[{"data":1,"prerenderedAt":862},["ShallowReactive",2],{"layer:light:deepen":3},{"layer":4,"contentHash":841,"dependencyHashes":842,"approval":856,"releaseId":861},{"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":836,"reviewStatus":837,"authoring":838},1,"light","en","deepen","Precise light: derivations, corrective lenses and the shape of a rainbow","Beyond the syllabus: derive the mirror formula, correct short and long sight, and see why a rainbow sits at 42 degrees.","Follow the speed of light to its modern exact definition, derive the mirror\u002Flens formula from similar triangles, work out lens powers for short and long sight, put numbers on fibre-optic latency, and see why the rainbow's angle is a genuine minimum.",[13,14,15,16,17],"Trace the improving precision of speed-of-light measurements from Rømer to the modern exact definition.","Derive the mirror\u002Flens formula 1\u002Fu + 1\u002Fv = 1\u002Ff from similar triangles in a ray diagram.","Calculate the lens power needed to correct a given case of short or long sight.","Explain why the rainbow's 42 degrees is a genuine minimum of deviation, not an arbitrary number.","State and apply Rayleigh's 1 ÷ λ⁴ scattering law to the colours of the sky.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Deepen",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Investigate: curved mirrors, TIR, rainbow geometry",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Sorting games, ray box, matching",{"label":38,"value":39},"Goes beyond syllabus","Yes — flagged throughout",[41,45,51,57,60,90,95,98,127,175,180,183,203,218,230,242,267,280,285,288,307,311,316,343,348,351,356,367,378,383,425,430,433,452,457,472,477,482,485,506,510,522,527,530,555,559,569,574,577,590,593,618,622,637,650,655,659,663,667,671,676,702,728,732,802,813,819,824],{"id":42,"type":43,"markdown":44},"intro-deepen","prose","This layer pushes further than a school syllabus usually goes. It asks *why* the formulas you have been using actually work, follows the history of measuring light's speed all the way to a precision method, and puts real numbers on total internal reflection, the rainbow and the colour of the sky.\n\nNone of this is harder to *read* than Investigate. It is harder in the sense that it asks for one more step of reasoning each time — not just what happens, but why it has to happen that way.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-read-d","callout","observation","What \"beyond the syllabus\" means here","Curved mirrors and lenses appear in the NCERT Class 10 Science syllabus, a few years ahead of the Class 7 chapter this topic is built around. Nothing here is invented physics — it is standard optics, just introduced earlier and pushed a little further than a typical Class 7 course would. If a chapter or number is new to your class, that is expected, not a mistake.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"d-ch1","chapter","Measuring light's speed: from a guess to a definition","Chapter 01","1 A precise speed",{"id":58,"type":43,"markdown":59},"history-precise","Rømer's astronomical estimate in 1676 and Fizeau's toothed wheel in 1849 both got light's speed roughly right using completely different apparatus. Neither was precise enough to be the last word. In 1879, the American physicist **Albert Michelson** built a rotating-mirror apparatus at the US Naval Academy that measured the speed to within a whisker of the modern value — 299,910 km\u002Fs, only about 0.04% above 299,792.458 km\u002Fs. Michelson kept refining his methods for decades and won the 1907 Nobel Prize in Physics, the first American to win a science Nobel, substantially for this work.",{"id":61,"type":62,"caption":63,"columns":64,"rows":69},"table-speed-history","table","Four measurements of the speed of light, in order of increasing precision",[65,66,67,68],"Method and year","Result","Error vs modern value","What limited its accuracy",[70,75,80,85],[71,72,73,74],"Rømer\u002FHuygens, 1676","2.27 × 10⁸ m\u002Fs","about 24% low","Rough 17th-century clocks and an imprecise value for the astronomical unit",[76,77,78,79],"Fizeau, 1849","3.133 × 10⁸ m\u002Fs","about 4.5% high","Hard to judge the exact instant the beam was fully extinguished",[81,82,83,84],"Michelson, 1879","299,910 km\u002Fs","about 0.04% high","Tiny remaining errors in the rotating mirror's speed and the measured baseline",[86,87,88,89],"Modern defined value","299,792.458 km\u002Fs","exact, by definition","None: since 1983 the metre itself is defined from this speed",{"id":91,"type":47,"variant":92,"title":93,"markdown":94},"aha-metre-defined","aha","The metre is now defined by the speed of