[{"data":1,"prerenderedAt":807},["ShallowReactive",2],{"layer:light:investigate":3},{"layer":4,"contentHash":786,"dependencyHashes":787,"approval":801,"releaseId":806},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":781,"reviewStatus":782,"authoring":783},1,"light","en","investigate","Chasing light: measuring, mirroring and bending it on purpose","How fast is light, and how would you find out? Predict and test curved mirrors, lenses, TIR and rainbows.","Step into the shoes of Rømer and Fizeau to measure something that seemed instant, then turn detective on curved mirrors, lenses pushed to a magnifier, total internal reflection in a diamond and a fibre-optic cable, and finally the exact geometry that puts a rainbow at 42 degrees from the Sun.",[13,14,15,16,17],"Explain how Rømer's and Fizeau's very different methods both measured the speed of light.","Predict whether a concave or convex mirror, or a lens, will give a real or virtual, magnified or diminished image.","Explain total internal reflection using the critical angle, and connect it to optical fibres and a diamond's sparkle.","Explain why a rainbow forms at 42 degrees from the antisolar point, with a fainter second bow and a dark band between.","Explain why the sky is blue and sunsets are red using scattering's strong dependence on wavelength.",42,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 42 minutes",{"label":29,"value":30},"Prior knowledge","Understand: reflection, refraction, refractive index",{"label":32,"value":33},"Chapters","9",{"label":35,"value":36},"Labs","Sorting games, ray box, matching, prism bench",{"label":38,"value":39},"Method","Predict first, then test",[41,45,51,57,60,78,91,96,141,146,149,162,174,179,199,215,220,223,228,241,253,266,316,321,333,338,341,354,379,413,416,421,426,429,442,452,456,461,464,477,490,514,518,523,526,539,551,555,569,573,578,581,594,607,620,624,629,634,666,679,749,759,765,769],{"id":42,"type":43,"markdown":44},"intro-investigate","prose","Discover gave you the facts. Understand gave you the rules. This layer asks you to act like the people who worked those rules out in the first place: **predict, test, and see whether the prediction survives.**\n\nYou will chase a genuinely hard historical question — how do you clock something that seems to arrive everywhere at once? — and then turn detective on curved mirrors, lenses, the bent-straw trick taken to its limit, and the rainbow. Every chapter starts with a prediction. Make it before you read on.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-read-i","callout","observation","How to use this lesson","Each chapter opens with a **prediction** block. Choose an answer and commit to it — in your head or out loud — before you scroll past it. Being wrong costs nothing and teaches more than being right by luck.\n\nWhere a lab appears, run it before reading the paragraph that follows: the paragraph is the debrief, not the instructions.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"i-ch1","chapter","A puzzle: is light instant, or just very fast?","Chapter 01","1 Instant or fast?",{"id":58,"type":43,"markdown":59},"roemer-setup","For most of history, nobody could tell the difference between \"light arrives instantly\" and \"light arrives so fast you cannot measure the delay.\" Galileo tried in 1638: two people with covered lanterns stood on hilltops a few kilometres apart, and each uncovered their lantern the instant they saw the other's light. He found no measurable delay at all — not because light is instant, but because even at hilltop distances the delay is a few **millionths** of a second, hopelessly too small for a human with a lantern and a pulse to catch.\n\nThe first real evidence came not from a lantern but from a **moon of Jupiter**, watched over years.",{"id":61,"type":62,"prompt":63,"options":64,"explanation":77},"pred-roemer","prediction","In 1676, the Danish astronomer Ole Rømer was timing the eclipses of Jupiter's innermost large moon, Io, which ducks behind Jupiter once every 42.5 hours with clockwork regularity. He noticed the eclipses sometimes ran up to about 10 minutes **late**, and the lateness depended on where Earth was in its orbit. What do you think was going on?",[65,68,71,74],{"id":66,"label":67},"a","Io's orbit genuinely speeds up and slows down through the year",{"id":69,"label":70},"b","Jupiter itself was moving, changing the eclipse timing",{"id":72,"label":73},"c","When Earth is farther from Jupiter, the light from each eclipse has extra distance to cross, so it arrives later than predicted",{"id":75,"label":76},"d","The telescopes of the time were simply too unreliable to trust","**c.** Rømer realised the eclipses were exactly on schedule; only the *light announcing them* was delayed. When Earth is on the far side of the Sun from Jupiter, the eclipse light has to cross roughly an extra Earth-orbit diameter — about 300 million kilometres — before it reaches us. It arrives late by exactly the time light takes to cross that extra distance. Io's orbit never changed at all.",{"id":79,"type":80,"title":81,"problem":82,"steps":83,"help":89},"we-roemer","worked_example","Turning Rømer's delay into a speed","Rømer (and later Christiaan Huygens, who did the arithmetic) estimated that light takes about 22 minutes to cross the diameter of Earth's orbit around the Sun — about 2 astronomical units, or roughly 299 million km. What speed does that give, and how does it compare with the true value?",[84,85,86,87,88],"Diameter of Earth's orbit ≈ 2 × 1.496 × 10⁸ km = 2.99 × 10⁸ km = 2.99 × 10¹¹ m.","Speed = distance ÷ time = 2.99 × 10¹¹ m ÷ (22 × 60 s) = 2.99 × 10¹¹ ÷ 1,320.","That gives about **2.27 × 10⁸ m\u002Fs**.","The modern value is 2.998 × 10⁸ m\u002Fs, so Rømer and Huygens's estimate was about **24% too low** — impressively close for a first attempt with 1670s clocks and telescopes.","Using the modern speed instead, that same 2 AU crossing actually takes 16 minutes 38 seconds, not 22 minutes: their timing of the eclipses, not their distance to Jupiter, was the weaker link.",{"simplerExplanation":90},"Distance ÷ time = speed. Rømer's numbers give about 2.3 × 10⁸ m\u002Fs, about a quarter slower than the real answer — a good first guess, not a wrong one.",{"id":92,"type":47,"variant":93,"title":94,"markdown":95},"nuance-roemer-doubt","nuance","Not everyone was convinced at once","Rømer never actually announced a value for the speed of light himself — he only argued that the delay existed and roughly estimated its size. Some astronomers of the time, including the influential Cassini (his own observatory director), were unconvinced and thought the timing anomalies might have another cause. It took decades, and a second, completely independent method — James Bradley's 1728 discovery of the **aberration of starlight** — before most astronomers accepted that light's speed, though enormous, was finite and measurable.",{"id":97,"type":98,"component":99,"componentVersion":5,"config":100,"objective":135,"textAlternative":136,"help":137},"lab-roemer-eclipse","interactive","sort-game",{"prompt":101,"bins":102,"items":109,"seconds":134},"Earth is somewhere in its orbit when an eclipse of Io happens. Will the light announcing it arrive early, on time, or late compared with the