iii says a laser is a common source of a single coherent, monochromatic beam of light that can be modeled as a ray, and that the wave nature of lasers will be considered in Unit 14. Then 13.1.A.2 adds that ray diagrams depict the path of light before and after an interaction with matter.

13.1.B, describe the reflection of light from a surface. Four statements sit under it. 13.1.B.1: light that is incident on a surface can be reflected. 13.1.B.2: the law of reflection, with the relevant equation $\theta_i = \theta_r$ . 13.1.B.3: diffuse reflection. 13.1.B.4: specular reflection. Sections below take each of these in the CED's own terms.

The suggested skills the CED lists for Topic 13.1 are 1.A (create diagrams, tables, charts, or schematics to represent physical situations), 2.B (calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway), 2.C (compare physical quantities between two or more scenarios or at different times and locations in a single scenario), and 3.B (apply an appropriate law, definition, theoretical relationship, or model to make a claim). The Unit at a Glance table and the topic page agree on all four.

Unit 13, Geometric Optics, is weighted at 12 to 15 percent of the multiple-choice section across a suggested 8 to 12 class periods. Only one topic in the unit prints a boundary statement, and it is not this one: the boundary statement under Topic 13.2 limits the study of mirrors to plane mirrors, convex spherical mirrors, and concave spherical mirrors.

A light ray is a modeling choice, and the CED says where it fails

A ray is not a thing light is made of. It is a line drawn perpendicular to the wavefront, pointing the way the wave travels, and 13.1.A.1 defines it exactly that way. Geometric optics is the part of the course where you are allowed to forget the wave and keep only the line.

The CED is unusually direct about the price of that simplification. 13.1.A.1.i grants the license: rays determine the behavior of light in geometric optics, where the wave nature of light can be neglected. 13.1.A.1.ii takes it back for a specific class of problem: rays are not sufficient to understand the spreading of light, and in interference and diffraction the wave nature of the light is important. So a ray diagram will tell you where an image sits and how big it is. It will not tell you why a beam through a narrow slit fans out, or why two beams can cancel.

That split is the reason Unit 13 and Unit 14 are separate units rather than one optics unit. Unit 13 is the ray model. The wave model comes later.

One more piece of vocabulary the CED bothers to define: a laser, per 13.1.A.1.iii, is a common source of a single coherent, monochromatic beam of light that can be modeled as a ray. That is why almost every optics lab in the course starts with a laser pointer. It is the closest physical object to the abstraction you are drawing. The CED then flags that lasers have a wave nature too, and defers it to Unit 14.

Finally, 13.1.A.2: ray diagrams depict the path of light before and after an interaction with matter. Note both halves. A ray diagram that shows only the incoming beam is not finished, and one that shows only the outgoing beam has hidden the reasoning.

The law of reflection, and the line you measure from

Convention for this whole page: every angle is measured from the normal, the line perpendicular to the surface at the point where the ray lands. Never from the surface itself. This is not a house preference. It is how 13.1.B.2 words the law.

13.1.B.1 sets the stage in seven words of physics: light that is incident on a surface can be reflected. Then 13.1.B.2 states the law of reflection as the angle between the incident ray and the normal (the line perpendicular to the surface) being equal to the angle between the reflected ray and the normal, and prints the relevant equation:

\theta_i = \theta_r

The parenthetical definition of the normal is part of the statement, and it is the part that decides whether you get the arithmetic right. A beam that grazes along a mirror at 5 degrees above the surface has an angle of incidence of 85 degrees, not 5.

Two things the equation does not say, which you still need:

The incident ray, the reflected ray and the normal all lie in one plane. The CED does not print this as a separate statement, so treat it as what the ray diagram in 13.1.A.2 shows rather than as a quotable rule.
The reflected ray leaves on the other side of the normal from the incoming ray. If both rays end up on the same side of the normal in your sketch, the diagram is wrong even when the two angles are equal.

The law holds at every point of every surface, curved or flat. A curved mirror does not obey a different law; it obeys this one at a surface whose normal direction changes from point to point. Hold that thought, because it is exactly what separates the next two sections.