light, not the other way round","Until 1960, the metre was the length of a specific metal bar kept in France. Then it was redefined using a wavelength of light emitted by krypton atoms. In 1983, scientists went one step further: they simply **declared** the speed of light to be exactly 299,792,458 metres per second, and defined the metre as whatever distance light travels in 1 ⁄ 299,792,458 of a second. Nothing about light's actual speed changed — but it became impossible, by definition, to measure it as anything other than exactly that number. Any future \"measurement\" of the speed of light is now really a measurement of the length of a metre.",{"id":96,"type":43,"markdown":97},"michelson-method","Michelson's apparatus was, in spirit, a much faster and much more precise cousin of Fizeau's toothed wheel. Instead of a wheel with teeth, he used an **octagonal rotating mirror**: light bounced off one face of the spinning mirror, travelled about 35 km to a fixed mirror on a distant mountain and back, and was only caught by the eyepiece if the octagon had turned by exactly one-eighth of a full turn (or a whole number of eighths) in the time the light was away. Spinning the mirror faster or slower shifted the returning beam out of alignment with the eyepiece, and finding the exact matching speeds — Michelson used several, as a cross-check — gave the round-trip time with far less guesswork than judging by eye exactly when a wheel's light had vanished.",{"id":99,"type":100,"title":101,"items":102},"steps-michelson","steps","Michelson's rotating-mirror method, in order",[103,107,111,115,119,123],{"title":104,"tag":105,"text":106},"Light leaves a slit","step 1","A bright, narrow beam is aimed at one face of a fast-spinning eight-sided mirror.",{"title":108,"tag":109,"text":110},"Reflects off the spinning mirror","step 2","The beam bounces off whichever face happens to be correctly angled at that instant, heading towards a distant fixed mirror.",{"title":112,"tag":113,"text":114},"Travels a long, carefully measured distance","step 3","Michelson's most famous later versions used a baseline of about 35 km between two Californian mountains, later refined to a measured vacuum path.",{"title":116,"tag":117,"text":118},"Reflects off a large, fixed mirror","step 4","The far mirror simply sends the beam straight back the way it came.",{"title":120,"tag":121,"text":122},"Returns to the spinning mirror","step 5","By now, if the mirror is spinning fast enough, a different face — or the same face turned by a whole number of eighths — is waiting to catch it.",{"title":124,"tag":125,"text":126},"Only lines up at the right speeds","step 6","The observer tunes the spin rate until the returning beam is seen in exactly its original position; several different matching speeds cross-check each other and pin down the round-trip time precisely.",{"id":128,"type":129,"component":130,"componentVersion":5,"config":131,"objective":169,"textAlternative":170,"help":171},"lab-speed-sort","interactive","sort-game",{"prompt":132,"bins":133,"items":143,"seconds":168},"Which historical method for measuring the speed of light does this clue describe?",[134,137,140],{"id":135,"label":136},"roemer","Rømer (1676)",{"id":138,"label":139},"fizeau","Fizeau (1849)",{"id":141,"label":142},"michelson","Michelson (1879)",[144,148,152,156,160,164],{"id":145,"label":146,"bin":135,"why":147},"jupiter","Used the changing arrival time of a moon's eclipses over a year","This is Rømer's method: timing Io's eclipses as the Earth–Jupiter distance changed.",{"id":149,"label":150,"bin":138,"why":151},"toothedwheel","Used a fast-spinning wheel with hundreds of teeth and a distant mirror","Fizeau's toothed wheel, over an 8,633 m path near Paris.",{"id":153,"label":154,"bin":141,"why":155},"octagon","Used a spinning eight-sided mirror and a mountain-to-mountain baseline","Michelson's rotating-mirror method, accurate to about 0.04%.",{"id":157,"label":158,"bin":135,"why":159},"wholesolar","Used the entire Earth's orbit as part of the measuring apparatus","Rømer's method needed no laboratory equipment at all — only the Solar System's own geometry.",{"id":161,"label":162,"bin":141,"why":163},"nobel","Its inventor later won a Nobel Prize substantially for this work","Michelson won the 1907 Nobel Prize in Physics.",{"id":165,"label":166,"bin":138,"why":167},"ground1849","The first measurement made entirely with equipment on the ground","Fizeau's was the first fully terrestrial measurement, needing no astronomy at all.",0,"Match six historical clues to the speed-of-light method they describe: Rømer, Fizeau or Michelson.","Six short clues about the three major