average?",[103,106],{"id":104,"label":105},"early","Earlier than average",{"id":107,"label":108},"late","Later than average",[110,114,118,122,126,130],{"id":111,"label":112,"bin":104,"why":113},"near1","Earth closest to Jupiter, moving towards it","Less distance for the light to cross than average, so the eclipse light arrives early.",{"id":115,"label":116,"bin":107,"why":117},"far1","Earth farthest from Jupiter, on the far side of the Sun","Up to an extra 2 AU (about 300 million km) for the light to cross: roughly 16.5 extra minutes.",{"id":119,"label":120,"bin":107,"why":121},"mid-out","Earth moving away from Jupiter, mid-orbit","Each successive eclipse has slightly farther to travel than the one before, so the arrivals drift later and later.",{"id":123,"label":124,"bin":104,"why":125},"mid-in","Earth moving towards Jupiter, mid-orbit","Each eclipse has slightly less far to travel than the one before, so the arrivals drift earlier and earlier.",{"id":127,"label":128,"bin":104,"why":129},"close2","Earth at its closest approach to Jupiter (opposition)","Minimum distance, minimum extra travel time: the earliest arrivals in the whole cycle.",{"id":131,"label":132,"bin":107,"why":133},"far2","Earth at its farthest point from Jupiter (conjunction)","Maximum distance: the latest arrivals in the whole 13-month cycle, up to about 10 minutes behind schedule.",0,"Work out whether Jupiter's moon Io will appear to eclipse early or late from six different points in Earth's orbit.","Six scenes describing where Earth sits in its year-long orbit relative to Jupiter. Sort each one into **earlier than average** or **later than average**, based only on whether the Earth–Jupiter distance is shrinking or growing at that point.\n\nThe pattern that emerges is Rømer's whole discovery: the eclipse times drift smoothly early and late over the year, exactly tracking the changing distance, and the total swing between the earliest and latest timings — about 16 minutes 38 seconds today — is the time light takes to cross the 2 AU diameter of Earth's orbit.",{"hints":138},[139,140],"Getting closer to Jupiter always means the light has a shorter trip.","The extreme early and late cases are opposition and conjunction, six months apart.",{"id":142,"type":53,"title":143,"eyebrow":144,"navLabel":145},"i-ch2","Catching light on Earth: Fizeau's spinning wheel","Chapter 02","2 Fizeau's wheel",{"id":147,"type":43,"markdown":148},"fizeau-setup","Rømer's method needed the whole Solar System as a stopwatch. In 1849, the French physicist Hippolyte Fizeau found a way to measure light's speed using equipment that fit on a hillside near Paris.\n\nHe shone a beam through a gap in the teeth of a fast-spinning toothed wheel, sent it 8,633 metres to a mirror on a distant hill, and back through the *same* wheel. If the wheel had turned by exactly one tooth's width while the light made its round trip, the returning beam would be blocked by the next tooth instead of passing back through the gap it left by.\n\nSpin the wheel just fast enough for the light to vanish behind the very next tooth, and you can calculate the round-trip time from the wheel's speed alone.",{"id":150,"type":62,"prompt":151,"options":152,"explanation":161},"pred-fizeau","Fizeau's wheel had 720 teeth. He found the light first vanished completely when the wheel spun at 12.6 revolutions per second. If the beam travels 8.633 km each way (17.27 km round trip), roughly how long did the round trip take?",[153,155,157,159],{"id":66,"label":154},"About 5.8 hundred-thousandths of a second (a few tens of microseconds)",{"id":69,"label":156},"About 5.8 thousandths of a second (a few milliseconds)",{"id":72,"label":158},"About 5.8 seconds",{"id":75,"label":160},"It cannot be worked out from this information alone","**a.** The round trip is 2 × 8,633 m = 17,266 m, and at roughly 3 × 10⁸ m\u002Fs that takes about 5.76 × 10⁻⁵ seconds — 57.6 microseconds, a few hundred-thousandths of a second. That is far too fast for any spinning wheel a person could watch directly; Fizeau's trick was to make the wheel spin fast enough that *many* teeth passed in that tiny time, so the light reliably met the next tooth instead of the gap it left through.",{"id":163,"type":80,"title":164,"problem":165,"steps":166,"help":172},"we-fizeau","From a spinning wheel to a speed of light","Wheel with 720 teeth, spinning at 12.6 revolutions per second, first blocks the returning beam completely. The mirror is 8,633 m away. What speed does this give for light?",[167,168,169,170,171],"In one revolution the wheel presents 720 teeth and 720 gaps, so 1,440 tooth-edges pass a fixed point.","At 12.6 rev\u002Fs, the time for the wheel to advance by one tooth-width is 1 ÷ (1,440 × 12.6) s ≈ 5.51 × 10⁻⁵ s.","That must equal the light's round-trip time: 2 × 8,633 m ÷ speed = 5.51 × 10⁻⁵ s.","Speed = 2 × 8,633 ÷ 5.51 × 10⁻⁵ ≈ **3.13 × 10⁸ m\u002Fs**.","That is about 4.5% above the modern value of 2.998 × 10⁸ m\u002Fs — remarkably good for gears, a stopwatch and a hillside, and the first measurement made entirely on the ground.",{"anotherExample":173},"It is the same idea as a fan: spin a ceiling fan fast enough and, lit by a flickering tube light, the blades can appear to freeze or crawl backwards. Fizeau used the freezing point deliberately, as a stopwatch.",{"id":175,"type":47,"variant":176,"title":177,"markdown":178},"careful-two-methods","careful","Two completely different methods, one answer","Rømer's astronomical method (1676, refined by Huygens) and Fizeau's toothed-wheel method (1849) share no equipment, no assumptions and 173 years. One used a moon hundreds of millions of kilometres away; the other used a wheel you could hold in two hands. Both landed within a few percent of 2.998 × 10⁸ m\u002Fs. When two independent methods agree, that is much stronger evidence than either alone — it is very hard to imagine two unrelated mistakes that happen to cancel out.",{"id":180,"type":181,"tone":182,"items":183},"spec-speed-history","spec","blue",[184,187,191,195],{"label":185,"value":186},"Galileo, 1638","Lanterns on hilltops: no delay detected. Correct conclusion, wrong reason — the delay was real but far too small to catch by eye.",{"label":188,"big":189,"value":190},"Rømer\u002FHuygens, 1676","2.27 × 10⁸ m\u002Fs","From the changing lateness of Io's eclipses across a year. About 24% below the true value.",{"label":192,"big":193,"value":194},"Fizeau, 1849","3.13 × 10⁸ m\u002Fs","From a toothed wheel and an 8,633 m round trip near Paris. About 5% above the true value.",{"label":196,"big":197,"value":198},"Modern defined value","2.99792 × 10⁸ m\u002Fs","Exact by definition since 1983: the metre is now defined from this speed, not the other way round.",{"id":200,"type":201,"itemId":202,"prompt":203,"check":204,"hints":209,"feedback":212},"prac-fizeau","practice","light.investigate-fizeau-teeth","A redesigned wheel has 1,440 teeth — double Fizeau's original 720 — turning at 6.3 rev\u002Fs (half his original 12.6), over the same 8,633 m path, and again the beam is first completely blocked. Using speed = 2 × distance × (2 × teeth × rev\u002Fs), what speed does this give, in units of 10⁸ m\u002Fs (to 2 decimal places)?",{"kind":205,"answer":206,"tolerance":207,"unit":208},"number",3.13,0.05,"×10⁸ m\u002Fs",[210,211],"2 × teeth × rev\u002Fs = 2 × 1,440 × 6.3. Compare that with 2 × 720 × 12.6 from the original setup.","Doubling the teeth and halving the speed leaves 2 × teeth × rev\u002Fs completely unchanged.",{"correct":213,"incorrect":214},"Right — 2 × 1,440 × 6.3 = 18,144, exactly the same as 2 × 720 × 12.6 from the original run, so this redesigned wheel gives precisely the same measured speed. Doubling the tooth count and halving the spin rate cancel out exactly, which is exactly the kind of self-consistency check that made Fizeau confident his method was measuring something real.","Compute 2 × teeth × rev\u002Fs for the new wheel and compare it with the original 720 and 12.6. If the two products match, the measured speed must match too.",{"id":216,"type":53,"title":217,"eyebrow":218,"navLabel":219},"i-ch3","Curved mirrors: predict the image","Chapter 03","3 Curved mirrors",{"id":221,"type":43,"markdown":222},"curved-setup","Every mirror you have met so far has been flat. Curve the reflecting surface and the law of reflection still applies at every single point — but because the surface tilts differently from point to point, the reflected rays no longer stay parallel. They can be gathered together or spread apart, and that changes everything about the image.\n\nThere are two shapes. A **concave** mirror curves inward, like the inside of a spoon, and can gather rays to a real point called the **focus**. A **convex** mirror curves outward, like the back of the spoon, and always spreads rays apart.",{"id":224,"type":47,"variant":225,"title":226,"markdown":227},"def-curved","definition","Concave, convex, focus, focal length","A **concave** mirror is a cave: it curves away from you, reflecting off its inner surface. Parallel rays (such as sunlight) reflect from it and converge to a point called the **principal focus**, at a distance called the **focal length** in front of the mirror.\n\nA **convex** mirror bulges towards you. Parallel rays reflect from it and spread out as if they came from a point *behind* the mirror — a virtual focus.\n\nA mirror's focal length is exactly **half** its radius of curvature: f = R ÷ 2. A concave shaving mirror with a 50 cm radius has a 25 cm focal length.",{"id":229,"type":62,"prompt":230,"options":231,"explanation":240},"pred-concave","You hold a concave shaving mirror close to your face, well within its focal length, and look at your reflection. What do you predict?",[232,234,236,238],{"id":66,"label":233},"Upright and magnified",{"id":69,"label":235},"Upside-down and magnified",{"id":72,"label":237},"Upright and smaller than life",{"id":75,"label":239},"No image forms that close","**a.** Held closer than the focal length, a concave mirror always gives an **upright, magnified, virtual** image — this is exactly why shaving and makeup mirrors are concave. Move your face back past the focal length and the image flips: it becomes real, upside-down, and can be projected onto a screen, which is how a concave mirror is used in a reflecting telescope or a solar cooker instead.",{"id":242,"type":80,"title":243,"problem":244,"steps":245,"help":251},"we-shave-mirror","The shaving mirror, worked out","A concave mirror has a focal length of 25 cm. A face is held 15 cm from it, closer than the focus. How big and what kind of image forms?",[246,247,248,249,250],"Use 1\u002Fv = 1\u002Ff − 1\u002Fu with f = 25 cm and u = 15 cm (both measured as positive distances from the mirror).","1\u002Fv = 1\u002F25 − 1\u002F15 = 3\u002F75 − 5\u002F75 = −2\u002F75, so v = −37.5 cm.","The **negative** sign means the image is **virtual**, sitting 37.5 cm *behind* the mirror, not in front of it.","Magnification = |v ÷ u| = 37.5 ÷ 15 = **2.5×**: the face looks two and a half times life size, and upright.","Move the same face out to 60 cm, beyond the focus, and the same formula gives v = +42.9 cm: **positive**, meaning real, upside-down, formed in front of the mirror — a completely different kind of image from the same piece of glass.",{"simplerExplanation":252},"Close to a concave mirror: big, upright, imaginary (behind the glass). Far from it: real, upside-down, hangs in the air in front of the glass.",{"id":254,"type":62,"prompt":255,"options":256,"explanation":265},"pred-convex","A convex mirror is used as a car's side mirror, or as a wide-angle security mirror in a shop. Whatever the object's distance, what kind of image does a convex mirror always give?",[257,259,261,263],{"id":66,"label":258},"Real, upright, magnified",{"id":69,"label":260},"Virtual, upright, diminished (smaller than life)",{"id":72,"label":262},"Real, upside-down, diminished",{"id":75,"label":264},"It depends entirely on the distance, like a concave mirror","**b.** Unlike a concave mirror, a convex mirror **never** changes character: it always gives a virtual, upright, diminished image, for any object distance. That is precisely why it is chosen for wide-angle jobs — cramming a wide scene into a small, upright, always-in-focus-looking image is worth more than getting the size right. A car with a wing mirror 200 cm behind it, seen in a convex mirror with a 20 cm focal length, appears only about 9% of its true size — which is also exactly why the mirror is stamped **\"Objects in mirror are closer than they appear\"**: your brain, used to plane mirrors, misjudges the shrunken image as being farther away than it really is.",{"id":267,"type":98,"component":99,"componentVersion":5,"config":268,"objective":310,"textAlternative":311,"help":312},"lab-mirror-sort",{"prompt":269,"bins":270,"items":277,"seconds":134},"Is this everyday mirror concave (curves inward, can focus and magnify) or convex (curves outward, always shrinks the view)?",[271,274],{"id":272,"label":273},"concave","Concave",{"id":275,"label":276},"convex","Convex",[278,282,286,290,294,298,302,306],{"id":279,"label":280,"bin":272,"why":281},"shave","A shaving or makeup mirror","Held close, it magnifies your face upright — a job only a concave mirror held within its focal length can do.",{"id":283,"label":284,"bin":275,"why":285},"wing","A car's side (wing) mirror","It shrinks the view to fit a wider field, so the driver sees more road with one small mirror.",{"id":287,"label":288,"bin":272,"why":289},"dentist","A dentist's mouth mirror","Held close to a tooth, it gives a magnified upright view of a small area.",{"id":291,"label":292,"bin":275,"why":293},"security","A wide-angle mirror in a shop corner","It packs a whole aisle into one small mirror, at the cost of making everything look smaller and farther away.",{"id":295,"label":296,"bin":272,"why":297},"torch","The reflector behind a torch or car headlight bulb","A bulb placed at the focus of a concave reflector sends its light out as a strong, roughly parallel beam.",{"id":299,"label":300,"bin":272,"why":301},"telescope","A reflecting telescope's main mirror","It gathers a wide beam of starlight from far away and focuses it to a bright, real image.",{"id":303,"label":304,"bin":275,"why":305},"adasroad","A curved mirror at a blind road junction","It shows a wide stretch of the crossing road to drivers who could otherwise see nothing coming.",{"id":307,"label":308,"bin":272,"why":309},"cooker","A large dish solar cooker","It concentrates sunlight from a wide dish down to a small, very hot spot at its focus.","Sort eight familiar curved mirrors into concave (can focus and magnify) and convex (always shrinks and widens the view).","Eight everyday curved mirrors — a shaving mirror, a car's side mirror, a dentist's tool, a shop security mirror, a torch reflector, a telescope mirror, a road-junction mirror and a solar cooker — sorted into **concave** and **convex** bins.