Specular and diffuse reflection are one law on two surfaces

The CED defines both, and it defines them by the normal, not by how the surface looks.

13.1.B.4: specular reflection is the reflection of light from a smooth surface and results in light uniformly reflected from the surface, because the line normal to the surface has an approximately constant direction over the area the light strikes.

13.1.B.3: diffuse reflection is the reflection of light from a rough surface and results in light reflected in many different directions, because the line normal to the surface varies over the area over which the light is incident.

Read the two causal clauses side by side and the physics is one sentence: parallel incoming rays stay parallel when every normal points the same way, and spray in all directions when the normals do not. Each individual ray still satisfies $\theta_i = \theta_r$ in both cases. Diffuse reflection is not a failure of the law of reflection, and a free-response answer that says light scatters because rough surfaces break the law has lost the point.

This is why you can read this page from any angle but cannot see your own face in it. Paper is rough on the scale of a wavelength of light, so its normals wander and it sends light everywhere. A mirror is smooth on that scale, so it sends the light one way and preserves the pattern of directions the light arrived with. That preserved pattern is what an image is, which is where Topic 13.2 picks up.

The word the CED uses for specular reflection is "uniformly", and the qualifier is "approximately constant". Both are deliberate. No real surface has a perfectly constant normal, so real mirrors scatter a little and real paper has a faint sheen at grazing angles.

Plane mirrors: what 13.1 sets up and what 13.2 owns

Students often meet plane mirrors inside a reflection lesson, so it is worth being precise about where the CED files the plane-mirror facts, because the essential knowledge codes matter on a free-response justification.

Topic 13.1 gives you reflection. It says nothing about images. Every image statement in the unit, including the ones about plane mirrors, sits under Topic 13.2, Images Formed by Mirrors:

13.2.A.3: the focal point of a plane mirror is an infinite distance from the mirror.
13.2.A.7.ii: the distance between the image formed and a plane mirror is equal to the distance between the object and the plane mirror.
13.2.A.6: a virtual image is formed by a mirror when reflected light rays diverge such that they appear to have originated from a common point.

What Topic 13.1 does contribute is the mechanism underneath all three. Apply $\theta_i = \theta_r$ to two rays leaving the same point on an object, extend the reflected rays backward behind the mirror, and they meet. That meeting point is the image, and the geometry that puts it as far behind the mirror as the object is in front is nothing more than the law of reflection applied twice.

The unit's Developing Understanding section names the misconception this addresses: students will be challenged to confront their misconceptions about light, including why objects are not always located where they are seen. A plane mirror is the cleanest example. Nothing is behind the mirror. The light only behaves as though something is.

On sign conventions, be careful about what you carry into this topic. The CED asserts in 13.2.A.7.i that the locations of a mirror's focal point, an object near the mirror, and the image of the object formed by the mirror follow sign conventions that are used to determine those locations relative to the mirror itself. It never prints what those conventions are, and no sign convention for optics appears in the list of conventions on the exam equation sheet. Topic 13.1 needs none of this: the law of reflection is an equality between two positive angles.

Drawing the diagram the CED asks for (skill 1.A)

Skill 1.A is listed first for this topic: create diagrams, tables, charts, or schematics to represent physical situations. On a reflection question that means a labeled ray diagram, and the labels are half the marks.

A diagram that earns credit has five parts:

The surface, drawn as a line, with hatching or shading on the back side so the reader knows which side the light is on.
The point of incidence marked.
The normal, drawn as a dashed line perpendicular to the surface at that point. Dashed, because it is a construction line and not a light path.
The incident ray and the reflected ray, each with an arrowhead showing the direction of travel.
Both angles marked between a ray and the normal, with the equality stated.

The two habits that cost marks are drawing the normal at some other point on the surface, and marking the angles to the surface because that is what a protractor sits flat against. If your protractor is easier to read against the surface, measure that angle and then subtract from 90 before you write anything down. The same discipline pays off in two-dimensional vector work, where the whole problem also turns on stating which line an angle is measured from.

For a curved mirror the recipe is unchanged except that the normal at the point of incidence is along the radius, pointing at the center of curvature. That construction belongs to Topic 13.2, but if you can draw the normal on a curved surface using only Topic 13.1, the next topic gets much shorter.

