AP Physics 2 · Unit 13 of 7

Unit 13: Geometric Optics

12-15% of the multiple-choice section4 topics

Topics in this unit

Geometric Optics is Unit 13 of AP Physics 2, 12 to 15 percent of the multiple-choice section across about 8 to 12 class periods. The four topics are reflection, images formed by mirrors, refraction, and images formed by lenses. One model runs all four: light goes straight until a surface bends it.

AP Physics: Unit 13 (topics 13.1 Reflection, 13.2 Images Formed by Mirrors, 13.3 Refraction, 13.4 Images Formed by Lenses). AP Physics 2 Unit 13, Geometric Optics, weighted at 12 to 15 percent of the multiple-choice section across a suggested 8 to 12 class periods (45-minute periods, five days a week). Four topics, five learning objectives in total: 13.1.A (describe light as a ray) and 13.1.B (describe the reflection of light from a surface) in Topic 13.1, then 13.2.A (describe the image formed by a mirror), 13.3.A (describe the refraction of light between two media) and 13.4.A (describe the image formed by a lens). The unit prints exactly one boundary statement, under Topic 13.2: AP Physics 2 limits the study of mirrors to plane mirrors, convex spherical mirrors, and concave spherical mirrors. Suggested skills by topic, identical on the topic pages and in the Unit at a Glance table: 13.1 uses 1.A, 2.B, 2.C and 3.B; 13.2 uses 1.A, 2.A, 2.C and 3.C; 13.3 uses 1.B, 2.B, 2.D, 3.A and 3.B; 13.4 uses 1.C, 2.B, 2.D, 3.A and 3.B. The unit opener flags 1.B, 2.B, 2.D and 3.A as the practices the unit develops, and ties them to the Experimental Design and Analysis free-response question. Four of the 15 equations in the Waves, Sound, and Optics group of the equation sheet are cited inside this unit; the law of reflection is a framework relevant equation that is not printed, and the critical angle equation is labeled a derived equation, which the CED defines as an equation not on the exam sheet. The CED asserts that sign conventions govern object, image and focal-point locations (13.2.A.7.i and 13.4.A.5.i) but never states them, and the printed magnification equation carries absolute value bars.

One model runs the whole unit

Unit 13 is built on a single simplification. Light travels in straight lines, and it changes direction only where it meets a surface. There are exactly two things a surface can do: send the light back (reflection) or let it through with a kink (refraction). Everything else in the unit is bookkeeping about where those redirected rays end up.

An image is the second half of the idea. Take many rays leaving one point of an object, redirect each one at the surface, and look at where they end up going. If they actually cross at a point, there is a real image there. If they only look as though they came from a point, there is a virtual image there. The image is a place rays meet, or appear to meet, and nothing more.

The CED's own framing of the unit, from the Developing Understanding section of the unit opener, is that Unit 13 demonstrates another distinct shift in both content and the models used to analyze physical scenarios, that students will be introduced to the different ways of thinking about and modeling light, and that the unit will focus on using the ray model of light to determine the images formed by mirrors as a result of reflection and the images formed by lenses as a result of refraction. It also names the misconception the unit exists to fix: students will be challenged to confront their misconceptions about light, including why objects are not always located where they are seen.

That sentence explains the whole unit's structure. Two mechanisms, reflection and refraction, and two applications of each, mirrors and lenses. The four topics come in exactly that order.

The unit opener also sets up what comes next: in Unit 14, students will continue to explore the behavior of light as an electromagnetic wave and examine additional ways of thinking about and modeling light. Unit 13 is the ray model. Unit 14 is the wave model.

What the CED says about the unit

The numbers, verified against the unit opener and the exam weighting table:

Exam weighting: 12 to 15 percent of the multiple-choice section, the same band as Units 12, 14 and 15. The three electricity and thermodynamics units, 9 through 11, carry 15 to 18 percent each.
Suggested pacing: about 8 to 12 class periods, based on 45-minute periods meeting five days a week. That is the shortest suggested span of any unit in AP Physics 2.
Four topics, one learning objective each except Topic 13.1, which has two.

