OpticsIntro

Concept module

Lens Imaging

Trace principal rays through converging and diverging lenses, connect the signed thin-lens equation to the diagram, and watch image distance and magnification respond to the same object setup.

Interactive lab

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

Lens Imaging

Drag the object arrow or use the controls. The ray diagram, signed distances, and response graphs stay on the same thin-lens model.

Lens state

lens: converging
f: 0.8 m
d_o: 2.4 m
d_i: 1.2 m
m: -0.5

real, inverted, smaller
Real image can land on a screen.

Graphs

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

Object to image map

Shows how the signed image distance responds when you move the object while keeping the same lens.

object distance d_o (m): 0.45 to 4signed image distance d_i (m): -4.4 to 4.4

Virtual branchReal branch

Hover or scrub to link the graph back to the stage.object distance d_o (m) / signed image distance d_i (m)

Controls

Adjust the physical parameters and watch the motion respond.

Converging lensTurn this off to switch to a diverging lens with the same focal-length magnitude.

Focal-length magnitude

f

0.8 m

Controls the focal points and how sharply the rays bend.

Object distance

d_{o}

2.4 m

Moves the object along the principal axis.

Object height

h_{o}

1 m

Changes the object size so the magnified image height can be compared directly.

Presets

Predict -> manipulate -> observe

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

Graph readingPrompt 1 of 2

Notice that the object-image graph is object-distance based, not time based: hovering the graph previews a different geometric setup, not a later moment.

Try this

Hover the image-distance graph and watch the ray diagram jump to the matching object position immediately.

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.

f

Focal length

0.8 m

Sets how strongly the lens bends the rays. The sign comes from the lens-type toggle: converging uses positive f, diverging uses negative f.

Graph: Object to image mapGraph: MagnificationOverlay: F and 2F markersOverlay: Principal rays

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

Use the live prompt to connect the signed equation, the rays, and the graph branch that the current setup lives on.

Graph readingPrompt 1 of 2

Graph: Object to image map

Notice that the object-image graph is object-distance based, not time based: hovering the graph previews a different geometric setup, not a later moment.

Try this

Hover the image-distance graph and watch the ray diagram jump to the matching object position immediately.

Why it matters

It keeps the graph honest: you are comparing families of setups, not stepping through time.

Graph: Object to image mapOverlay: F and 2F markers

Guided overlays

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

2 visible

Overlay focus

F and 2F markers

Shows the focal points and twice-focal points on both sides of the lens.

What to notice

Crossing the focal mark changes the image from real to virtual for a converging lens.

Why it matters

The focus markers anchor the standard lens-regime language directly on the stage.

Control: Converging lensControl: Focal-length magnitudeControl: Object distanceGraph: Object to image mapEquation

f

Equation

d_{o}

Challenge mode

Use the real lens controls and graphs to hit image targets. The checks read the live signed distances and magnification instead of a separate puzzle state.

0/2 solved

TargetCore

2 of 4 checks

Real-image target

Starting from the converging reference, tune the setup until the image distance lands between 1.0 and 1.2 m and the magnification lands between -1.4 and -1.1.

Graph-linkedGuided start2 hints

Suggested start

Use the image-map graph and the ray intersection together.

Matched

Open the object-image map.

Object to image map

Matched

Keep the principal rays visible.

Pending

Make d_i land between 1.0 and 1.2 m.

1.2 m

Pending

Keep magnification between -1.4 and -1.1.

-0.5

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

The converging lens uses signed focal length 0.8 m. An object at 2.4 m with height 1 m forms a inverted, smaller real image at 1.2 m, with magnification -0.5.

Equation detailsDeeper interpretation, notes, and worked variable context.

Thin-lens equation

Relates signed focal length, object distance, and image distance for a thin lens.