What the equation sheet prints for reflection

Nothing.

The AP Physics 2 equation sheet groups its optics content under Waves, Sound, and Optics, and that group holds 15 equations. Four of them are cited as relevant equations inside Unit 13:

$n = \dfrac{c}{v}$ and $n_1 \sin\theta_1 = n_2 \sin\theta_2$ , both from Topic 13.3.
$\dfrac{1}{s_i} + \dfrac{1}{s_o} = \dfrac{1}{f}$ and $|M| = \left|\dfrac{h_i}{h_o}\right| = \left|\dfrac{s_i}{s_o}\right|$ , both from Topics 13.2 and 13.4.

The remaining 11 belong to waves, sound, interference and diffraction, which is Unit 14 material. $\theta_i = \theta_r$ is not among the 15. The CED prints it as a relevant equation inside 13.1.B.2, and the CED is explicit that not every equation in the course framework appears on the exam equation sheet.

That is not a problem, and it is worth understanding why rather than adding a line to a flashcard. The law of reflection is a statement of equality between two angles that you have already been asked to define. There is nothing to substitute into. What you are really being asked to remember is the definition of the normal, and the sheet does not print definitions.

One related detail from the same sheet: the variable list for the Waves, Sound, and Optics group defines $\theta$ simply as "angle". It does not say from where. The measuring convention lives in the CED's essential knowledge, not on the sheet, so the sheet cannot rescue you if you have learned the law with the angles measured from the surface.

How Topic 13.1 shows up on the exam

The AP Physics 2 exam is 3 hours long: Section I is 42 multiple-choice questions in 85 minutes for 50 percent of the score, and Section II is 4 free-response questions in 95 minutes for the other 50 percent. The four free-response questions are Mathematical Routines, Translation Between Representations, Experimental Design and Analysis, and Qualitative/Quantitative Translation. A four-function, scientific, or graphing calculator is allowed on both sections. The 12 to 15 percent weighting quoted for Unit 13 applies to the multiple-choice section.

The three non-diagram skills listed for this topic tell you the shapes the questions take.

2.B, calculations. One angle in, one angle out, usually with a conversion between the surface and the normal buried in the wording. These are fast marks and the arithmetic is trivial, which is precisely why the question writers hide the trap in the phrasing rather than in the numbers.

2.C, comparisons. Two scenarios, or one scenario at two moments: rotate the mirror, move the source, tilt the surface, and say how the outgoing beam compares. Worked example 2 below is this shape.

3.B, make a claim. Apply an appropriate law, definition, theoretical relationship, or model to make a claim. Here that usually means naming the law of reflection or the specular and diffuse definitions and then saying what follows. The claim must be tied to the normal to be complete: "the surface is rough, so the normal direction varies across the illuminated area, so the reflected rays leave in many directions" is a full answer, while "the surface scatters light" restates the observation.

Unit 13's opener also notes that the third free-response question is the Experimental Design and Analysis question, and three of the unit's five sample instructional activities are Desktop Experiment Tasks. None of the five is listed against Topic 13.1: two point at 13.3 and three at 13.4. Reflection is the topic that sets up the measurement habits those tasks lean on rather than the topic they are set on.

Errors that turn a right idea into a wrong answer

Measuring from the surface. The most common single-line mistake in the topic. If a question says a beam strikes a mirror at 30 degrees to the surface, the angle of incidence is 60 degrees. Write down which line you measured from before you write the number.

Saying diffuse reflection breaks the law of reflection. It does not. Both 13.1.B.3 and 13.1.B.4 attribute the difference to the behavior of the surface normal, not to the law. Every single ray obeys $\theta_i = \theta_r$ .

Treating a matte surface as one that absorbs rather than reflects. Diffuse reflection is reflection. A white wall reflects most of the light that lands on it; it just does so in many directions at once.

Using the ray model on a diffraction question. 13.1.A.1.ii rules this out in advance. If a question involves a slit comparable to a wavelength, spreading, or two beams that cancel, the ray model is the wrong tool and the topic is Unit 14.