The unit opener lists four essential questions to hang the topics on: why can't a flat lens focus light, why does a mirror flip words, how can we make things invisible, and why does a straw in a glass of water look bent.

The science practices the opener flags for the unit are 1.B, 2.B, 2.D and 3.A. Its guidance is that in Unit 13 students can be asked to practice collecting data and determining appropriate experimental procedures to answer scientific questions (3.A), and that once students have designed a procedure and collected data they can practice analyzing that data (1.B, 2.B, 2.D) by plotting linearized graphs and using the best fit line to the plotted data to make claims about the physical scenario. The unit is unusually lab-flavored for that reason.

The opener connects that to the exam directly: the third free-response question on the AP Physics 2 exam is the Experimental Design and Analysis question, and students often struggle with knowing where to start when designing an experiment. The unit's five sample instructional activities bear that out: three are Desktop Experiment Tasks, one is a paired reasoning task called Working Backward, and one is a Quickwrite on how the human eye works. Two are listed against Topic 13.3 and three against Topic 13.4, and none against 13.1 or 13.2.

Topic 13.1: Reflection

Topic 13.1 is the only topic in the unit with two learning objectives, and the first of them is about the model rather than about mirrors.

13.1.A, describe light as a ray. A light ray is defined in 13.1.A.1 as a straight line perpendicular to the wavefront of a light wave, pointing in the direction the wave travels. The sub-statements are candid about the limits: rays work in geometric optics where the wave nature of light can be neglected (13.1.A.1.i), but rays are not sufficient to understand the spreading of light, and in interference and diffraction the wave nature is important (13.1.A.1.ii). 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. The law of reflection, in 13.1.B.2, states that the angle between the incident ray and the normal (the line perpendicular to the surface) equals the angle between the reflected ray and the normal:

\theta_i = \theta_r

The rest of 13.1.B is the specular and diffuse distinction, and the CED defines both by what the surface normal does. Specular reflection (13.1.B.4) comes from a smooth surface where the normal has an approximately constant direction over the illuminated area. Diffuse reflection (13.1.B.3) comes from a rough surface where the normal varies over that area. Same law, different surfaces.

Suggested skills: 1.A, 2.B, 2.C, 3.B. No boundary statement.

Topic 13.2: Images Formed by Mirrors

Topic 13.2 takes the law of reflection and asks where the reflected rays end up. It carries the unit's only boundary statement: AP Physics 2 limits the study of mirrors to plane mirrors, convex spherical mirrors, and concave spherical mirrors.

The focal point comes first. Rays parallel to the principal axis of a concave (converging) mirror reflect toward a common location called the focal point (13.2.A.1). For a convex (diverging) mirror they reflect so that they appear to have originated from a common location behind the mirror (13.2.A.2). A plane mirror's focal point is an infinite distance from the mirror (13.2.A.3), and a spherical mirror's focal point may be approximated as a point on the principal axis halfway between the mirror surface and the center of the mirror's radius of curvature (13.2.A.4).

Then the real and virtual distinction, defined by rays rather than by a sign. A real image is formed when reflected rays emanating from a common point intersect at a common point (13.2.A.5). A virtual image is formed when reflected rays diverge such that they appear to have originated from a common point (13.2.A.6).

The quantitative statements are 13.2.A.7, that the image location depends on the focal length and the object distance, with the equation $\frac{1}{s_i} + \frac{1}{s_o} = \frac{1}{f}$ , and 13.2.A.8, the magnification. 13.2.A.7.ii gives the plane-mirror special case: the image distance equals the object distance. Ray diagrams get their own statement, 13.2.A.9, with the three principal rays spelled out in 13.2.A.9.i.

Suggested skills: 1.A, 2.A, 2.C, 3.C. This is the only topic in Unit 13 that lists 2.A, derive a symbolic expression, and the only one that lists 3.C, justify a claim with evidence.