\frac{1}{f} = \frac{1}{d _{o}} + \frac{1}{d _{i}}

f

Focal length 0.8 m

d_{o}

Object distance 2.4 m

Magnification

Gives the size ratio and the orientation sign of the image.

m = \frac{h _{i}}{h _{o}} = - \frac{d _{i}}{d _{o}}

d_{o}

Object distance 2.4 m

h_{o}

Object height 1 m

Sign rule

Positive image distance means the refracted rays actually meet and the image can land on a screen.

d_{i} > 0 \Rightarrow real image on the far side

Negative image distance means the rays only appear to meet by backward extension.

f

Focal length 0.8 m

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

Thin-lens imaging works when many rays from one object point leave the lens in a pattern that either meets at one image point or appears to meet there when you extend the rays backward. The ray diagram and the thin-lens equation are two views of that same geometry.

This module keeps the setup compact on purpose. You change lens type, focal length, object distance, and object height, then the signed image distance, image orientation, and magnification update together on the stage and in the response graphs.

Key ideas

01A converging lens can make a real inverted image when the object is outside the focal length, but it flips to a virtual upright image when the object moves inside the focal length.

02A diverging lens always makes a virtual upright reduced image for a real object on the left.

03The thin-lens equation gives the signed image distance, and magnification tells both the image size ratio and whether the image is upright or inverted.

Live worked example

Solve the exact state on screen.

Solve the current lens setup, not a detached worksheet. The substitutions follow the live controls, and the same signed values appear in the ray diagram and the graphs.

Live valuesFollowing current parameters

For the current converging lens, what signed image distance follows from the thin-lens equation?

f

Signed focal length

0.8 m

d_{o}

Object distance

2.4 m

1. Start from the thin-lens relation

Use

\frac{1}{f} = \frac{1}{d _{o}} + \frac{1}{d _{i}}

2. Rearrange for the signed image distance

\frac{1}{d _{i}} = \frac{1}{f} - \frac{1}{d _{o}} = 1.25 - 0.42 = 0.83

3. Invert the result

d_{i} = 1.2 m

Signed image distance

d_{i} = 1.2 m

The positive image distance means the refracted rays actually meet on the far side of the lens, so the image can be projected onto a screen.

Common misconception

A diverging lens spreads light out, so it does not make an image at all.

A diverging lens still makes an image. The refracted rays separate, but their backward extensions meet at a virtual image on the object side.

That is why the image is upright, reduced, and cannot be projected onto a screen even though it is still a legitimate image point in the geometry.

Mini challenge

If you move a converging-lens object inward toward the focal point, what does the ray diagram predict before the object crosses inside the focus?

Prediction prompt

Decide what happens to image distance and image size.

Check your reasoning

The real image moves farther away and grows in magnitude.

As the object approaches the focal point from outside, the refracted rays need more distance to meet. The signed image distance shoots outward and the magnitude of the magnification grows.

Quick test

Reasoning

Question 1 of 4

Use the sign of d_i, the sign of m, and the ray behavior together. The goal is to reason from the live lens model, not to recite a slogan.

Which setup can place the image on a screen to the right of the lens?

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 thin lens at the center of the principal axis, an object arrow to the left, and an image arrow that moves according to the signed thin-lens equation. Depending on the setup, the image arrow appears on the far side as a real inverted image or on the object side as a virtual upright image.

Optional overlays show the focal markers, the principal rays, and the distance-and-height guide used in magnification.

Graph summary

The object-image graph plots signed image distance against object distance for the current lens family and focal length.

The magnification graph plots m against object distance, so the sign and magnitude of the image scaling are visible without leaving the ray diagram.

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 mirrorsOptics

Lens Imaging

See each variable before you move it.

F and 2F markers

Real-image target

What the system is doing

Solve the exact state on screen.

For the current converging lens, what signed image distance follows from the thin-lens equation?

Which setup can place the image on a screen to the right of the lens?

Keep moving through the concept map.

Mirrors

Optical Resolution / Imaging Limits

Refraction / Snell's Law