Glass that gathers light — and glass that spreads it
Convex & concave lenses · 1/v − 1/u = 1/f · magnification · real vs virtual
By Pawan Nayar · LLOS.ai · ~10 min read · Lenses & the Lens Formula
A thin lens does one quiet job: it bends every ray that passes through it. Thicker in the middle and rays meet; thinner in the middle and they part. One formula, with honest signs, tells you where the image will sit.
Lens formula · 1/v − 1/u = 1/fSigns · New Cartesianm = v/u47 questions · 7 formats · layered hints
What you'll learn
Lenses & the lens formula — the whole toolkit
This page covers convex and concave thin lenses, the lens formula1/v − 1/u = 1/f, magnificationm = v/u, and real vs virtual images under the New Cartesian sign convention. By the end you'll be able to:
Use 1/v − 1/u = 1/f with consistent signs to find u, v or f.
Predict when a convex lens magnifies (object inside f) versus when it inverts on a screen.
Read magnification — size and upright vs inverted from m = v/u.
Drive the Lens Lab and match what the rays and formula predict.
Why it matters · where it's tested
Cameras, eyes and every board's ray-optics paper
Your eye is a lens. Your phone is a stack of them. Class 10–12 papers lean hard on the lens formula and ray diagrams. We go beyond the syllabus, but we never skip it:
Helper for a single spherical surface story (context)
m
🟡
convex f > 0 · concave f < 0
New Cartesian (light left → right)
—
🟢
object on left ⇒ u < 0
Against the incident light = negative
—
🟢
See it live
Predict image distance v, lock it, then dial |u| and |f| and watch the rays. Convex / concave · lens formula · m readout. 🟢 real thin-lens engine
Lens Lab
|u|30 cm
v60 cm
f+20 cm
m−2
Predict first: for this |u| and |f|, what is v in cm? (signed OK; or |v| if unsure)
Guess, then change |u| — the rays wait until you commit.
Lens
1/v − 1/u = 1/f → v = +60 cm
Set the lens and |u| — the formula 1/v − 1/u = 1/f places the image.
Lens Challenge — hit the target
Dial |u| for a convex lens with fixed |f| = 20 cm. Image distance v must land in the amber screen zone — 3 attempts.
3-move focus
Attempts left: 3
Dial |u|, then fire. Physics places v from 1/v − 1/u = 1/f.
Impossible lenses — what is wrong?
Each diagram breaks thin-lens physics. Tap the broken idea. Expert thinking starts when something looks almost right.
What is impossible?
Inside f a convex lens makes a virtual upright enlarged image — rays diverge; a far-side screen stays dark.
What is impossible?
For a real object, a diverging lens always forms a virtual diminished upright image — you look into the lens.
What is impossible (as correct method)?
Thin lenses use 1/v − 1/u = 1/f under New Cartesian. The plus form is the mirror story.
Always · Sometimes · Never
Thinking beats memorising. Sort the universe of lenses into three boxes.
Always
1/v − 1/u = 1/f (with a chosen sign convention)
A ray through the optical centre goes undeviated (thin-lens model)
A lens bends light — it does not invent new photons
Sometimes
Convex lens forms a real inverted image (depends on u)
Convex lens magnifies (object inside f)
|m| > 1 (enlarged) or |m| < 1 (diminished)
Never
Concave lens casting a real image of a real object alone
Mixing mirror (+) and lens (−) formulas in one sum
Putting f in centimetres inside P = 1/f for dioptres
Ray Builder — place the image
Convex lens, |f| = 20 cm. Guess |v| for the given |u|. Check against the formula.
You place v
|u| locked until Check
Place the image yourself — the engine grades the formula, not hope.
Design your own lens
Pick a job. Dial kind, |u| and |f|. The formula tells you if you built the right tool.
Goal: virtual enlarged face
Not yet
For a reading glass: convex, object inside the focus → virtual, upright, |m| > 1.
Everyday detective — name that lens
No upload needed — read the scene. What lens (and image behaviour) is at work?
Page looks huge & upright — why?
Classic magnifier: u between O and F → virtual, upright, |m| > 1.
Inverted picture on the wall — why?
Projector throw: object beyond f (often beyond 2f) → real inverted image at v on the screen.
Wide hallway in a tiny porthole — why?
Door peep-holes use a strong diverging lens — virtual, diminished, wide field of view.
What's going on
In plain terms: a thin lens adds a bend to every ray. Parallel rays meet at the focus of a convex lens — or appear to diverge from the focus of a concave lens. The formula is the map of where those bends put the image.
What it is
A convex lens is thicker in the middle and converges light. A concave lens is thinner in the middle and diverges light. The optical centre is the mid-point: a ray through it goes straight on. The principal focus is where parallel rays meet (or appear to come from).
How the formula works
1/v − 1/u = 1/f links object distance u, image distance v and focal length f. With New Cartesian signs (light left→right), a real object on the left has u < 0; a convex lens has f > 0; a real image on the right has v > 0. Magnification is m = v/u.