Topic 13.3: Refraction

Topic 13.3 is the other mechanism. Refraction is the change in direction of a light ray as it passes from one medium into another (13.3.A.1), and it happens because the speed of light changes when light enters a new medium (13.3.A.2).

Two equations carry the topic. The index of refraction is inversely proportional to the speed of light in the medium (13.3.A.3):

n = \frac{c}{v}

and Snell's law relates the two angles to the two indices (13.3.A.4):

n_1 \sin\theta_1 = n_2 \sin\theta_2

Both angles are measured from the normal. The direction of the bend has its own two statements: higher index into lower index refracts away from the normal (13.3.A.4.i), lower index into higher index refracts toward the normal (13.3.A.4.ii), and a ray arriving along the normal is not refracted at all (13.3.A.4.iii).

Total internal reflection closes the topic. It may occur when light passes into a medium with a lower index of refraction (13.3.A.5), it happens beyond a critical angle of incidence (13.3.A.5.i), at exactly the critical angle the ray refracts at 90 degrees and travels along the surface of the material (13.3.A.5.ii), and beyond that angle all light is reflected with none transmitted into the other medium (13.3.A.5.iii).

Suggested skills: 1.B, 2.B, 2.D, 3.A, 3.B. No boundary statement. Worth knowing what is absent: dispersion appears nowhere in Unit 13, and neither does any statement about the index of refraction varying with wavelength.

Topic 13.4: Images Formed by Lenses

Topic 13.4 is Topic 13.2 rebuilt on refraction, and the parallel between them is close enough to be worth using as a study tool.

Rays parallel to the principal axis of a thin convex (converging) lens are refracted and converge toward a common location on the transmitted side, called the focal point (13.4.A.1). For a thin concave (diverging) lens they are refracted and diverge as if they originated from a focal point on the incident side (13.4.A.2). Note the difference from a mirror: a lens has light on both sides of it, so the CED has to say which side each focal point is on, and 13.4.A.5.ii adds that lenses have a focal point on both sides of the lens, depending on the shape of the respective side.

Real and virtual images are defined the same way as for mirrors, with refracted in place of reflected: a real image forms when refracted rays from a common point intersect at another common point (13.4.A.3), and a virtual image forms when refracted rays diverge such that they appear to have originated from a common point (13.4.A.4).

The thin-lens equation in 13.4.A.5 is the same equation as the mirror equation, with the object distance measured to the midline of the lens. Magnification (13.4.A.6) and the three principal rays (13.4.A.7.i) match their mirror counterparts too. 13.4.A.7.ii closes the unit: images formed by a lens can be upright or inverted, virtual or real, and reduced, enlarged, or the same size as the object.

Suggested skills: 1.C, 2.B, 2.D, 3.A, 3.B. No boundary statement. Topic 13.4 is the only place in Unit 13 that lists 1.C, create qualitative sketches of graphs.

The Unit 13 equations, counted against the sheet

The AP Physics 2 equation sheet prints 129 equations across seven groups. Optics lives in the Waves, Sound, and Optics group, which holds 15. Four of those 15 are cited as relevant equations inside Unit 13:

Printed equation	Cited in
$n = \dfrac{c}{v}$	13.3.A.3
$n_1 \sin\theta_1 = n_2 \sin\theta_2$	13.3.A.4
$\dfrac{1}{s_i} + \dfrac{1}{s_o} = \dfrac{1}{f}$	13.2.A.7 and 13.4.A.5
$\vert M \vert = \left\vert \dfrac{h_i}{h_o} \right\vert = \left\vert \dfrac{s_i}{s_o} \right\vert$	13.2.A.8 and 13.4.A.6

The other 11 belong to waves, sound, interference, diffraction and beats, which is Unit 14 material.

Two Unit 13 equations are in the course framework but not on the sheet. $\theta_i = \theta_r$ , the law of reflection, is a relevant equation in 13.1.B.2 and is absent from the printed sheet. The critical angle equation $\theta_{critical} = \sin^{-1}(n_2/n_1)$ is labeled a derived equation in 13.3.A.5.i, and the CED explains that derived equations are provided for reference and guidance, or to demonstrate the final results of derivations expected of students on the exam, and that not all equations in the framework appear on the exam sheet. You are expected to be able to rebuild that one from Snell's law in a single step.