Assuming reflection means a mirror. 13.1.B.1 says light incident on a surface can be reflected, with no qualifier about what the surface is. Partial reflection at a window, at the surface of a pond, or at the face of a glass block is still reflection, and it happens at the same surface where refraction is bending the transmitted part of the beam.

Carrying a sign convention into this topic. Angles of incidence and reflection are both positive angles measured from the normal. Signs enter the unit only where object, image and focal distances do, in Topics 13.2 and 13.4.

A beam given at 22 degrees to the mirror surface

A laser beam in air strikes a flat mirror. The beam makes an angle of $22^\circ$ with the surface of the mirror. Find (a) the angle of incidence, (b) the angle of reflection, and (c) the angle between the incident and reflected rays.

Identify the reference line. The law of reflection in 13.1.B.2 is stated between each ray and the normal, which is perpendicular to the surface. The problem has given the angle to the surface, so it must be converted first.
(a) Angle of incidence: $\theta_i = 90^\circ - 22^\circ = 68^\circ$ from the normal.
(b) Apply $\theta_i = \theta_r$ directly: $\theta_r = 68^\circ$ from the normal, on the opposite side of the normal from the incident ray.
(c) The incident and reflected rays sit $68^\circ$ either side of the normal, so the angle between them is $68^\circ + 68^\circ = 136^\circ$ .
Check the result against the surface instead. The reflected ray makes $90^\circ - 68^\circ = 22^\circ$ with the surface, matching the incident beam on the other side. The angle between the rays measured around through the surface is $180^\circ - 22^\circ - 22^\circ = 136^\circ$ , which agrees.

(a) $\theta_i = 68^\circ$ from the normal. (b) $\theta_r = 68^\circ$ from the normal. (c) The rays are $136^\circ$ apart.

Rotate the mirror, and the beam turns twice as far (skill 2.C)

A laser is fixed in place and its beam strikes a plane mirror at an angle of incidence of $30^\circ$ . The mirror is then rotated by $12^\circ$ about the point where the beam lands, in the plane of the beam and in the direction that increases the angle of incidence. Find the new angle of incidence, and the angle through which the reflected beam has turned.

The laser has not moved, so the incident ray direction is unchanged. What changes is the direction of the normal, because the normal is fixed to the mirror and the mirror has rotated.
Rotating the mirror by $12^\circ$ rotates its normal by the same $12^\circ$ . The angle between the fixed incident ray and the normal therefore changes by $12^\circ$ : $\theta_i = 30^\circ + 12^\circ = 42^\circ$ .
Apply the law of reflection in the new configuration: $\theta_r = 42^\circ$ , measured from the new normal.
Track the reflected ray in fixed laboratory directions rather than relative to the mirror. Before the rotation the incident and reflected rays were $2 \times 30^\circ = 60^\circ$ apart. After the rotation they are $2 \times 42^\circ = 84^\circ$ apart.
The incident ray never moved, so the whole change in separation is a change in the reflected ray's direction: the beam has turned by $84^\circ - 60^\circ = 24^\circ$ .
Generalize it, which is what skill 2.C is really asking for: rotating a plane mirror by an angle $\alpha$ turns the reflected beam by $2\alpha$ , for any fixed incident ray. Here $\alpha = 12^\circ$ and $2\alpha = 24^\circ$ .

The new angle of incidence is $42^\circ$ , and the reflected beam has turned through $24^\circ$ , twice the rotation of the mirror.

Two mirrors at right angles send the beam straight back

Two plane mirrors are joined so that their surfaces are perpendicular to each other. A ray in the plane of both mirrors strikes mirror A at an angle of incidence of $35^\circ$ , reflects, and then strikes mirror B. Find the angle of incidence on mirror B, and describe the direction of the ray that leaves mirror B compared with the ray that arrived at mirror A.