How it works in the lab
The Rays tab eases object rays through the lens and shows where they meet (motion-feel). Switch Convex / Concave, set |u| and |f|, and readouts follow the formula. The 1/v vs 1/u tab is a straight line of slope 1 shifted by 1/f — the formula drawn as a graph. 🟡 maths of the picture
Edge cases
Object at focus (convex) → rays leave parallel; no sharp finite image (v → ∞).
Object inside f (convex) → virtual, upright, enlarged (magnifying glass).
Concave → f negative; image of a real object always virtual and diminished.
Mixing signs → pick New Cartesian and stick to it for the whole sum.
Three quantities & units
u, v, f (cm or m) — object, image, focal distances.
Magnification m — m = v/u (sign flags inversion).
Power P (D) — P = 1/f with f in metres.
Solve it with me, step by step 🟢 Class 8–10
Exams reward the method, not just the answer. Work it out one step at a time — read the thought, predict the line, then reveal it. Switch to practice to type your own numbers and check them.
See the maths
Drag the numbers — watch v, m and power update.
🔢 P = 1/f
Lens power — f in metres, P in dioptres.
∠r
+2.0 D
P = 1/f = +2.0 D
📏 1/v − 1/u = 1/f
Convex lens — New Cartesian (u negative, f positive).
v
−60 cm
v from 1/v = 1/f − 1/u → −60 cm
🔎 m = −v/u
Magnification from object and image distances.
|m|
2.0
|m| = |v|/|u| → 2.0 (sign separate)
Myth-busting
Things lenses are not
Sticky wrong ideas about lenses deserve their own warning label. Tap a card to bust the myth.
From burning glass to phone cameras
Same bend. Wilder stacks. Civilization is a gallery of lenses.
~700 BCE
Rock-crystal burning glasses
Assyrian polished rock — sunlight gathered to a focus long before the word "lens".
1280s · Europe
Spectacles arrive
Convex reading glasses open books to ageing eyes — power before the formula.
1600s
Microscope & telescope
Stacked lenses invent new worlds — cells and moons in the same century.
1800s–today
Cameras & cinema
Compound objectives, apertures, film then sensors — everyday 1/v − 1/u craft.
Now
Phone modules
Tiny multi-element stacks + software — still thin-lens geometry at heart.
Physics illusions
The brain meets fair refraction and calls it magic. Each trick is still a lens (or a stack).
Fish-eye peep
A strong concave door lens packs a wide hallway into a tiny porthole — always virtual, always diminished.
Diverging · wide field
Magnifying glass
Hold a convex lens inside its focal length — the page leaps upright and huge (virtual image).
Object inside f
Projector flip
Slide beyond 2f → real inverted image on the wall. Upside-down is not a bug; it is m < 0.
Real image on screen
Camera bokeh
Defocus is image distance missing the sensor plane — the formula still holds; the plane does not.
v ≠ sensor
In the real world
Your eye · living camera
The cornea and lens bend light onto the retina. Glasses and contacts only tweak f so the image lands where the sensors are.
Phone camera stacks
Several tiny elements share the work of one fat lens — still 1/v − 1/u at heart, plus coatings and software.
Microscopes & projectors
Objective and eyepiece (or lamp and lens) place real and virtual images in careful order — school ray diagrams at civilization scale.
March of technology
How lenses escaped the textbook
Once humans could grind glass, they built spectacles, cameras, microscopes and sensors that all start from 1/v − 1/u = 1/f.
From magnifier to projector — same lens 🟡
Drag the ladder — everyday convex jobs that still obey 1/v − 1/u = 1/f.
glasseyecamscopehall
Reading glass
object inside f
Job changes; the formula stays the habit.
Build it yourself — science-fair projects
Physics you can see. Each project shows refraction through a lens — and the measuring is what turns a demo into a science-fair winner.
🔍Beginner
Magnifier distance
Shows · object inside f
Build: hold a reading glass over text; move until the letters look largest and sharp.
Measure: distance lens-to-page — it should be less than f for a virtual upright image.
🪟Beginner
Window focus
Shows · distant object → f
Build: aim a convex lens at a bright distant window; catch a sharp image on a card.
Measure: lens-to-card ≈ f for a far object.
🕯️Intermediate
Candle on a screen
Shows · real inverted image
Build: candle, convex lens beyond f, white card.
Measure: u and v; check 1/v − 1/u ≈ 1/f.
👓Intermediate
Spectacle power
Shows · P = 1/f
Build: borrow a known reading glass labelled in dioptres.
Measure: focus a distant lamp; compare 1/f (metres) with the stamped power.
🚪Intermediate
Peephole vs plane
Shows · concave wide field
Build: compare a door peep-hole view with looking through a clear flat pane.
Measure: count distinct hallway objects — diverging lens should win on width.
📐Champion
u–v graph for f
Shows · lens formula slope
Build: several (u,v) pairs for one convex lens; plot 1/v against 1/u.
Measure: intercept gives 1/f — compare with a single distant-object trial.
Glossary — the 10 words that unlock it
Optical centre
What it means
Mid-point of a thin lens on the axis.