historical speed-of-light methods, sorted into the correct scientist and year. The pattern that emerges: each method traded one kind of difficulty for another — Rømer needed no equipment but relied on astronomical data of uncertain precision; Fizeau needed only a hillside but had to judge an instant by eye; Michelson needed a very long, carefully surveyed baseline but achieved by far the best precision of the three.",{"hints":172},[173,174],"Only one method used no laboratory apparatus at all.","The most precise of the three used a spinning mirror with eight faces, not a toothed wheel.",{"id":176,"type":53,"title":177,"eyebrow":178,"navLabel":179},"d-ch2","The mirror and lens equation, and why it works","Chapter 02","2 Mirror and lens maths",{"id":181,"type":43,"markdown":182},"mirror-math","The relationship 1\u002Fv = 1\u002Ff − 1\u002Fu (with u, v and f all measured as positive distances for a real object and, respectively, a real image or a positive focal length) is not an arbitrary rule to memorise — it falls directly out of similar triangles, in almost the same way as the shadow-size rule from Understand.\n\nDrawing a ray diagram, three rays are enough to fix an image exactly, and each obeys a simple, memorable rule:",{"id":184,"type":100,"title":185,"items":186},"steps-three-rays","The three rays that build any ray diagram",[187,191,195,199],{"title":188,"tag":189,"text":190},"The parallel ray","rule 1","Any ray travelling parallel to the main axis reflects (or refracts) through the principal focus.",{"title":192,"tag":193,"text":194},"The focal ray","rule 2","Any ray that passes through the principal focus on the way in reflects (or refracts) out parallel to the axis — the reverse of rule 1.",{"title":196,"tag":197,"text":198},"The centre ray","rule 3","For a mirror, a ray aimed at the centre of curvature hits the mirror square-on and reflects straight back along itself. For a lens, a ray aimed at the very centre of the lens passes straight through, undeviated.",{"title":200,"tag":201,"text":202},"Where two rays cross","the image","Any two of the three rays from the same point on the object cross (or appear to cross, extended backwards) at the image of that point. The third ray is only a check.",{"id":204,"type":205,"title":206,"problem":207,"steps":208,"help":216},"we-mirror-derive","worked_example","Deriving the shape of the mirror formula","Why should 1\u002Fv and 1\u002Fu add up to a constant 1\u002Ff, rather than, say, v and u adding to a constant? Sketch the reasoning.",[209,210,211,212,213,214,215],"Take an object of height h₀ standing on the axis at distance u from a concave mirror of focal length f.","The parallel ray from its tip reflects through the focus; the centre ray from its tip (aimed at the point f in this simplified sketch) reflects back parallel to the axis. Where they cross fixes the image, of height hᵢ, at distance v.","Two pairs of similar triangles fall out of that crossing: one pair comparing the object and image heights across the mirror's pole, giving hᵢ\u002Fh₀ = v\u002Fu (this defines magnification), and a second pair comparing the same heights across the focus, giving hᵢ\u002Fh₀ = (v − f)\u002Ff.","Setting the two expressions for hᵢ\u002Fh₀ equal: v\u002Fu = (v − f)\u002Ff.","Multiply both sides by uf: vf = u(v − f) = uv − uf.","Rearrange: vf + uf = uv. Divide every term by uvf: 1\u002Fu + 1\u002Fv = 1\u002Ff.","That is exactly the formula — it is a direct consequence of similar triangles, the same tool you used for shadow sizes, applied to a reflected image instead of a projected one.",{"simplerExplanation":217},"Two different pairs of similar triangles both describe the same magnification. Setting them equal and rearranging is all it takes to get 1\u002Fu + 1\u002Fv = 1\u002Ff.",{"id":219,"type":205,"title":220,"problem":221,"steps":222,"help":228},"we-glass-water","Refraction from one medium straight into another","Light travels inside a block of glass (n = 1.50) and strikes a glass–water boundary (water n = 1.33) at 30° to the normal, water filling the space beyond. What angle does it take in the water?",[223,224,225,226,227],"This is refraction between two real materials, not from air, so use the general form of Snell's law: n₁ sin θ₁ = n₂ sin θ₂.","n₁ = 1.50 (glass), θ₁ = 30°, n₂ = 1.33 (water).","1.50 × sin 30° = 1.33 × sin θ₂, so sin θ₂ = 1.50 × 0.5 ÷ 1.33 = 0.564.","θ₂ = arcsin(0.564) ≈ **34.3°**.","The ray bends *away* from the normal, because water is the less dense (lower refractive index) of the two materials here — the same rule as air-to-water, just applied between two materials that both refract light.",{"anotherExample":229},"Snell's