\n\nThe rule that solves every card: anything designed to **magnify a nearby object** or **concentrate light to a hot or bright point** is concave. Anything designed to **widen the field of view** at the cost of making things look smaller and farther away is convex.",{"hints":313},[314,315],"Concave mirrors can form both magnified and real, upside-down images depending on distance.","Convex mirrors only ever do one job: shrink and widen the view.",{"id":317,"type":47,"variant":318,"title":319,"markdown":320},"nuance-mirror-limit","model_limit","Real mirrors are not perfect cones of rays","The formula 1\u002Fv = 1\u002Ff − 1\u002Fu assumes every ray close to the mirror's centre line focuses to exactly the same point — true only approximately, and only for rays that stay close to the axis. Wide mirrors (a satellite dish, a big telescope) are shaped as a **parabola**, not a sphere, specifically to fix this and bring even the outermost rays to the same sharp focus, a problem called **spherical aberration** in a plain spherical mirror.",{"id":322,"type":201,"itemId":323,"prompt":324,"check":325,"hints":327,"feedback":330},"prac-mirror","light.investigate-mirror-radius","A concave mirror has a radius of curvature of 60 cm. What is its focal length, in centimetres?",{"kind":205,"answer":326,"tolerance":134},30,[328,329],"Focal length is always half the radius of curvature.","f = R ÷ 2.",{"correct":331,"incorrect":332},"Right: f = 60 ÷ 2 = 30 cm.","Use f = R ÷ 2, so f = 60 ÷ 2 = 30 cm.",{"id":334,"type":53,"title":335,"eyebrow":336,"navLabel":337},"i-ch4","Lenses: bending an image into being","Chapter 04","4 Lenses",{"id":339,"type":43,"markdown":340},"lens-setup","A lens bends light by **refraction** instead of reflection, at both of its curved surfaces. A **convex** (converging) lens is thicker in the middle and bends parallel rays inward to a focus, just like a concave mirror bends them by reflection. A **concave** (diverging) lens is thinner in the middle and always spreads rays apart, exactly mirroring what a convex mirror does.\n\nThe formula is identical in form to the mirror formula you just used: 1\u002Fv = 1\u002Ff − 1\u002Fu. Only the geometry — refraction through glass instead of reflection off a coated surface — is different.",{"id":342,"type":62,"prompt":343,"options":344,"explanation":353},"pred-lens-magnify","A convex lens has a focal length of 10 cm. An object sits 30 cm away — well beyond the focus. What kind of image forms?",[345,347,349,351],{"id":66,"label":346},"Virtual, upright, magnified — a magnifying glass",{"id":69,"label":348},"Real, upside-down, smaller than the object",{"id":72,"label":350},"Real, upright, the same size as the object",{"id":75,"label":352},"No image at all; the object is too far away","**b.** Placed well beyond the focal length (here 3 times it), a convex lens forms a **real, inverted, diminished** image — exactly what a camera lens does with a distant scene. Using 1\u002Fv = 1\u002Ff − 1\u002Fu: 1\u002Fv = 1\u002F10 − 1\u002F30 = 1\u002F15, so v = 15 cm, magnification = 15 ÷ 30 = 0.5×, half life size. Only when the object is closer than the focal length does the same lens flip into a magnifying glass, giving a virtual, upright, enlarged image — try option (a) for u = 5 cm and you would find v = −10 cm, virtual and magnified.",{"id":355,"type":356,"caption":357,"columns":358,"rows":363},"table-lens-cases","table","The same convex lens (f = 10 cm), three object distances, all from the same formula 1\u002Fv = 1\u002Ff − 1\u002Fu",[359,360,361,362],"Object distance","Image distance","Image type","Real-world example",[364,369,374],[365,366,367,368],"30 cm (beyond 2f)","15 cm","Real, inverted, smaller (0.5×)","A camera photographing something far away",[370,371,372,373],"15 cm (between f and 2f)","30 cm","Real, inverted, larger (2×)","A slide projector enlarging a small slide",[375,376,377,378],"5 cm (inside the focus)","10 cm (virtual)","Virtual, upright, larger","A magnifying glass held close to text",{"id":380,"type":98,"component":381,"componentVersion":5,"config":382,"objective":407,"textAlternative":408,"help":409},"lab-lens-match","match-pairs",{"prompt":383,"mode":384,"pairs":385},"Match each optics term to what it means.","connect",[386,389,392,395,398,401,404],{"a":387,"b":388},"Real image","Light rays actually meet there; can be caught on a screen",{"a":390,"b":391},"Virtual image","Rays only appear to come from there; cannot be caught on a screen",{"a":393,"b":394},"Converging (convex) lens","Thicker in the middle; bends parallel rays to a real focus",{"a":396,"b":397},"Diverging (concave) lens","Thinner in the middle; spreads parallel rays apart",{"a":399,"b":400},"Principal focus","Where parallel rays meet, or seem to come from, after a lens or mirror",{"a":402,"b":403},"Magnification","Image size divided by object size; less than 1 means diminished",{"a":405,"b":406},"Dioptre","Unit of lens power, equal to 1 ÷ focal length in metres","Match seven optics terms — real image, virtual image, converging and diverging lenses, focus, magnification and dioptre — to their meanings.","A connect-the-pairs game with seven optics terms on one side and their plain-language meanings on the other: real versus virtual image, converging versus diverging lens, principal focus, magnification and the dioptre (a lens-power unit used on spectacle prescriptions, equal to 1 divided by the focal length in metres).",{"hints":410},[411,412],"A real image can be projected onto a screen; a virtual one cannot.","A stronger (more powerful) lens has a shorter focal length and more dioptres.",{"id":414,"type":43,"markdown":415},"eye-lens","Your own eye is a living example. Light entering it is bent mostly by the curved front surface, the **cornea**, with the flexible internal **lens** doing the fine adjustment. Together they act as a single converging lens system with a power of about **59 dioptres** — roughly a focal length of 17 mm — squeezing a real, upside-down image of the world onto the light-sensing **retina** at the back of the eyeball. Your brain, entirely used to this, simply learns to treat that upside-down signal the right way up; nothing in the eye itself ever flips it back.",{"id":417,"type":47,"variant":418,"title":419,"markdown":420},"aha-eye-focus","aha","The lens does less work than the cornea","Of the eye's roughly 59 dioptres of focusing power, the fixed, curved front surface (the cornea) supplies about two-thirds on its own, simply because light bends most sharply when it first crosses from air (refractive index 1.0) into a denser material. The flexible internal lens supplies the rest, and does the one job the cornea cannot: changing its own shape to bring both distant and close objects into focus, a trick called **accommodation**. A magnifying glass held close to a page does mechanically what your eye's lens does automatically.",{"id":422,"type":53,"title":423,"eyebrow":424,"navLabel":425},"i-ch5","Two lenses together: telescopes and microscopes","Chapter 05","5 Two lenses",{"id":427,"type":43,"markdown":428},"two-lens-setup","One lens can magnify. Put a **second** lens in the path and you can multiply the effect — which is exactly how both a microscope and a refracting telescope work, using nothing more exotic than two convex lenses lined up on the same axis.