Why it matters
A ray through it goes undeviated (school model).
Example
Point O in every ray diagram.
Key question
Does every ray bend at O?
Convex lens
What it means
Thicker in the middle — converging.
Why it matters
Can form real or virtual images depending on u.
Example
Reading glass, camera objective.
Key question
When is the image virtual?
Concave lens
What it means
Thinner in the middle — diverging.
Why it matters
Real-object image is always virtual & diminished.
Example
Door peep-hole, some spectacle lenses.
Key question
Can it cast a candle on a screen?
Focal length f
What it means
Distance from optical centre to principal focus.
Why it matters
Sets the scale of every image location.
Example
|f| = 20 cm in the lab.
Key question
How does f relate to power?
Lens formula
What it means
1/v − 1/u = 1/f with a sign convention.
Why it matters
Predicts where the image sits.
Example
u = −30, f = +20 → v = +60 cm.
Key question
Why the minus, not a plus?
Magnification m
What it means
m = v/u ≈ h′/h.
Why it matters
Size and often orientation of the image.
Example
m = −2 means twice as tall, inverted.
Key question
What does |m| < 1 mean?
Power P
What it means
P = 1/f with f in metres; unit dioptre.
Why it matters
Spectacle labels and quick strength talk.
Example
+2.0 D → f = 0.5 m.
Key question
Is negative power converging?
New Cartesian signs
What it means
Against incident light = −; along = +.
Why it matters
Makes u, v, f consistent in one sum.
Example
Object on left ⇒ u < 0.
Key question
What is the sign of f for concave?
Real vs virtual
What it means
Real: rays meet on a screen. Virtual: rays only appear to meet.
Why it matters
Projectors need real; magnifiers need virtual.
Example
Candle on card vs page under a glass.
Key question
Which side is a virtual image for a lens?
Principal focus
What it means
Where parallel rays meet (convex) or appear to diverge from (concave).
Lenses look simple until the first sign convention. Straight answers:
What is the lens formula?
What
1/v − 1/u = 1/f links object distance u, image distance v and focal length f for a thin lens (with a chosen sign convention). Know two, solve for the third. Schools usually use New Cartesian signs.
Why minus for lenses but plus for mirrors?
Why
Reflection and refraction place object and image differently once signs are chosen. Do not mix 1/v + 1/u = 1/f (mirrors) with 1/v − 1/u = 1/f (lenses) in one sum.
When does a convex lens magnify?
How
When the object is between the optical centre and the focus, the image is virtual, upright and enlarged — the magnifying-glass case. Beyond the focus you get real inverted images whose size depends on exact u.
Can a concave lens cast a real image of a candle?
Scenario
Not for a real object in air — the image is always virtual, upright and diminished. You look into the lens; a screen on the far side stays without a sharp flame image from that lens alone.
How is power related to focal length?
How
P = 1/f with f in metres; unit dioptre (D). A +2.0 D glass has f = 0.5 m. Positive power means converging.
What does m = −2 mean?
What
The image is twice as tall as the object and inverted. Magnitude is size; the minus sign is orientation for the usual lens convention m = v/u.
Where is the optical centre?
What
For a thin lens it is the mid-point on the principal axis. A ray through it goes straight on undeviated in the school model — one of the three standard construction rays.
Lens vs mirror — same idea?
Comparative
Both map object distance to image distance with a focal length. Mirrors bounce; lenses bend. Formulas differ by a sign under New Cartesian — learn each device on its own page.
Common mistakes — and the fix 🟢 Class 10
Spot these before the exam does.
The slip
The fix
Using 1/v − 1/u = 1/f for a lens
Thin lens: 1/v − 1/u = 1/f
Flipping signs mid-sum
Pick New Cartesian and stick to it
Writing P = f or f in cm inside P = 1/f
f in metres for dioptres
Saying concave always magnifies
Concave (real object) → diminished virtual
Forgetting object-at-f → no finite image
Convex, u = −f → rays || · v → ∞
Test yourself — a mixed set
Seven question formats, the way Beyond Dictionary serves them — multiple choice, multiple-correct, fill-in-the-blank, match, sequence, read-think-connect, and write-your-own. Every question has layered hints: a quick nudge, the reasoning, then a deeper connection — so a wrong answer opens a door, never a dead end. 🟢 received from a board-tagged question bank · seed toward 2,000
Pick your board — the set re-tunes to its wording and emphasis. Competitive draws the JEE / NEET / Olympiad lane.
Loading the question bank…
Question 1 of 16
Multiple choice
Key takeaways
Convex gathers, concave spreads — thicker middle vs thinner middle.
1/v − 1/u = 1/f with New Cartesian signs places every thin-lens image.
m = v/u — size and upright vs inverted in one number.
P = 1/f (f in metres) is how spectacle strength is spoken.
Object inside f (convex) → virtual magnifier; beyond f → real images possible.
🪜 Where this lesson leads
Lenses are the floor of cameras, eyes and instruments. Master them and you have already started climbing toward: