A straw appears bent at the water surface even though the straw itself has not developed a hinge. The path of light changes, and your eye interprets the arriving rays. Drawing that path carefully is the start of refraction, apparent depth and total internal reflection.
We build from one interface to slabs, guided rays and prisms, keeping angle conventions, wavelength dependence and approximation limits visible throughout.
What you will learn to do
- Measure optical angles from the local normal.
- Apply Snell’s law and explain the direction of bending.
- Distinguish a parallel slab’s angular effect from its lateral displacement.
- Derive near-normal apparent depth.
- Test both conditions for total internal reflection.
- Relate guide-axis angles to side-wall incidence.
- Use wavelength-specific indices and derive prism deviation relations.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Learning goals: - Measure optical angles from the local normal. - Apply Snell’s law and explain the direction of bending. - Distinguish a parallel slab’s angular effect from its lateral displacement. - Derive near-normal apparent depth. - Test both conditions for total internal reflection. - Relate guide-axis angles to side-wall incidence. - Use wavelength-specific indices and derive prism deviation relations. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
From which line is the angle in Snell’s law measured? Why does that choice matter?
Compare your reasoning
It is measured from the normal to the interface, not from the surface. Complementary angles have different sines, so using the wrong reference can reverse or distort a prediction.
Revisit the explanation →01 / Refraction, internal reflection and dispersion
A ray describes direction; a normal defines the angle
Geometrical optics models light by rays that follow the direction of propagation. It works well when the structures and apertures of interest are large compared with wavelength and diffraction can be neglected. A ray is a direction model, not a narrow material thread.
At a smooth boundary, draw the normal: the line perpendicular to the surface at the point where the ray meets it. Both incidence and refraction angles are measured from this normal. Angles marked from the surface must first be converted to their complements.
A reflected ray remains in the incident medium and makes the same angle with the normal as the incident ray. A transmitted ray enters the other medium and may change direction. Reflection and transmission can occur together; refraction does not imply that every incident photon or all incident energy is transmitted.
Refractive index n = c/vphase describes the phase speed at the specified frequency. A larger index means a smaller phase speed. Optical density in this sense is not a claim about mass density.
The elementary ray model uses positive refractive indices and smooth stationary boundaries between transparent isotropic media.
WORKED EXAMPLE
Convert a surface angle before using an optical law
A ray meets a flat surface at 25° to the surface. What is its incidence angle, and what angle does the reflected ray make with the normal?
The normal is perpendicular to the surface.
The incidence angle is the complement of the given angle.
The reflection law uses that same normal-based angle.
What this gives usBoth normal-based angles are 65°.
Check the result another way
The reflected ray also makes 25° with the surface, providing a direct geometry check.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Convert a surface angle before using an optical law Problem: A ray meets a flat surface at 25° to the surface. What is its incidence angle, and what angle does the reflected ray make with the normal? Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion
Focus: A ray describes direction; a normal defines the angle
Learning goals:
- Measure optical angles from the local normal.
- Apply Snell’s law and explain the direction of bending.
- Distinguish a parallel slab’s angular effect from its lateral displacement.
- Derive near-normal apparent depth.
- Test both conditions for total internal reflection.
- Relate guide-axis angles to side-wall incidence.
- Use wavelength-specific indices and derive prism deviation relations.
Equations in this section (LaTeX):
n=\frac{c}{v_{\mathrm{phase}}},\qquad\theta_{\mathrm{reflection}}=\theta_{\mathrm{incidence}}
Meaning: The elementary ray model uses positive refractive indices and smooth stationary boundaries between transparent isotropic media.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Refraction, internal reflection and dispersion
Match the wave along the boundary to obtain the bending law
At a stationary interface, the oscillation frequency is the same on both sides. The wave phase must also match along the interface. For a plane wave this means equal tangential wave-vector components: k1 sin i = k2 sin r.
Because k = 2π/λ = nω/c at the fixed frequency, this becomes n1 sin i = n2 sin r. This is Snell’s law. It relates the two directions without requiring an assumed speed for material particles.