The sheet's variable list for this group is short and worth reading once: $f$ is frequency or focal length, $h$ is height, $M$ is magnification, $n$ is index of refraction, $s$ is position, and $\theta$ is angle. No table of refractive index values is printed anywhere on the sheet, so any index a question needs, it gives you.

Sign conventions: what the CED specifies, and what it does not

This is the part of Unit 13 where textbooks disagree with each other, so it is worth being exact about what the College Board has actually committed to.

Specified. Angles are measured from the normal. 13.1.B.2 defines the law of reflection between each ray and the normal, and defines the normal as the line perpendicular to the surface. 13.3.A.4.iii then states that a ray incident along the normal is not refracted, which pins the same convention onto Snell's law: normal incidence has to mean an angle of zero for that to be true. Every angle in Unit 13 is a positive angle from the normal.

Asserted but not printed. Distance signs. 13.2.A.7.i says 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. 13.4.A.5.i says the same for a lens. Neither statement says which distances are positive. The list of conventions printed on the exam equation sheet covers inertial frames, ideal components, ohmic resistors, monatomic ideal gases, conventional current, air-filled capacitors and the small-angle approximation for diffraction. There is no optics sign convention on it.

Sidestepped, deliberately. The magnification equation is printed with absolute value bars on both sides, as $|M|$ . It gives you the size ratio of the image and carries no information about orientation at all.

What that means in practice: get the orientation and the type of image from the ray diagram, not from a sign. The CED defines real and virtual by ray behavior in 13.2.A.5, 13.2.A.6, 13.4.A.3 and 13.4.A.4, and 13.2.A.9.ii and 13.4.A.7.ii both say images can be upright or inverted, virtual or real, and reduced, enlarged, or the same size as the object. Those are the words to answer in. If your class uses a particular sign convention, keep using it, state it at the top of your work, and hold it to the end of the question. Do not import one from a textbook mid-problem, and do not assume a multiple-choice option is wrong because its sign disagrees with the one you learned.

How Unit 13 is tested, and how to work through it

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. Section II is 4 free-response questions in 95 minutes for the other 50 percent: Question 1 Mathematical Routines, Question 2 Translation Between Representations, Question 3 Experimental Design and Analysis, Question 4 Qualitative/Quantitative Translation. A four-function, scientific, or graphing calculator is allowed on both sections. The 12 to 15 percent figure applies to the multiple-choice section.

The CED's own case study for the science practices happens to be a thin lens, which tells you how the same optics content gets asked six different ways: derive a symbolic expression for the magnification (2.A), calculate an image position (2.B), compare image heights between two object positions (2.C), read a focal length off a graph of inverse image distance against inverse object distance (2.D), name the image type (3.B), and explain why the image is virtual in terms of what the rays do (3.C).

A workable order for the unit:

Learn the two mechanisms first, in 13.1 and 13.3. Both are one equation and a couple of definitions, and both are fast marks.
Then do 13.2 and 13.4 together rather than in CED order. They share an equation, share a magnification relationship, share the real and virtual definitions, and share the three-principal-ray construction. Studying them side by side halves the work and makes the one genuine difference, that a lens has two focal points and a mirror has one, stand out.
Draw before you calculate, every time. The equation gives a number; the diagram tells you whether the number is a real image in front or a virtual image behind.
Practice the lab shape. Three of the unit's five sample activities are Desktop Experiment Tasks: measure the index of refraction of gelatin without touching it, write and follow a procedure to determine the speed of light in water, and find the focal length of a magnifying glass from a graph of object and image distances.

Unit 13 sits inside the wider AP Physics 2 course, and it is the one unit that shares almost no machinery with the units before it. Treat it as a fresh start rather than as an extension of the electricity units.

