Topic 13.3: Refraction

What Topic 13.3 requires

Topic 13.3 carries one learning objective, 13.3.A, describe the refraction of light between two media. It prints no boundary statement. Five essential knowledge statements sit under it, and three of them carry sub-statements.

• 13.3.A.1 Refraction is the change in direction of a light ray as the ray passes from one medium into another.
• 13.3.A.2 Refraction is a result of the speed of light changing when light enters a new medium.
• 13.3.A.3 The index of refraction of a given medium is inversely proportional to the speed of light in the medium. Relevant equation: $n = \dfrac{c}{v}$.
• 13.3.A.4 Snell's law relates the angles of incidence and refraction of a light ray passing from one medium into another to the indices of refraction of the two media. Relevant equation: $n_1 \sin\theta_1 = n_2 \sin\theta_2$. Sub-statements: 13.3.A.4.i (higher index into lower index refracts away from the normal), 13.3.A.4.ii (lower index into higher index refracts toward the normal), 13.3.A.4.iii (a ray incident along the normal is not refracted).
• 13.3.A.5 Total internal reflection may occur when light passes from one medium into another medium with a lower index of refraction. Sub-statements: 13.3.A.5.i (total internal reflection occurs beyond a critical angle of incidence, with the derived equation for $\theta_{critical}$), 13.3.A.5.ii (at the critical angle the ray refracts at 90 degrees and travels along the surface of the material), 13.3.A.5.iii (beyond the critical angle all light is reflected, with no light transmitted into the other medium).

The suggested skills are 1.B (create quantitative graphs with appropriate scales and units, including plotting data), 2.B (calculate or estimate an unknown quantity with units from known quantities), 2.D (predict new values or factors of change of physical quantities using functional dependence between variables), 3.A (create experimental procedures that are appropriate for a given scientific question) and 3.B (apply an appropriate law, definition, theoretical relationship, or model to make a claim). Five skills, more than any other topic in the unit except 13.4, which lists the same five with 1.C in place of 1.B. The topic page and the Unit at a Glance table agree.

Unit 13 is weighted at 12 to 15 percent of the multiple-choice section across a suggested 8 to 12 class periods.

Light bends because it changes speed, not because it hits something

13.3.A.1 and 13.3.A.2 are two short statements that should be read as one causal chain. Refraction is the change in direction of a light ray as the ray passes from one medium into another, and refraction is a result of the speed of light changing when light enters a new medium.

The change of speed is the cause. The change of direction is the consequence. Nothing about the surface pushes the light sideways.

The wrong mental model, that light is deflected by bumping into the material, predicts the wrong things. It predicts that a beam hitting the surface head on would be knocked off course, and 13.3.A.4.iii says it is not: when a light ray is incident along the normal to a surface, the transmitted ray is not refracted. The speed still changes at normal incidence. The direction does not.

The picture that does work is a wavefront arriving at an angle. One edge of the front crosses into the slow medium first and falls behind while the other edge is still traveling fast, so the whole front pivots. The ray, defined in 13.1.A.1 as the line perpendicular to the wavefront, pivots with it. That also explains why normal incidence is the exception: both edges cross the boundary at the same instant, so nothing pivots.

Index of refraction: n = c/v

13.3.A.3 says the index of refraction of a given medium is inversely proportional to the speed of light in the medium, and prints the relevant equation:

Here $c$ is the speed of light in a vacuum, printed in the constants block of the AP Physics 2 equation sheet as $c = 3.00 \times 10^8 \text{ m/s}$, and $v$ is the speed of light in the medium. The variable list on the sheet defines $n$ as index of refraction and $v$ as speed.

Three consequences worth holding on to:

• $n$ has no units. It is a ratio of two speeds.
• A bigger $n$ means slower light. "High index" and "optically dense" both mean the same thing here, and neither has anything to do with mass density.
• A vacuum has $n = 1$ exactly, because $v = c$ there. Air is treated as $n = 1.00$ in nearly every AP problem, and problems say so.

The AP Physics 2 equation sheet prints no table of refractive indices. There is no list of values for water, glass, or diamond anywhere on it. Any index you need in a question has to come out of the question itself, out of a data table you are given, or out of measurements you took. That is a relief rather than a gap: nothing here is worth memorizing, and a question that wants you to compare two media will tell you both numbers.

One small trap in the wording of 13.3.A.3: "inversely proportional" is a precise phrase, and skill 2.D will test it as one. Halving the speed of light in a medium doubles that medium's index.