If n2 > n1, then sin r < sin i and the transmitted ray bends toward the normal. If n2 < n1 and transmission is possible, it bends away from the normal. At normal incidence the direction stays straight even though speed and wavelength can change.
Solve for the sine first and check that its magnitude does not exceed one. A value greater than one is not an instruction to force an inverse-sine calculation; it signals that no propagating refracted ray exists in this elementary model.
Frequency is unchanged at the stationary interface. The angles are measured from the local normal.
WORKED EXAMPLE
Predict the direction before calculating the angle
A ray travels from a medium of index 1.20 into one of index 1.60 at incidence 40°. Find the refraction angle.
The higher destination index predicts bending toward the normal.
Rearrange Snell’s law for the transmitted sine.
Take the inverse sine only after checking its allowed range.
What this gives usThe refracted angle is approximately 28.8°, smaller than the 40° incidence angle.
Check the result another way
Substitution gives 1.20 sin 40° ≈ 1.60 sin 28.82°, so the tangential phase condition is satisfied.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Predict the direction before calculating the angle Problem: A ray travels from a medium of index 1.20 into one of index 1.60 at incidence 40°. Find the refraction angle. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
TRY THE IDEA
Follow the incident, reflected and transmitted rays
Start with air into glass. Increase the angle, then reverse the two media. In the reverse direction, set the critical angle and move just beyond it.
Refracted angle from normal19.47°
Critical incidence angleNone in this direction
Follow the green incident ray into the interface. Orange is reflected; purple is refracted. Compare each angle with the normal, not the surface.
Ray directions only: equal stroke widths do not represent equal powers. Ideal transparent media; indices fixed at the chosen wavelength.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion
Focus: Match the wave along the boundary to obtain the bending law
Learning goals:
- Measure optical angles from the local normal.
- Apply Snell’s law and explain the direction of bending.
- Distinguish a parallel slab’s angular effect from its lateral displacement.
- Derive near-normal apparent depth.
- Test both conditions for total internal reflection.
- Relate guide-axis angles to side-wall incidence.
- Use wavelength-specific indices and derive prism deviation relations.
Equations in this section (LaTeX):
n_1\sin i=n_2\sin r,\qquad\lambda_2=\lambda_1\frac{n_1}{n_2}
Meaning: Frequency is unchanged at the stationary interface. The angles are measured from the local normal.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.03 / Refraction, internal reflection and dispersion
A parallel slab restores direction but can shift the ray sideways
Consider a slab with parallel faces, surrounded by the same medium on both sides. At the first face, n0 sin i = ns sin r. At the second face, ns sin r = n0 sin e because the face normals are parallel. Hence e = i for the transmitted emerging ray.
The final direction is parallel to the incident direction, but the two lines need not coincide. Inside the slab the ray followed a different angle, so it emerges displaced sideways.
For slab thickness t measured normally between faces, the perpendicular separation between the incident-line extension and emerging ray is d = t sin(i − r)/cos r. This is a geometric displacement, not a change of outgoing angle.
At normal incidence the displacement vanishes. If the slab index matches its surroundings, i = r and it also vanishes. For several parallel layers, the quantity n sin θ is the same through the stack wherever propagating transmission occurs.
The displacement is perpendicular to the incident/emergent ray direction; t is the normal thickness.
WORKED EXAMPLE
Separate emerging angle from lateral displacement
A parallel slab has thickness 6.0 cm. A ray enters at i = 45° and travels inside at r = 30°. The medium outside both faces is the same. Find the emerging angle and lateral displacement.
Use parallel normals for the direction and slab geometry for the offset.
The two Snell relations cancel the slab index.
Insert the internal path angle in the displacement expression.
What this gives usThe ray emerges at 45° and is displaced by about 1.79 cm.