One ray, both mechanisms, at the same surface

A laser beam traveling in air ( $n_1 = 1.00$ ) strikes the flat top surface of a glass block ( $n_2 = 1.52$ ) at $55^\circ$ from the normal. Part of the beam reflects and part refracts into the glass. Find (a) the angle of the reflected ray, (b) the angle of the refracted ray, and (c) the speed of light inside the glass.

Set the convention before touching numbers: every angle here is measured from the normal, the line perpendicular to the surface at the point where the beam lands. This holds for both mechanisms.
(a) Reflection. The law of reflection (13.1.B.2) gives $\theta_r = \theta_i = 55^\circ$ , on the opposite side of the normal from the incoming beam. The glass being there at all makes no difference to this half of the answer.
(b) Refraction. Snell's law (13.3.A.4): $n_1 \sin\theta_1 = n_2 \sin\theta_2$ , so $\sin\theta_2 = \dfrac{n_1 \sin\theta_1}{n_2} = \dfrac{(1.00)(0.8192)}{1.52} = 0.5389$ .
Take the inverse sine: $\theta_2 = \sin^{-1}(0.5389) = 32.6^\circ$ from the normal, inside the glass.
Check the direction against 13.3.A.4.ii before moving on. Air to glass is lower index into higher index, so the ray must bend toward the normal, and $32.6^\circ$ is smaller than $55^\circ$ . Correct.
(c) Speed inside the glass, from $n = c/v$ (13.3.A.3) rearranged to $v = c/n$ , with $c = 3.00 \times 10^8 \text{ m/s}$ from the sheet: $v = \dfrac{3.00 \times 10^8 \text{ m/s}}{1.52} = 1.97 \times 10^8 \text{ m/s}$ .
Notice what one surface did to one beam. It sent part of the light back at $55^\circ$ , bent the rest to $32.6^\circ$ , and slowed the transmitted part to about 66 percent of its vacuum speed. That is the whole physics content of the unit's first half.

(a) The reflected ray leaves at $55^\circ$ from the normal. (b) The refracted ray travels at $32.6^\circ$ from the normal inside the glass. (c) $v = 1.97 \times 10^8 \text{ m/s}$ .

Where the rays meet: the image equation used once

An object $4.0 \text{ cm}$ tall is placed $0.30 \text{ m}$ from a converging optical element of focal length $0.20 \text{ m}$ , on the axis. Find (a) the image distance, (b) the magnification, and (c) the height of the image. State what the calculation can and cannot tell you about the image.

Declare the convention. Every distance in this example is a positive distance measured from the element: the object is outside the focal point, so this is the case the printed equation handles with no signs at all. Object distance $s_o = 0.30 \text{ m}$ , focal length $f = 0.20 \text{ m}$ .
(a) Rearrange the printed equation $\frac{1}{s_i} + \frac{1}{s_o} = \frac{1}{f}$ for the image distance: $\frac{1}{s_i} = \frac{1}{f} - \frac{1}{s_o}$ .
Substitute: $\frac{1}{s_i} = \dfrac{1}{0.20 \text{ m}} - \dfrac{1}{0.30 \text{ m}} = 5.00 \text{ m}^{-1} - 3.33 \text{ m}^{-1} = 1.67 \text{ m}^{-1}$ .
Invert, and only now: $s_i = \dfrac{1}{1.67 \text{ m}^{-1}} = 0.60 \text{ m}$ . Inverting each term separately instead of adding the reciprocals first is the single most common arithmetic slip in this unit.
(b) Magnification from the printed equation: $|M| = \left|\dfrac{s_i}{s_o}\right| = \dfrac{0.60 \text{ m}}{0.30 \text{ m}} = 2.0$ . It is a pure number.
(c) Image height from the other half of the same equation: $|h_i| = |M| \, |h_o| = (2.0)(4.0 \text{ cm}) = 8.0 \text{ cm}$ .
Now the honest limit. The equations located the image and sized it. They said nothing about whether it is upright or inverted, because the printed magnification equation is an absolute value on both sides. The CED defines real and virtual by what the rays do (13.2.A.5, 13.2.A.6, 13.4.A.3, 13.4.A.4), so that part of the answer comes from a ray diagram, not from arithmetic.