Snell's law, and the convention that makes it work

Convention for this page, declared once and held to the end: every angle in Snell's law is measured from the normal, the line perpendicular to the surface at the point where the ray crosses. Subscript 1 always labels the medium the light is in before the boundary, and subscript 2 the medium it enters. Both are positive angles. No signs anywhere in this topic.

13.3.A.4 states that Snell's law relates the angles of incidence and refraction of a light ray passing from one medium into another to the indices of refraction of the two media, and prints:

The CED does not spell out the measuring convention next to that equation, and it does not have to. Topic 13.1 already defined the normal, in 13.1.B.2, as the line perpendicular to the surface, and 13.3.A.4.iii states that a ray incident along the normal is not refracted. Put those together and the convention is forced: a ray along the normal must give $\theta_1 = 0$, since $\sin 0 = 0$ makes $\sin\theta_2 = 0$ as well, and only the from-the-normal convention does that. Measuring from the surface would make normal incidence $\theta_1 = 90^\circ$, and Snell's law would then predict the largest possible bend at exactly the angle where the CED says there is none.

So the two conventions are not interchangeable, and the sheet cannot settle it for you: the Waves, Sound, and Optics variable list defines $\theta$ only as "angle".

The subscript convention matters just as much, because Snell's law is symmetric and the equation will not complain if you swap the media. Decide which medium is 1 by asking where the light is coming from, then do not revisit it inside a single calculation.

Toward the normal or away from it: the two-line rule

13.3.A.4.i and 13.3.A.4.ii turn Snell's law into a qualitative check you can run before touching a calculator.

• 13.3.A.4.i When a light ray travels from a medium with a higher index of refraction into a medium with a lower index of refraction, the ray refracts away from the normal.
• 13.3.A.4.ii When a light ray travels from a medium with a lower index of refraction into a medium with a higher index of refraction, the ray refracts toward the normal.

The algebra behind them is one line. Rearranged, $\sin\theta_2 = (n_1/n_2)\sin\theta_1$. If $n_2 > n_1$ the factor is less than one, so $\theta_2 < \theta_1$ and the ray has swung closer to the normal. If $n_2 < n_1$ the factor exceeds one and the angle opens out.

The useful habit is to run the rule first and the arithmetic second. Decide from the two indices whether the answer must be bigger or smaller than the angle you started with, then check that your calculator agrees. An inverted ratio is the easiest slip to make here and the easiest to catch this way.

The rule also answers the CED's own essential question for the unit, "why does a straw in a glass of water look bent?" Light leaves the submerged part of the straw and travels from water into air, higher index to lower index, so it bends away from the normal. Your eye traces those rays straight back and the submerged part appears displaced from where it actually is. The unit's Developing Understanding section names this directly: students will be challenged to confront their misconceptions about light, including why objects are not always located where they are seen.

And 13.3.A.4.iii is the third case: at normal incidence there is no bend at all, in either direction, whatever the two indices are.

Total internal reflection and the critical angle

13.3.A.5 states that total internal reflection may occur when light passes from one medium into another medium with a lower index of refraction. Read "may" as a real hedge, because it takes two conditions, not one.

Condition one, from 13.3.A.5 itself: the light must be going from higher index to lower index, $n_1 > n_2$. If it is going the other way there is no critical angle at all and total internal reflection cannot happen at that boundary.

Condition two, from 13.3.A.5.i: total internal reflection of light occurs beyond a critical angle of incidence, and the CED gives the derived equation

with $n_1$ the medium the light starts in and $n_2$ the medium beyond the boundary, following the same subscript convention declared above. Because $n_1 > n_2$ is required, the argument of the inverse sine is less than one and the equation returns a real angle. If you ever compute a critical angle and the ratio comes out greater than one, you have the media the wrong way round.

The CED labels this a derived equation, not a relevant equation, and that label carries a specific meaning. The CED explains that not all equations in the course framework appear on the exam equation sheet, and that equations denoted as derived equations are provided for reference and guidance, or to demonstrate the final results of derivations expected of students on the exam. Checking the sheet confirms it: the Waves, Sound, and Optics group has 15 entries and no critical-angle formula among them. You are expected to be able to produce it, which takes one step from Snell's law: set $\theta_2 = 90^\circ$ so $\sin\theta_2 = 1$, and $n_1\sin\theta_1 = n_2$ rearranges to $\sin\theta_1 = n_2/n_1$.

That derivation is also exactly what 13.3.A.5.ii describes physically: for incident rays at the critical angle, the ray refracts at 90 degrees and travels along the surface of the material. The critical angle is the last angle at which anything gets through.