Check the result another way
The outgoing ray is parallel, not necessarily collinear, with the incident ray. At zero thickness this offset would vanish.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Separate emerging angle from lateral displacement Problem: A parallel slab has thickness 6.0 cm. A ray enters at i = 45° and travels inside at r = 30°. The medium outside both faces is the same. Find the emerging angle and lateral displacement. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion
Focus: A parallel slab restores direction but can shift the ray sideways
Learning goals:
- Measure optical angles from the local normal.
- Apply Snell’s law and explain the direction of bending.
- Distinguish a parallel slab’s angular effect from its lateral displacement.
- Derive near-normal apparent depth.
- Test both conditions for total internal reflection.
- Relate guide-axis angles to side-wall incidence.
- Use wavelength-specific indices and derive prism deviation relations.
Equations in this section (LaTeX):
d=\frac{t\sin(i-r)}{\cos r}\quad\text{for a parallel slab with identical surrounding media}
Meaning: The displacement is perpendicular to the incident/emergent ray direction; t is the normal thickness.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.04 / Refraction, internal reflection and dispersion
Near-normal viewing traces rays back to an apparent location
Light from a submerged object changes direction on leaving a flat surface. An observer in the second medium mentally extends the emerging rays backward along straight lines. Those extensions locate an apparent image, which can differ from the true object position.
For small angles near the normal, tan θ ≈ sin θ ≈ θ. If the object is at depth d in index n1 and viewed from index n2, the apparent depth is dapp = d n2/n1. Viewing from air into water therefore makes the object appear shallower.
This is a paraxial result. At larger viewing angles, different rays need not extend back to the same simple apparent depth. The object itself has not moved; the inferred location changes because the outgoing directions changed.
Always label which medium contains the object and which contains the observer. Reversing those roles in the ratio gives the reciprocal result and can incorrectly predict a shallower image when the geometry requires a deeper one.
WORKED EXAMPLE
Use the indices in the order set by the viewing geometry
An object lies 24 cm below a flat surface in a liquid of index 4/3. It is viewed nearly normally from air, with index 1. Find its apparent depth.
Use the paraxial flat-interface relation with the liquid as the object medium.
The observer-medium index is divided by the object-medium index.
What this gives usThe object appears 18 cm below the surface, a 6 cm apparent upward shift.
Check the result another way
For equal indices the expression gives the true depth. The higher-index object medium here produces the expected shallower appearance.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Use the indices in the order set by the viewing geometry Problem: An object lies 24 cm below a flat surface in a liquid of index 4/3. It is viewed nearly normally from air, with index 1. Find its apparent depth. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Focus: Near-normal viewing traces rays back to an apparent location Learning goals: - Measure optical angles from the local normal. - Apply Snell’s law and explain the direction of bending. - Distinguish a parallel slab’s angular effect from its lateral displacement. - Derive near-normal apparent depth. - Test both conditions for total internal reflection. - Relate guide-axis angles to side-wall incidence. - Use wavelength-specific indices and derive prism deviation relations. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Refraction, internal reflection and dispersion
Total internal reflection requires two separate conditions
When light travels from larger index n1 to smaller index n2, the refracted angle grows faster than the incident angle. At a particular incidence, the refracted direction reaches 90° to the normal and lies along the interface. This is the critical angle ic.
Setting sin r = 1 in Snell’s law gives sin ic = n2/n1. A critical angle exists only for n1 > n2. For incidence greater than ic, there is no propagating refracted ray and the ideal lossless boundary gives total internal reflection.
The word internal refers to incidence from the higher-index side. A large incidence angle from air into ordinary glass does not satisfy the first condition and cannot produce total internal reflection at that single interface.
At the critical angle the refracted ray is the grazing limiting case. Beyond it, wave theory still allows an evanescent field close to the boundary; saying there is no propagating transmitted ray is more precise than saying there is no field at all.
The standard total-power reflection statement assumes an ideal lossless boundary and no nearby coupling structure.
WORKED EXAMPLE
Check the direction and then compare angles
Light in a medium of index 1.50 reaches air at incidence 50°. Determine whether a propagating refracted ray exists.
First verify higher-to-lower index, then calculate the critical angle.
