OpticsIntermediate

Concept module

Total Internal Reflection

Push a ray from a higher-index medium toward a lower-index boundary, watch the critical angle emerge, and see the same live diagram hand off from ordinary refraction to full internal reflection.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Drag the incoming ray or use the sliders. The boundary diagram, critical-angle readouts, and response graphs stay on the same Snell-law model.

Graphs

Switch graph views without breaking the live stage and time link.

Threshold transition

The transmitted branch rises toward 90 degrees at the critical angle. Beyond that threshold, the reflected branch takes over because the ray stays inside medium 1.

incident angle theta_1 (°): 0 to 80angle from the normal (°): 0 to 90

Refracted branchCritical angleReflected branch

Hover or scrub to link the graph back to the stage.incident angle theta_1 (°) / angle from the normal (°)

Controls

Adjust the physical parameters and watch the motion respond.

Incident angle

θ_{1}

46°

Changes how steeply the incoming ray meets the normal.

Incident-medium index n1

n_{1}

1.52

Higher n means slower light in the incident medium.

Transmitted-medium index n2

n_{2}

Lowering n2 against n1 makes the critical angle smaller.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Try thisPrompt 1 of 5

Notice that just below theta_c the transmitted ray is still real, but it is flattening hard along the boundary as theta_2 approaches 90 degrees.

Try this

Use Near critical and nudge theta_1 upward one degree at a time until the transmitted ray nearly skims the boundary.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

θ_{1}

Incident angle

46°

Sets how steeply the incoming ray approaches the normal. Raising it drives the live transition from ordinary refraction toward and then beyond the critical angle.

Graph: Threshold transitionGraph: Transmitted angle below thresholdOverlay: Normal and angle guideOverlay: Critical-angle guideOverlay: Reflected-angle guide

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Stay with one boundary and one threshold. The point is to watch the refracted branch end honestly, then see why the reflected branch takes over.

Try thisPrompt 1 of 5

Notice that just below theta_c the transmitted ray is still real, but it is flattening hard along the boundary as theta_2 approaches 90 degrees.

Try this

Use Near critical and nudge theta_1 upward one degree at a time until the transmitted ray nearly skims the boundary.

Why it matters

That is the clean geometric limit you cross before the transmitted branch disappears.

Control: Incident angleGraph: Transmitted angle below thresholdOverlay: Critical-angle guide

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

4 visible

Overlay focus

Normal and angle guide

Shows the interface normal and the angle markers for the incident and transmitted branches.

What to notice

Every angle on this page is measured from the normal, not from the surface itself.

Why it matters

TIR mistakes usually start with measuring the wrong angle.

Control: Incident angleGraph: Threshold transitionGraph: Transmitted angle below thresholdEquation

θ_{1}

Challenge mode

Use the live threshold model rather than a detached puzzle. The checks read the same incident angle, index pair, and graph state you see on the stage.

0/2 solved

TargetCore

2 of 4 checks

Land just below the threshold

Starting from Glass to air near critical, tune the setup until the incident angle stays just below theta_c: keep theta_1 - theta_c between -2 and -0.4 degrees while theta_2 remains between 74 and 89 degrees.

Graph-linkedGuided start2 hints

Suggested start

Use the critical-margin row and the transition map together.

Matched

Open the threshold transition graph.

Threshold transition

Matched

Keep the critical-angle guide visible.

Pending

Keep theta_1 - theta_c slightly negative.

4.86°

Pending

Keep the transmitted angle steep but real.

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

Light crosses from n1 = 1.52 to n2 = 1 at 46°. Snell's law would require a sine above 1, so there is no real refracted ray and the ray reflects internally at 46°. The critical angle is about 41.14°.

Equation detailsDeeper interpretation, notes, and worked variable context.

Snell's law

The transmitted angle follows from the same relation used in ordinary refraction.

n_{1} sin (θ_{1}) = n_{2} sin (θ_{2})

θ_{1}

Incident angle 46°

n_{1}

Incident-medium index 1.52

n_{2}

Transmitted-medium index 1

Critical angle

This threshold marks the last incident angle that still allows a real transmitted ray when light tries to leave the higher-index medium.

sin (θ_{c}) = \frac{n _{2}}{n _{1}} (n_{1} > n_{2})

θ_{1}

Incident angle 46°

n_{1}

Incident-medium index 1.52

n_{2}

Transmitted-medium index 1

Reflected angle

Once the setup crosses into total internal reflection, the reflected ray stays in medium 1 and leaves at the same angle from the normal.