Draw the two mirrors, the normal to each, and the ray path. The two normals are perpendicular to each other because the two surfaces are.
At mirror A: $\theta_i = 35^\circ$ , so $\theta_r = 35^\circ$ by the law of reflection.
The ray leaving mirror A makes $35^\circ$ with A's normal. Since B's normal is perpendicular to A's normal, the angle that same ray makes with B's normal is $90^\circ - 35^\circ = 55^\circ$ . So the angle of incidence on mirror B is $55^\circ$ .
At mirror B: $\theta_r = 55^\circ$ as well.
Add the deviations. A single reflection turns a ray through $180^\circ - 2\theta_i$ , the angle between its old and new directions. At A that is $180^\circ - 2(35^\circ) = 110^\circ$ . At B it is $180^\circ - 2(55^\circ) = 70^\circ$ . Total turn: $110^\circ + 70^\circ = 180^\circ$ .
A total deviation of $180^\circ$ means the outgoing ray travels exactly opposite to the incoming ray. Note that $35^\circ$ never appeared in the total: the two angles of incidence always sum to $90^\circ$ , so the deviations always sum to $360^\circ - 2(90^\circ) = 180^\circ$ whatever angle you start with.

The angle of incidence on mirror B is $55^\circ$ , and the ray leaves traveling antiparallel to the ray that arrived, for any starting angle.

Frequently asked questions

What is the law of reflection in AP Physics 2?

The law of reflection states that the angle between the incident ray and the normal, which is the line perpendicular to the surface, is equal to the angle between the reflected ray and the normal. The AP Physics 2 course framework writes it as theta_i = theta_r in essential knowledge 13.1.B.2. It applies at every point of every surface, curved or flat, and the reflected ray leaves on the opposite side of the normal from the incoming ray.

Is the angle of reflection measured from the mirror or from the normal?

From the normal. The AP Physics 2 CED defines both angles against the normal, the line perpendicular to the surface at the point of incidence. A beam that makes 22 degrees with the mirror face has an angle of incidence of 68 degrees, because 90 minus 22 is 68. Questions frequently give the angle to the surface on purpose, so check which line the number was measured from before using it.

What is the difference between specular and diffuse reflection?

Specular reflection is reflection from a smooth surface and results in light uniformly reflected, because the normal to the surface has an approximately constant direction over the area the light strikes. Diffuse reflection is reflection from a rough surface and results in light reflected in many different directions, because the normal varies over the illuminated area. Those are the AP Physics 2 definitions in 13.1.B.4 and 13.1.B.3. Both obey the same law of reflection ray by ray.

Is the law of reflection on the AP Physics 2 equation sheet?

No. The Waves, Sound, and Optics group on the AP Physics 2 equation sheet holds 15 equations and theta_i = theta_r is not one of them. The course framework prints it as a relevant equation inside essential knowledge 13.1.B.2, and the CED states that not all equations in the framework appear on the exam sheet. Of the 15 printed, only four are cited inside Unit 13: n = c/v, Snell's law, the thin-lens and mirror equation, and the magnification equation.

Why can you see a sheet of paper from any angle but not your reflection in it?

Paper is rough on the scale of a light wavelength, so the direction of the surface normal varies across the spot the light lands on. Parallel rays therefore leave in many different directions, which is diffuse reflection, and light reaches your eye from every viewing angle. A mirror is smooth, so its normal points essentially the same way across the whole illuminated area, and the reflected rays keep the pattern of directions they arrived with. That preserved pattern is what forms an image.

If you rotate a mirror, how much does the reflected beam turn?

Twice as far. With the light source fixed, rotating a plane mirror through an angle turns the reflected beam through double that angle. Rotating the mirror by 12 degrees swings the reflected beam by 24 degrees. The reason is that the normal is attached to the mirror, so rotating the mirror by 12 degrees changes the angle of incidence by 12 degrees, and the law of reflection changes the angle of reflection by another 12 degrees on the far side.

Does AP Physics 2 Topic 13.1 cover mirrors and images?

Not images. Topic 13.1 covers the ray model of light and the reflection of light from a surface: what a light ray is, when the ray model is valid, the law of reflection, and the specular versus diffuse distinction. Every statement about images formed by mirrors, including the plane-mirror rule that the image sits as far behind the mirror as the object is in front, belongs to Topic 13.2. Topic 13.2 also carries the only boundary statement in Unit 13, which limits mirrors to plane, convex spherical, and concave spherical.