The index ratio defines a critical incidence below 90°.
The actual incidence exceeds the critical value.
What this gives usThe ray undergoes total internal reflection in the ideal interface model.
Check the result another way
Direct substitution would demand sin r = 1.50 sin 50° ≈ 1.149, which lies outside the real-angle range.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Check the direction and then compare angles Problem: Light in a medium of index 1.50 reaches air at incidence 50°. Determine whether a propagating refracted ray exists. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion
Focus: Total internal reflection requires two separate conditions
Learning goals:
- Measure optical angles from the local normal.
- Apply Snell’s law and explain the direction of bending.
- Distinguish a parallel slab’s angular effect from its lateral displacement.
- Derive near-normal apparent depth.
- Test both conditions for total internal reflection.
- Relate guide-axis angles to side-wall incidence.
- Use wavelength-specific indices and derive prism deviation relations.
Equations in this section (LaTeX):
\sin i_c=\frac{n_2}{n_1},\qquad n_1>n_2,\qquad i>i_c\ \text{for total internal reflection}
Meaning: The standard total-power reflection statement assumes an ideal lossless boundary and no nearby coupling structure.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.06 / Refraction, internal reflection and dispersion
Geometry determines whether repeated internal reflections stay trapped
A prism or optical guide can redirect a ray by internal reflection. The relevant incidence is measured from each local face normal, so a ray’s angle to a guide axis must often be converted before applying the critical condition.
For a straight core with parallel side boundaries, a meridional ray making angle α with the axis meets a side boundary at incidence 90° − α. With core index nc and cladding index ns < nc, guidance requires sin(90° − α) > ns/nc, or cos α > ns/nc in the ray model.
If the ray enters through a face perpendicular to the axis from index n0, Snell’s law gives n0 sin θentry = nc sin α. Combining this with the internal-reflection condition gives n0 sin θentry < √(nc² − ns²) for the accepted meridional rays, with equality as the limiting case.
This acceptance-cone model neglects wave modes, bending losses and finite boundary effects. It is a useful geometric-optics description when the core is large compared with wavelength, not a complete theory of every optical fibre.
WORKED EXAMPLE
Convert an axial ray angle into a boundary incidence
A ray inside a straight guide has core index 1.50 and cladding index 1.40. It travels at 15° to the guide axis. Does it satisfy the ideal side-wall total-reflection condition?
The side-wall normal is perpendicular to the guide axis.
The incidence at a side wall is the complementary angle.
Compare with the core-to-cladding critical angle.
What this gives usSince 75° exceeds about 68.96°, the ray is guided by total internal reflection in this ideal geometry.
Check the result another way
Equivalently, cos 15° ≈ 0.966 exceeds 1.40/1.50 ≈ 0.933. Both comparisons express the same condition.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Convert an axial ray angle into a boundary incidence Problem: A ray inside a straight guide has core index 1.50 and cladding index 1.40. It travels at 15° to the guide axis. Does it satisfy the ideal side-wall total-reflection condition? Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Focus: Geometry determines whether repeated internal reflections stay trapped Learning goals: - Measure optical angles from the local normal. - Apply Snell’s law and explain the direction of bending. - Distinguish a parallel slab’s angular effect from its lateral displacement. - Derive near-normal apparent depth. - Test both conditions for total internal reflection. - Relate guide-axis angles to side-wall incidence. - Use wavelength-specific indices and derive prism deviation relations. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Refraction, internal reflection and dispersion
A wavelength-dependent index separates colours
A material’s refractive index generally depends on frequency. Dispersion is this dependence of phase speed or index on frequency. Different colour components can therefore refract through different angles at the same interface.
In many transparent materials over the visible range away from strong absorption bands, the index is larger for shorter vacuum wavelengths. This is called normal dispersion. It is a common regime, not a universal inequality across every material and spectral band.
When comparing colours at a stationary boundary, each component keeps its own frequency. Its wavelength inside the material changes according to λmedium = λvacuum/n at that frequency. Use the index belonging to the component being traced.
A parallel slab returns the colours to parallel outgoing directions when the surrounding media match, although their lateral shifts can differ. A prism has nonparallel faces and can leave different colours travelling in different outgoing directions.
WORKED EXAMPLE
Compare two colours without treating the index as a universal constant
Two monochromatic rays enter a material from air at the same 45° incidence. The material indices at their frequencies are 1.50 and 1.60. Which ray has the smaller refracted angle? Estimate both angles.
Apply Snell’s law separately to each frequency.
The lower-index component has the larger transmitted sine.
The higher-index component bends closer to the normal.
What this gives usThe index-1.60 component has the smaller refraction angle, about 26.2° compared with 28.1°.
Check the result another way
The result follows from the specified indices even without assigning colour names. Colour ordering requires information about the material’s dispersion.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Compare two colours without treating the index as a universal constant Problem: Two monochromatic rays enter a material from air at the same 45° incidence. The material indices at their frequencies are 1.50 and 1.60. Which ray has the smaller refracted angle? Estimate both angles. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Focus: A wavelength-dependent index separates colours Learning goals: - Measure optical angles from the local normal. - Apply Snell’s law and explain the direction of bending. - Distinguish a parallel slab’s angular effect from its lateral displacement. - Derive near-normal apparent depth. - Test both conditions for total internal reflection. - Relate guide-axis angles to side-wall incidence. - Use wavelength-specific indices and derive prism deviation relations. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
08 / Refraction, internal reflection and dispersion
A prism links two refractions through its apex geometry
For a ray passing through the two refracting faces of a prism, let the apex angle be A, internal refraction angles be r1 and r2, and external incidence/emergence angles be i and e. Geometry gives r1 + r2 = A and total deviation δ = i + e − A.
With the same surrounding medium on both sides, the minimum-deviation path is symmetric: i = e and r1 = r2 = A/2. Snell’s law then gives the relative index nprism/nsurroundings = sin[(A + δmin)/2]/sin(A/2).
Minimum deviation is a specific orientation. If a problem gives an arbitrary incidence, trace both surfaces instead of using the minimum-deviation formula without justification. The second face must also allow emergence rather than total internal reflection.
For a thin prism with small angles in radians and modest refraction angles, δ ≈ (nrelative − 1)A. Two components then have angular separation Δδ ≈ AΔnrelative. This approximation makes the connection between dispersion and prism separation explicit.
The minimum-deviation relation assumes identical surrounding media; the final approximation uses small angles.
WORKED EXAMPLE
Infer an index from a symmetric prism path
A prism in air has apex angle 60° and measured minimum deviation 30°. Find its refractive index at that wavelength.
The minimum-deviation condition makes the internal path symmetric.
The internal angles are half the apex angle and the external angles follow the deviation geometry.
Use Snell’s law at either face.
What this gives usThe index is approximately 1.414 at the measured wavelength.
Check the result another way
Using δ = i + e − A gives 45° + 45° − 60° = 30°, reproducing the observed minimum deviation.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Infer an index from a symmetric prism path Problem: A prism in air has apex angle 60° and measured minimum deviation 30°. Find its refractive index at that wavelength. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
WORKED EXAMPLE
Estimate angular colour separation with an explicit approximation
A thin prism has apex angle 0.050 rad. Its relative indices for two specified components differ by 0.020. Estimate their angular separation using the small-angle model.
Subtract the two thin-prism deviation expressions.
The common geometrical term cancels, leaving the index difference.
What this gives usThe approximate angular separation is 1.0 mrad.
Check the result another way
The product is dimensionless with its angle interpreted in radians. A zero index difference would produce no separation in this model.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Estimate angular colour separation with an explicit approximation Problem: A thin prism has apex angle 0.050 rad. Its relative indices for two specified components differ by 0.020. Estimate their angular separation using the small-angle model. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion
Focus: A prism links two refractions through its apex geometry
Learning goals:
- Measure optical angles from the local normal.
- Apply Snell’s law and explain the direction of bending.
- Distinguish a parallel slab’s angular effect from its lateral displacement.
- Derive near-normal apparent depth.
- Test both conditions for total internal reflection.
- Relate guide-axis angles to side-wall incidence.
- Use wavelength-specific indices and derive prism deviation relations.
Equations in this section (LaTeX):
\delta=i+e-A,\qquad n_{\mathrm{rel}}=\frac{\sin[(A+\delta_{\min})/2]}{\sin(A/2)},\qquad\delta\approx(n_{\mathrm{rel}}-1)A\ \text{for a thin prism}
Meaning: The minimum-deviation relation assumes identical surrounding media; the final approximation uses small angles.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.BRING THE IDEAS TOGETHER
What should stay with you
- Snell’s law matches n sin θ across a stationary interface.
- Frequency is preserved at a stationary boundary; wavelength and phase speed can change.
- A parallel slab with identical surrounding media restores the direction but can shift the ray.
- Total internal reflection requires higher-to-lower index and incidence beyond the critical angle.
- A guide’s axial angle must be converted to the boundary incidence angle.
- Dispersion belongs to the material’s frequency-dependent index.
- Minimum-deviation and thin-prism formulas have different, explicit conditions.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A ray enters from index 1.20 into index 1.50 at incidence 30°. What is sin r?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Problem: A ray enters from index 1.20 into index 1.50 at incidence 30°. What is sin r? Requested answer unit: Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 2
An object lies 30 cm deep in a medium of index 1.50 and is viewed nearly normally from air. What is its apparent depth?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Problem: An object lies 30 cm deep in a medium of index 1.50 and is viewed nearly normally from air. What is its apparent depth? Requested answer unit: cm Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 3
For incidence from index 1.60 toward index 1.20, what is sin of the critical angle?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Problem: For incidence from index 1.60 toward index 1.20, what is sin of the critical angle? Requested answer unit: Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 4
A thin prism in air has apex angle 0.040 rad and index 1.50. What approximate deviation does the small-angle formula give?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Problem: A thin prism in air has apex angle 0.040 rad and index 1.50. What approximate deviation does the small-angle formula give? Requested answer unit: rad Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 5
Which case can produce total internal reflection at a single ideal interface?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Problem: Which case can produce total internal reflection at a single ideal interface? Choices: 1. Air into glass at a very large incidence 2. Glass into air above the critical angle 3. Any interface at normal incidence Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 6
A ray emerges through a parallel slab into the same medium it started in. Which statement is generally correct?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Problem: A ray emerges through a parallel slab into the same medium it started in. Which statement is generally correct? Choices: 1. It is parallel to the incident ray but can be laterally displaced 2. Its outgoing angle is always zero 3. It must follow the identical original line Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 7
Why can a prism separate two frequencies into different outgoing directions?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Refraction, internal reflection and dispersion Problem: Why can a prism separate two frequencies into different outgoing directions? Choices: 1. The prism must change their frequencies 2. The refractive index can differ at those frequencies 3. All materials have the same index for every frequency Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
A ray crosses into glassIntermediate · Free sample · Hints and guided reasoningUse the internal angle for the film phaseAdvanced · Hints and guided reasoningSeparate colour spread from deviationIntermediate · Hints and guided reasoningWhich angle separates refraction from total reflection?Intermediate · Hints and guided reasoningA coin looks closer to the surfaceIntermediate · Hints and guided reasoningA weak prism bends a rayIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Why is there no total internal reflection when light goes from a lower to a higher refractive index in the simple interface model?
Compare your reasoning
Snell’s law can still supply a transmitted angle for every allowed incidence angle. A critical angle requires travel toward the lower-index medium and incidence beyond the corresponding boundary value.
Revisit the explanation →A few days later: change the situation
A ray passes through a parallel-sided glass slab back into air. Can it emerge parallel to its original direction but shifted sideways?
Compare your reasoning
Yes. The two refractions restore the direction when the external media match and the faces are parallel. The path inside the slab still creates a lateral displacement.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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