θ_{r} = θ_{1}

θ_{1}

Incident angle 46°

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Short explanation

What the system is doing

Total internal reflection is not a separate boundary law sitting beside refraction. It is the point where Snell's law stops giving a real transmitted angle because the light is trying to leave a higher-index medium for a lower-index one too steeply.

This concept keeps one compact boundary in charge. The incident angle, refractive-index contrast, critical-angle readout, reflected path, graph previews, prediction mode, compare mode, quick test, and worked examples all stay tied to the same live boundary state so the handoff from ordinary refraction to full internal reflection stays honest.

Key ideas

01Total internal reflection only becomes possible when light starts in the higher-index medium and tries to enter a lower-index medium, so n_1 must be larger than n_2 in the chosen travel direction.

02The critical angle satisfies sin(theta_c) = n_2 / n_1, so a larger index contrast lowers theta_c and makes internal reflection happen at smaller incident angles.

03As theta_1 approaches theta_c from below, the transmitted ray opens until it skims along the boundary at theta_2 = 90 degrees. Above that limit there is no real transmitted angle.

04Once the ray keeps reflecting inside the higher-index region, you have the same local event that fiber-optic guidance relies on, even though this page stays focused on one boundary.

Live worked example

Solve the exact state on screen.

Work from the boundary you are already editing. The substitutions pull from the live indices and incident angle, and the same state stays visible on the ray diagram and threshold graphs.

Live valuesFollowing current parameters

For the current media pair, what critical angle does this boundary allow?

n_{1}

Incident-medium index

1.52

n_{2}

Transmitted-medium index

θ_{c}

Critical angle

41.14 °

1. Check the travel direction

A critical angle only exists when light starts in the higher-index medium, so first compare

n_{1} = 1.52

with

n_{2} = 1

2. Use the critical-angle relation

When

n_{1} > n_{2}

, use

sin (θ_{c}) = \frac{n _{2}}{n _{1}}

3. Evaluate the threshold

θ_{c} = 41.1 4^{\circ}

Critical-angle result

θ_{c} = 41.1 4^{\circ}

The current incident angle is 4.86^\circ above the critical angle, so the refracted branch has already ended.

Common misconception

Any large incident angle can give total internal reflection if the boundary is steep enough.

A large angle by itself is not enough. The light must be going from higher n to lower n so that a real critical angle exists in the first place.

If n_1 is not larger than n_2, Snell's law still gives a real transmitted angle no matter how far you raise theta_1 within this model's range.

Mini challenge

Keep the incident angle fixed above the glass-to-air critical angle, then raise n_2 closer to n_1. Before you test it, what should happen to the threshold?

Prediction prompt

Decide whether theta_c rises or falls, and whether the same ray is more or less likely to keep reflecting internally.

Check your reasoning

The critical angle rises, so the same incident angle becomes less likely to stay in total internal reflection.

Raising n_2 closer to n_1 reduces the index contrast, so n_2 / n_1 gets larger and theta_c increases. A setup that used to be above the threshold can drop back into ordinary refraction.

Quick test

Reasoning

Question 1 of 4

Reason from the threshold, not from a slogan. Use the index contrast, the critical angle, and the branch that still exists on the live boundary.

Which condition must be true before total internal reflection can happen at all?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a horizontal boundary between a top medium and a bottom medium, with a dashed normal line through the contact point. One incoming ray approaches from the upper left. Below the critical angle a transmitted ray leaves into the lower right, and above the critical angle the outgoing branch stays in the top medium as a reflected ray.

Optional overlays show the normal and angle markers, the critical-angle threshold, the equal-angle reflection cue used after the threshold, and the speed labels tied to the current refractive indices.

Graph summary

The threshold transition graph plots angle from the normal against incident angle for the current media pair. The transmitted branch rises toward 90 degrees at the critical angle, and beyond that point the reflected branch continues because the ray remains in the first medium.

The transmitted-angle graph isolates the ordinary-refraction branch below threshold, so hovering it previews another boundary setup rather than a later time.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Compare with reflectionOptics

Total Internal Reflection

See each variable before you move it.

Normal and angle guide

Land just below the threshold

What the system is doing

Solve the exact state on screen.

For the current media pair, what critical angle does this boundary allow?

Which condition must be true before total internal reflection can happen at all?

Keep moving through the concept map.

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