Past it, 13.3.A.5.iii is absolute: for incident rays beyond the critical angle, all light is reflected, with no light transmitted into the other medium. "All" is the word that makes total internal reflection useful. It is also why the reflected ray obeys the law of reflection from Topic 13.1 at that point, at an angle equal to the angle of incidence.

The graph the CED wants you to plot (skill 1.B)

Skill 1.B is listed first for this topic, and it is the only topic row in Unit 13's Unit at a Glance table that carries it: create quantitative graphs with appropriate scales and units, including plotting data. Snell's law is the reason. It is the one relationship in the unit that a class can measure directly with a protractor, and it is not linear in the angles.

The CED's Instructional Approaches section makes the intended activity explicit under skill 1.B: when learning about Snell's law, have students measure the incident angle and refracted angle of a beam of light from a laser incident on a piece of transparent plastic, repeat for various angles of incidence, and have students determine what they should graph so that the data is linear.

The answer to "what should we graph" comes from rearranging the printed equation rather than from remembering a rule. Write $n_1\sin\theta_1 = n_2\sin\theta_2$ as

and it is $y = mx$ with $y = \sin\theta_1$, $x = \sin\theta_2$, and slope $n_2/n_1$. Plot the sines, not the angles. Plotting $\theta_1$ against $\theta_2$ gives a curve that looks nearly straight over small angles and misleads you.

The line goes through the origin, which is 13.3.A.4.iii showing up as a graph feature: zero in, zero out. That gives you a free check on your data and a reason not to force an intercept.

Two of the unit's five sample instructional activities point at Topic 13.3, and both are Desktop Experiment Tasks. One gives groups orange gelatin dessert in a rectangular prism shaped container, a red laser, a protractor and paper, and asks them to determine the index of refraction of the gelatin without touching it directly. The other gives them a plastic hemispherical dish of water, three pins and graph paper, and asks for a procedure to determine the speed of light in water: that one is $n = c/v$ run backwards. Skill 3.A, create experimental procedures that are appropriate for a given scientific question, is what both are training.

What the equation sheet gives you, and what it does not

The AP Physics 2 equation sheet prints 129 equations in seven groups. The Waves, Sound, and Optics group holds 15 of them, and exactly two are Topic 13.3 equations:

Two more of the 15, the image location equation and the magnification equation, belong to Topics 13.2 and 13.4. The remaining 11 are Unit 14 material.

Not printed, and worth listing because students expect them:

• The critical angle equation. It is a derived equation in 13.3.A.5.i, which by the CED's own definition means it does not appear on the sheet.
• Any table of refractive index values.
• $\theta_i = \theta_r$. The law of reflection is a relevant equation in 13.1.B.2 and is also absent from the sheet.
• Any statement of where an angle is measured from. The variable list gives $\theta$ as "angle" and stops there.

What is printed and useful alongside them: $c = 3.00 \times 10^8 \text{ m/s}$ in the constants block, and $\lambda = v/f$ at the top of the same optics group, which is the bridge into Unit 14 rather than anything Topic 13.3 requires. The wave speed guide covers that relationship if you want it now.

How Topic 13.3 is tested, and what it does not include

The AP Physics 2 exam runs 3 hours: 42 multiple-choice questions in 85 minutes for half the score, and 4 free-response questions in 95 minutes for the other half. Unit 13's 12 to 15 percent weighting applies to the multiple-choice section, and a calculator is allowed on both.

The five suggested skills map onto five recognizable question shapes. 2.B is a Snell's law substitution. 2.D is functional dependence with no numbers at all, of the form "if the index of the second medium is doubled, what happens to $\sin\theta_2$". 1.B is the linearized plot from the previous section. 3.A is designing a procedure to measure something you cannot measure directly. 3.B applies the toward-or-away rule, or the two conditions for total internal reflection, to make a claim.

Two things are worth naming because plenty of non-AP refraction material includes them and this course does not.

Dispersion is not in Unit 13. The word does not appear anywhere in Unit 13 of the AP Physics 2 CED, and neither does any statement about the index of refraction depending on wavelength or color. The only place the CED covers white light splitting into colors is essential knowledge 14.8.A.5, and there it is a diffraction grating rather than a prism. So a prism spectrum is not a Topic 13.3 requirement, however often it appears on a refraction worksheet.

Lenses are not in Topic 13.3. Refraction is the mechanism a lens uses, but every statement about focal points, image location, real and virtual images and magnification sits in Topic 13.4, Images Formed by Lenses. Topic 13.3 stops at the single boundary, and signs stop there with it: object and image distances carry sign conventions in 13.2 and 13.4, which the CED refers to without ever printing them, while Snell's law and the critical angle equation use positive angles from the normal and nothing else.