\n\nThe first lens (the **objective**) forms a real image of the object. The second lens (the **eyepiece**) is then used as a simple magnifying glass to examine *that* image, rather than the original object. The two magnifications multiply together.",{"id":430,"type":62,"prompt":431,"options":432,"explanation":441},"pred-two-lens","A microscope's objective lens forms a real image magnified 10 times. The eyepiece then magnifies that image a further 5 times, acting as a simple magnifying glass. What is the overall magnification of the object as finally seen?",[433,435,437,439],{"id":66,"label":434},"15 times (the two magnifications add)",{"id":69,"label":436},"50 times (the two magnifications multiply)",{"id":72,"label":438},"10 times (only the objective matters)",{"id":75,"label":440},"5 times (only the eyepiece matters)","**b.** Because the eyepiece magnifies the objective's *already-magnified* image, not the original object, the two factors multiply rather than add: 10 × 5 = 50 times overall. This multiplying trick is exactly why compound microscopes reach far higher magnifications than a single hand lens ever could — a hand lens is limited to roughly a magnifier's power (25 cm ÷ focal length), typically at most about 10 to 20 times, while stacking two or more lenses easily reaches hundreds of times.",{"id":443,"type":80,"title":444,"problem":445,"steps":446,"help":450},"we-telescope","How a simple refracting telescope magnifies","A basic refracting telescope's magnification is the objective lens's focal length divided by the eyepiece's focal length. An objective has a focal length of 100 cm and the eyepiece 5 cm. What magnification does the telescope give, and what does changing the eyepiece do?",[447,448,449],"Magnification = objective focal length ÷ eyepiece focal length = 100 ÷ 5 = **20 times**.","Swap in a shorter eyepiece, say 2 cm, and the same telescope gives 100 ÷ 2 = 50 times — a stronger eyepiece (shorter focal length) always means more magnification from the same telescope tube.","This also explains why a telescope's objective, not its eyepiece, mainly decides how much *light* it gathers and how sharp the image can be: a bigger objective lens (or mirror) collects more light regardless of which eyepiece is plugged in afterwards.",{"anotherExample":451},"Binoculars use the same idea as a telescope, folded into a shorter tube using extra prisms — which is also why binoculars use total internal reflection internally, the very topic of the next chapter.",{"id":453,"type":47,"variant":93,"title":454,"markdown":455},"nuance-telescope-light","Magnification is not the same as a good telescope","A cheap telescope can be built to give a huge magnification number and still show a dim, blurry mess, because magnifying a faint, fuzzy image just gives you a bigger faint, fuzzy image. What actually limits how much fine detail a telescope can show is the **size of its objective lens or mirror**: a bigger objective gathers more light and bends it more precisely, which is why serious telescopes are described first by their aperture (objective diameter) — as in Hubble's 2.4 m mirror — and only second by their magnification.",{"id":457,"type":53,"title":458,"eyebrow":459,"navLabel":460},"i-ch6","The bent straw, taken to its limit","Chapter 06","6 Total internal ref.",{"id":462,"type":43,"markdown":463},"tir-setup","You already know a ray bends away from the normal when it leaves a slow material (like water or glass) for a faster one (like air). Now push that idea harder: what if the ray hits the boundary at a very steep slant?\n\nAs the angle inside the water or glass increases, the refracted ray outside bends further and further from the normal — until, at one particular angle, the refracted ray would have to bend a full **90°**, skimming exactly along the surface. Push past that angle and refraction simply **stops working**: no ray can escape at all, and instead **all** of the light reflects back inside, as if the boundary had become a perfect mirror. This is **total internal reflection**, and the angle where it begins is the **critical angle**.",{"id":465,"type":62,"prompt":466,"options":467,"explanation":476},"pred-tir","A beam of light travels inside water (critical angle 48.8°) and hits the water–air surface from below at 60°, measured from the normal. What happens to it?",[468,470,472,474],{"id":66,"label":469},"It refracts out into the air, bending away from the normal",{"id":69,"label":471},"It reflects entirely back into the water; none escapes",{"id":72,"label":473},"Half escapes as light, half reflects, exactly like a plane mirror",{"id":75,"label":475},"It is absorbed and disappears","**b.** 60° is greater than water's critical angle of 48.8°, so **total internal reflection** happens: every bit of the light reflects back into the water, obeying the ordinary law of reflection, and none crosses into the air. This is exactly why, looking up at a calm pool's surface from underwater at a shallow, glancing angle, the surface looks like a perfect mirror rather than a window — you are looking at the surface beyond its critical angle.",{"id":478,"type":98,"component":479,"componentVersion":5,"config":480,"objective":484,"textAlternative":485,"help":486},"lab-tir-ray","light-ray",{"initialAngle":481,"showNormal":482,"showAngles":482,"challengeAngle":483},35,true,49,"Slide the angle inside water past its critical angle of 48.8° and watch refraction switch off completely.","A ray box aimed up at a water–air boundary from below, with a slider for the angle of incidence (measured from the normal) and a target challenge angle of 49°, just past water's critical angle of 48.8°.\n\nBelow the critical angle, most of the light refracts out into the air (bending away from the normal), with only a faint reflected ray staying inside. Right at 48.8°, the refracted ray grazes along the surface at a full 90°. Past that, the refracted ray vanishes entirely and **all** the light reflects back inside at an equal angle — this is total internal reflection, and it switches on with no warning exactly at the critical angle, not gradually.",{"hints":487},[488,489],"Water's critical angle is about 48.8°; glass's is smaller at about 41.8°.","A denser material always has a smaller critical angle.",{"id":491,"type":356,"caption":492,"columns":493,"rows":498},"table-critical","Critical angles for light leaving three materials into air (critical angle = arcsin(1 ÷ refractive index))",[494,495,496,497],"Material","Refractive index","Critical angle","What it means",[499,504,509],[500,501,502,503],"Water","1.33","48.8°","A fairly wide \"escape window\" looking up from underwater",[505,506,507,508],"Glass","1.50","41.8°","A narrower window; more angles trap light inside",[510,511,512,513],"Diamond","2.42","24.4°","A very narrow window — most light entering a cut diamond bounces around inside repeatedly before escaping, which is most of its sparkle",{"id":515,"type":47,"variant":418,"title":516,"markdown":517},"aha-fibre","A pipe made of total internal reflection","An optical fibre is a hair-thin strand of glass carrying light down its length by total internal reflection, bouncing off the glass–cladding boundary again and again at an angle steeper than the critical angle, losing almost none of it at each bounce. A signal sent by fibre from Chennai to Delhi (cable route about 2,200 km) takes only about **10.8 milliseconds** — barely 3.5 milliseconds longer than if it travelled the same distance through a vacuum at the full speed of light, because light in glass fibre moves at only about 68% of its vacuum speed.",{"id":519,"type":53,"title":520,"eyebrow":521,"navLabel":522},"i-ch7","Where does a rainbow actually come from?","Chapter 07","7 Rainbow investigation",{"id":524,"type":43,"markdown":525},"rainbow-setup","You have already dispersed white light with a prism into a spectrum, always violet-to-red across a flat band. A rainbow does something stranger: it forms a curved **arc**, always at the same angle from the point directly opposite the Sun, and it needs no glass at all — only raindrops.",{"id":527,"type":62,"prompt":528,"options":529,"explanation":538},"pred-rainbow","Millions of raindrops are falling all around you when the Sun is low behind you. Only some of them contribute to the rainbow you see. What determines which raindrops send colour to your eye?",[530,532,534,536],{"id":66,"label":531},"Only the raindrops directly in front of the Sun",{"id":69,"label":533},"Only raindrops at a very particular angle — about 42° from the direction opposite the Sun — measured from your eye",{"id":72,"label":535},"Every raindrop in the sky sends you some colour; that is why rainbows are so wide",{"id":75,"label":537},"Only raindrops that are unusually large or unusually round","**b.** Inside every sunlit raindrop, light refracts entering, reflects once off the back, and refracts again leaving — and this three-step journey happens to redirect light most strongly at a deviation that puts red light back towards you at almost exactly **42°** from the point directly opposite the Sun (the antisolar point), and violet at about 40.5°. Only the particular raindrops sitting on that 42°-and-40.5° cone, seen from where you stand, send their colour to your eye — which is also why a rainbow moves with you: it is not a fixed object in the sky but a fixed *angle* relative to you and the Sun.",{"id":540,"type":80,"title":541,"problem":542,"steps":543,"help":549},"we-rainbow-angle","Why 42 degrees and not some other angle","Light enters a raindrop, refracts (bending towards the normal, since water is denser than air), reflects once off the inside of the back surface, and refracts again leaving. Different entry points on the drop give different total deviations from the ray's original direction. Why does one particular deviation dominate, and what is it?",[544,545,546,547,548],"As the entry point moves from the centre of the drop's face out towards its edge, the total deviation angle first decreases, reaches a **minimum**, and then increases again.","Rays entering across a whole range of points near that minimum all emerge within a narrow spread of angles — they bunch up, while rays elsewhere spread thinly across many angles.","That bunching concentrates enormous numbers of rays into one narrow angular band, which is intense enough to see as a bright arc; every other angle is too faint to notice.","For water, that minimum-deviation angle works out to about 138° of total turning, which corresponds to **about 42°** measured back from the antisolar point.","Violet light bends slightly more than red in water (higher refractive index for shorter wavelengths), so violet's bunched angle is about 40.6° and red's about 42.4° — violet sits on the *inside* of the bow, red on the *outside*.",{"simplerExplanation":550},"One particular bending angle collects far more rays than any other, so that is the only angle bright enough to see — and it happens to be about 42°.",{"id":552,"type":47,"variant":418,"title":553,"markdown":554},"aha-secondary","The fainter second bow, and the dark band between them","Look carefully above a bright rainbow and you can sometimes see a fainter **secondary bow**, with its colours reversed (red on the inside this time), at about **51°** from the antisolar point instead of 42°. It comes from light that reflects **twice** inside each raindrop instead of once, losing some brightness at the extra bounce.\n\nBetween the two bows, from about 42° to 51°, the sky looks distinctly **darker** than everywhere else — a real, named effect called **Alexander's dark band**, after the Greek philosopher who first described it around 200 CE. No raindrop sends light back to your eye from that particular band of angles at all; every ray gets bunched into one bow or the other, none in between.",{"id":556,"type":98,"component":557,"componentVersion":5,"config":558,"objective":563,"textAlternative":564,"help":565},"lab-rainbow-prism","prism-lab",{"modes":559,"rounds":562},[560,561],"prism","mixing",8,"Investigate how a prism separates white light, then how mixed coloured lights recombine — the two halves of a rainbow's story.","A prism mode showing dispersion (splitting white light into a spectrum) and a mixing mode showing coloured lights recombining (red, green and blue light overlapping to white).\n\nA rainbow performs a version of both in one raindrop: it disperses the Sun's white light into colours (the prism half), and because each raindrop actually sends out a slightly overlapping smear of all wavelengths rather than a pure single colour, our eyes perceive continuous, gently blended bands rather than hard-edged stripes (a gentler version of the mixing half).",{"hints":566},[567,568],"A rainbow's violet band sits at a slightly smaller angle than its red band.","No two people ever see exactly the same rainbow: it depends on the angle from your own eye.",{"id":570,"type":47,"variant":176,"title":571,"markdown":572},"careful-rainbow-photo","Why you can never walk up to a rainbow","A rainbow is not a location in the sky above a particular field — it is a **direction**, fixed relative to the Sun and your own eyes, made of light from a different set of raindrops for every observer and every step you take. Walk towards it and the raindrops that were making your rainbow fall behind you, while new ones ahead take over the job. This is also why no two people, even standing side by side, ever see quite the same rainbow: each eye traces out its own 42° cone.",{"id":574,"type":53,"title":575,"eyebrow":576,"navLabel":577},"i-ch8","Why is the sky blue, and sunsets red?","Chapter 08","8 Blue sky, red sun",{"id":579,"type":43,"markdown":580},"sky-setup","White sunlight is a mix of every visible wavelength, from violet (about 400 nanometres) to red (about 700 nanometres). Air is almost — but not quite — perfectly transparent: its molecules, far too small to see, still nudge passing light a little off course, an effect called **scattering**. The amount of scattering is ferociously sensitive to wavelength: it goes as **1 ÷ wavelength⁴**, so a wavelength that is only slightly shorter scatters *far* more than one slightly longer.",{"id":582,"type":62,"prompt":583,"options":584,"explanation":593},"pred-sky","Using scattering ∝ 1 ÷ wavelength⁴, roughly how much more strongly does air molecules scatter blue light (about 450 nm) than red light (about 700 nm)?",[585,587,589,591],{"id":66,"label":586},"About 1.5 times as much",{"id":69,"label":588},"About 6 times as much",{"id":72,"label":590},"About 60 times as much",{"id":75,"label":592},"They scatter equally; only our eyes are more sensitive to blue","**b.** (700 ÷ 450)⁴ works out to roughly 5.9 — around six times more scattering for blue than for red, and roughly 9.4 times more for violet than red. Looking at the sky away from the Sun, you are seeing sunlight that has been scattered sideways into your eye, and blue dominates that scattered light by a wide margin — even though violet scatters even more strongly, our eyes are more sensitive to blue and there is less violet in sunlight to begin with, so the sky reads as blue rather than violet.",{"id":595,"type":98,"component":557,"componentVersion":5,"config":596,"objective":601,"textAlternative":602,"help":603},"lab-sky-scatter",{"modes":597,"rounds":600},[598,599],"scattering","filters",6,"Test how strongly small particles scatter different colours of light, and see coloured filters remove light of the wrong wavelength.","A scattering mode showing tiny particles deflecting beams of different colours by different amounts, and a filters mode showing coloured filters blocking everything except their own colour.\n\nIn the scattering mode, blue and violet light visibly scatter far more than red as they pass among the particles — a direct model of what real air molecules do to sunlight. The filters mode shows the flip side: a red filter looks red because it **absorbs** every colour except red and lets red through, which is a completely different mechanism from scattering, even though both end up changing the colour you see.",{"hints":604},[605,606],"Scattering redirects light sideways; a filter absorbs the colours it does not transmit.","Shorter wavelengths always scatter more strongly than longer ones.",{"id":608,"type":62,"prompt":609,"options":610,"explanation":619},"pred-sunset","At sunset, sunlight has to pass through far more atmosphere at a low, slanting angle than it does at noon. What do you predict happens to the Sun's own colour as seen directly, and why?",[611,613,615,617],{"id":66,"label":612},"It turns blue, because blue scatters most",{"id":69,"label":614},"It turns orange or red, because so much blue and green have already been scattered away out of the direct beam by the time it reaches your eye",{"id":72,"label":616},"It stays perfectly white; only the sky around it changes colour",{"id":75,"label":618},"It gets brighter, because scattering adds light to the direct beam","**b.** At sunset the sunlight's path through the atmosphere is many times longer than at noon (skimming through far more air near the horizon than shining straight down). Along that long path, blue and violet are scattered out of the beam so thoroughly that what is left, still travelling straight at your eye, is dominated by orange and red — the colours that scatter least and survive the trip. The sky far from the Sun, meanwhile, is full of exactly the blue light that got scattered away from someone else's direct sunbeam.",{"id":621,"type":47,"variant":418,"title":622,"markdown":623},"aha-sky-dust","Why sunsets vary so much","A very clean sky gives a fairly mild orange sunset, because only air molecules are doing the scattering. Add dust, smoke or volcanic ash — all bigger than a single molecule — and the scattering becomes stronger and less wavelength-selective, often producing spectacular, deep red and purple sunsets. Some of history's most vivid recorded sunsets, including across Europe for months after the 1883 eruption of Krakatoa, were caused by exactly this: volcanic particles high in the atmosphere scattering sunlight long after the eruption itself was over.",{"id":625,"type":47,"variant":626,"title":627,"markdown":628},"misc-sky-reflect","misconception","\"The sky is blue because it reflects the ocean\"","A common guess, and a tidy one, but wrong: the sky over a desert, with no ocean for thousands of kilometres, is exactly as blue as the sky over the sea. It is Rayleigh scattering of sunlight by the air itself, not any reflection of water, that colours the sky — and in fact the causation half-runs the other way: the ocean itself looks blue partly because it reflects the blue sky above it.",{"id":630,"type":53,"title":631,"eyebrow":632,"navLabel":633},"i-ch9","Pulling it together","Chapter 09","9 Pulling it together",{"id":635,"type":636,"title":637,"terms":638},"gloss-investigate","glossary","Terms from this lesson",[639,642,646,650,654,658,662],{"term":496,"meaning":640,"example":641},"The angle of incidence inside a denser material above which no light can refract out; total internal reflection takes over.","Water: 48.8°. Diamond: 24.4°.",{"term":643,"meaning":644,"example":645},"Total internal reflection","When light hitting a boundary from the denser side, at an angle greater than the critical angle, reflects completely instead of refracting out.","Why light stays trapped inside an optical fibre.",{"term":647,"meaning":648,"example":649},"Focal length","The distance from a mirror or lens to the point where parallel rays meet (or seem to meet).","A concave shaving mirror often has f ≈ 25 cm.",{"term":651,"meaning":652,"example":653},"Concave mirror","A mirror curving inward like the inside of a spoon; can give either a magnified virtual image or a real, inverted image.","A torch or headlight reflector.",{"term":655,"meaning":656,"example":657},"Convex mirror","A mirror curving outward; always gives a virtual, upright, diminished image.","A car's side mirror.",{"term":659,"meaning":660,"example":661},"Dispersion","Splitting white light into its component colours because a material's refractive index is slightly different for each wavelength.","A prism, or a raindrop making a rainbow.",{"term":663,"meaning":664,"example":665},"Antisolar point","The point in the sky directly opposite the Sun from your own shadow's position.","A rainbow is always centred on this point, 42° away.",{"id":667,"type":62,"prompt":668,"options":669,"explanation":678},"pred-wrap","You place your face progressively closer to a concave mirror, starting from far beyond its focal length. In what order do you pass through the different kinds of image?",[670,672,674,676],{"id":66,"label":671},"Magnified virtual upright → real inverted → diminished virtual upright",{"id":69,"label":673},"Real inverted, shrinking → (at the focus) no clear image at all → virtual, upright, growing",{"id":72,"label":675},"Diminished virtual → real inverted → magnified virtual, all in one smooth blend with no boundary",{"id":75,"label":677},"The image never changes kind, only its size","**b.