(a) $s_i = 0.60 \text{ m}$ from the element. (b) $|M| = 2.0$ . (c) The image is $8.0 \text{ cm}$ tall. Whether it is upright or inverted, real or virtual, has to come from the ray diagram: the printed magnification equation carries absolute value bars and no orientation information.

Frequently asked questions

How much of the AP Physics 2 exam is Unit 13 Geometric Optics?

Unit 13 carries a 12 to 15 percent weighting on the multiple-choice section of the AP Physics 2 exam, the same band as Units 12, 14 and 15. That works out to roughly 5 to 6 of the 42 multiple-choice questions. The CED suggests about 8 to 12 class periods for the unit, based on 45-minute periods five days a week, which is the shortest suggested span of any unit in the course. Optics can also appear in any of the four free-response questions.

What topics are in AP Physics 2 Unit 13?

Four: 13.1 Reflection, 13.2 Images Formed by Mirrors, 13.3 Refraction, and 13.4 Images Formed by Lenses. They pair up. Topics 13.1 and 13.3 are the two mechanisms by which a surface can redirect light, and Topics 13.2 and 13.4 apply those mechanisms to find where images form. Topic 13.1 is the only one with two learning objectives, because it also has to define what a light ray is before any of the rest can be stated.

Which optics equations are on the AP Physics 2 equation sheet?

The Waves, Sound, and Optics group prints 15 equations and four of them are cited inside Unit 13: n = c/v, Snell's law n1 sin(theta1) = n2 sin(theta2), the image equation 1/si + 1/so = 1/f, and the magnification equation |M| = |hi/ho| = |si/so|. The other 11 are wave, sound, interference and diffraction equations belonging to Unit 14. Two Unit 13 equations are not printed: the law of reflection, and the critical angle equation, which the CED labels a derived equation.

What sign convention does AP Physics 2 use for mirrors and lenses?

The CED does not print one. Essential knowledge 13.2.A.7.i and 13.4.A.5.i both state that the locations of the focal point, object and image follow sign conventions used to determine those locations relative to the mirror or lens itself, without saying which distances are positive, and no optics sign convention appears in the conventions list on the exam equation sheet. The magnification equation is printed with absolute value bars, so it carries no orientation information. Get real versus virtual and upright versus inverted from the ray diagram, and if your class uses a particular sign convention, state it and hold it for the whole question.

What is the difference between a real image and a virtual image?

A real image forms where light rays actually intersect. A virtual image forms where rays only appear to have come from, because they are diverging when they reach your eye. The AP Physics 2 CED defines both by ray behavior rather than by a sign: 13.2.A.5 and 13.4.A.3 define a real image as one where rays from a common point intersect at another common point, and 13.2.A.6 and 13.4.A.4 define a virtual image as one where rays diverge such that they appear to have originated from a common point.

Is dispersion or a prism spectrum part of AP Physics 2 Unit 13?

No. The word dispersion does not appear anywhere in Unit 13 of the AP Physics 2 CED, and no essential knowledge statement in the unit mentions the index of refraction depending on wavelength or color. The only place the CED covers white light splitting into colors is 14.8.A.5, where the higher-order maxima from a diffraction grating spread white light into a rainbow with the longest-wavelength light farthest from the central maximum. Unit 13 refraction stops at Snell's law and total internal reflection.

What is the best order to study AP Physics 2 Unit 13?

Learn the two mechanisms first, Topic 13.1 reflection and Topic 13.3 refraction, since each is one equation plus a few definitions. Then study Topics 13.2 and 13.4 together rather than in numerical order: mirrors and lenses share the image equation, the magnification equation, the real and virtual definitions, and the three-principal-ray construction, so covering them side by side halves the work and highlights the real difference, which is that a lens has a focal point on both sides while a mirror has one.