** Far out, the image is real and inverted and roughly the same size as at 2f (where it matches the object exactly). Moving closer, the real image grows and moves farther in front of the mirror until, right at the focal point, the reflected rays leave the mirror parallel and never converge to any image at all. Move closer still, inside the focal length, and the image reappears — now virtual, upright and magnified, sitting behind the mirror. The switch at the focus is sudden, not gradual, which is exactly the same kind of \"nothing, then everything\" behaviour you saw at the critical angle for total internal reflection.",{"id":680,"type":681,"title":682,"questions":683},"quiz-investigate","quiz","Test what you worked out",[684,697,710,723,736],{"itemId":685,"prompt":686,"options":687,"correct":69,"why":696},"light.investigate-q-roemer","Rømer's observation of Io's eclipses being late or early was best explained by:",[688,690,692,694],{"id":66,"label":689},"Io's orbit genuinely changing speed",{"id":69,"label":691},"The changing distance light has to cross as Earth orbits the Sun",{"id":72,"label":693},"Faulty 17th-century telescopes",{"id":75,"label":695},"Jupiter's own motion around the Sun","The eclipses themselves were exactly on schedule; only the light announcing them took longer or shorter to reach Earth depending on the Earth–Jupiter distance.",{"itemId":698,"prompt":699,"options":700,"correct":69,"why":709},"light.investigate-q-fizeau","Fizeau's toothed-wheel method measured the speed of light by timing:",[701,703,705,707],{"id":66,"label":702},"How long a candle takes to burn down",{"id":69,"label":704},"A round trip to a distant mirror and back through the same spinning wheel",{"id":72,"label":706},"How fast Jupiter's moon orbits",{"id":75,"label":708},"The cooling of a hot filament","The wheel's known spin rate converted the round-trip travel time into a distance-over-time speed calculation.",{"itemId":711,"prompt":712,"options":713,"correct":69,"why":722},"light.investigate-q-convex","Why is a convex mirror, not a concave one, used for a car's side mirror?",[714,716,718,720],{"id":66,"label":715},"It magnifies distant traffic for safety",{"id":69,"label":717},"It always gives an upright, wide-field view, even though objects look smaller and farther than they are",{"id":72,"label":719},"It is cheaper to manufacture",{"id":75,"label":721},"It reflects only certain colours","A convex mirror's constantly diminished, upright, wide-angle image trades true distance judgement for a much wider field of view — worth the trade-off for spotting traffic.",{"itemId":724,"prompt":725,"options":726,"correct":72,"why":735},"light.investigate-q-tir","Total internal reflection happens when light inside a denser material hits a boundary:",[727,729,731,733],{"id":66,"label":728},"At any angle at all",{"id":69,"label":730},"At an angle smaller than the critical angle",{"id":72,"label":732},"At an angle equal to or greater than the critical angle",{"id":75,"label":734},"Only if the material is glass, never water","Below the critical angle light refracts out; at and beyond it, no refracted ray can exist, and all the light reflects back inside.",{"itemId":737,"prompt":738,"options":739,"correct":69,"why":748},"light.investigate-q-rainbow","A rainbow's primary bow appears at about 42° from:",[740,742,744,746],{"id":66,"label":741},"The Sun itself",{"id":69,"label":743},"The point directly opposite the Sun (the antisolar point)",{"id":72,"label":745},"The horizon, regardless of the Sun's position",{"id":75,"label":747},"Wherever the nearest cloud happens to be","Refraction, one internal reflection, and refraction again inside raindrops bunches light most strongly at about 42° from the point directly away from the Sun.",{"id":750,"type":751,"title":752,"points":753},"cheat-investigate","summary","Cheat sheet",[754,755,756,757,758],"**Rømer (1676):** the changing Earth–Jupiter distance changes the arrival time of Io's eclipse light. First evidence that light's speed, though vast, is finite.","**Fizeau (1849):** a spinning toothed wheel timed a round trip to a mirror 8,633 m away and back, the first ground-based measurement, landing within about 5% of the true value.","**Mirror\u002Flens formula:** 1\u002Fv = 1\u002Ff − 1\u002Fu. Concave mirrors and convex lenses can give either a magnified virtual image (object closer than f) or a real, inverted image (object farther than f). Convex mirrors and concave lenses always give a virtual, upright, diminished image.","**Critical angle:** arcsin(1 ÷ refractive index). Above it, total internal reflection traps all the light — the principle behind optical fibres and a diamond's sparkle.","**Rainbow:** refraction, one internal reflection, refraction again in raindrops, bunching most strongly at about 42° (red) and 40.5° (violet) from the antisolar point. A fainter secondary bow, colours reversed, sits near 51°, with Alexander's dark band between them.",{"id":760,"type":761,"conceptId":762,"relation":763,"explanation":764},"conn-investigate-eclipses","connection","eclipses","related_to","Rømer's method used real eclipses of a real moon as a natural clock — the same geometry of one body's shadow falling on light's path that you meet again with the Moon and Earth.",{"id":766,"type":761,"conceptId":767,"relation":763,"explanation":768},"conn-investigate-electricity","electricity","Optical fibres now carry most of the world's long-distance data using total internal reflection instead of electric current in a copper wire, because light in glass loses far less energy over long distances.",{"id":770,"type":771,"sourceIds":772},"sources-investigate","sources",[773,774,775,776,777,778,779,780],"light-wikipedia-speed-of-light","light-wikipedia-roemer","light-hyperphysics-mirror","light-hyperphysics-lens","light-hyperphysics-totint","light-hyperphysics-rainbow","light-physicsclassroom-refraction","light-wikipedia-rayleigh",[773,774,775,776,777,778,779,780],"needs_review",{"generatedBy":784,"notes":785},"claude-code","Draft generated locally; every number computed and asserted in scratchpad\u002Flight\u002Fnumbers.py. Pending owner review.","9f3c835776dfb722b2c779e1d1fac75e324e3b4b8ad5f79cc88d5edf98dd338c",{"component:sort-game@1":788,"logic:practice":789,"component:match-pairs@1":790,"component:light-ray@1":791,"component:prism-lab@1":792,"source:light-hyperphysics-lens":793,"source:light-hyperphysics-mirror":794,"source:light-hyperphysics-rainbow":795,"source:light-hyperphysics-totint":796,"source:light-physicsclassroom-refraction":797,"source:light-wikipedia-rayleigh":798,"source:light-wikipedia-roemer":799,"source:light-wikipedia-speed-of-light":800},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","d66d44021dc28328ce2ea82e0d6cefc9466b5dc65dbb93c92b05b88161725be6","835c8ba7fd707c60ded46494f4b672aa095199ba9a2628409905335d0b3434a9","4cdd337c59a170b52016937cc2cc4327db640e3f7c8e883689608119257931d4","cc4f04df90efb4f54357cdc43554d566fcd0bb5efb3303655a1579a15d292259","dce907528f01e833f82d68150b423cc68c60d1c9c88673a96a84eb269bbbd3ce","fbea9871167acaf5df4f2d491a5417bc634467e447ecc5463c5d581b47e11cc6","43e2d25a3db005014fd7d491471e9e47242ce863ec6c79370b74a25bfe96acfb","4c7f726ffcab1b913b628c2df33237f78d9472f867c4ab3682ea2c2f080d0ea0","b3b2ba6a172784734e5a829204e49f841e19c0b1cd6b52aad9835a967d919179","e6620f59bd5a24800f47530a4ed586390c383a730742e4e6bdf5fda99401c43a",{"state":802,"reviewer":803,"selfReview":482,"reviewedAt":804,"method":805},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597765]