law never actually needs air as one of the two materials. Air-to-water and air-to-glass are simply the two special cases most often taught first.",{"id":231,"type":205,"title":232,"problem":233,"steps":234,"help":240},"we-2f-case","The special case at twice the focal length","A concave mirror has a 20 cm focal length. An object stands exactly at 2f, 40 cm away. Where does the image form, and what size is it?",[235,236,237,238,239],"Use 1\u002Fv = 1\u002Ff − 1\u002Fu with f = 20 cm and u = 40 cm.","1\u002Fv = 1\u002F20 − 1\u002F40 = 2\u002F40 − 1\u002F40 = 1\u002F40, so v = 40 cm.","The image forms at exactly 40 cm too — the object and image swap no distance at all, both sitting at 2f.","Magnification = v\u002Fu = 40\u002F40 = 1: the image is exactly the same size as the object, real and inverted.","This particular case, object and image both at 2f, is the single dividing line between an *enlarged* real image (object between f and 2f) and a *diminished* real image (object beyond 2f) — worth memorising as a landmark, not just a special case.",{"simplerExplanation":241},"At exactly twice the focal length, a concave mirror or convex lens always gives a same-size, real, upside-down image — neither bigger nor smaller than the object.",{"id":243,"type":62,"caption":244,"columns":245,"rows":249},"table-mirror-lens-cases","The complete map of where an object can sit, for a concave mirror or convex lens of focal length f",[246,247,248],"Object distance from f","Image type","Image size",[250,254,257,260,264],[251,252,253],"Beyond 2f","Real, inverted","Smaller than the object",[255,252,256],"Exactly at 2f","Exactly the same size",[258,252,259],"Between f and 2f","Larger than the object",[261,262,263],"Exactly at f","No image forms","Rays leave perfectly parallel",[265,266,259],"Closer than f","Virtual, upright",{"id":268,"type":129,"component":269,"componentVersion":5,"config":270,"objective":274,"textAlternative":275,"help":276},"lab-ray-general","light-ray",{"initialAngle":271,"showNormal":272,"showAngles":272,"challengeAngle":273},30,true,42,"See a ray bend as it crosses a boundary, and check the angle against Snell's law by hand.","A ray box with a slider for the angle of incidence and a challenge target of 42°. Whatever pair of materials the boundary represents, the same rule governs the bend: n₁ sin θ₁ = n₂ sin θ₂, always bending towards the normal on entering the denser material and away from it on leaving.",{"hints":277},[278,279],"Work out sin θ₂ = (n₁ ÷ n₂) × sin θ₁ before checking the diagram.","A ratio of refractive indices greater than 1 bends the ray towards the normal; less than 1 bends it away.",{"id":281,"type":53,"title":282,"eyebrow":283,"navLabel":284},"d-ch3","Huygens' wavelets: why refraction happens at all","Chapter 03","3 Huygens' wavelets",{"id":286,"type":43,"markdown":287},"huygens-setup","The ray model explains *what* refraction does, but not *why* a boundary bends light in the first place. In 1678, the Dutch physicist **Christiaan Huygens** — the same Huygens who turned Rømer's observation into a speed — proposed a picture that answers exactly that: every point on an advancing wavefront of light can itself be treated as a source of tiny new wavelets, and the overall wavefront a moment later is simply the outer edge of all those wavelets added together.",{"id":289,"type":100,"title":290,"items":291},"steps-huygens","Huygens' construction, applied to a wavefront hitting a slower medium at a slant",[292,295,298,301,304],{"title":293,"tag":105,"text":294},"A straight wavefront approaches at a slant","Picture the wavefront as a straight line of dots, each about to act as its own tiny wave source, all still travelling in air.",{"title":296,"tag":109,"text":297},"One edge arrives first","Because the wavefront is tilted, one end of it reaches the water's surface before the other end does.",{"title":299,"tag":113,"text":300},"That edge slows down immediately","The moment a dot crosses into water, its own tiny wavelet spreads out more slowly than its neighbours still in air — water's higher refractive index means a lower wave speed.",{"title":302,"tag":117,"text":303},"The far edge is still moving at full speed","While one end has slowed, the other end is still in air, still spreading its wavelets at the faster speed.",{"title":305,"tag":121,"text":306},"The new wavefront tilts","Joining up all the new little wavelets — some big (still in air, fast), some small (now in water, slow) — gives a new wavefront at a different angle: the wave, and the ray drawn perpendicular to it, has bent towards the normal.",{"id":308,"type":47,"variant":92,"title":309,"markdown":310},"aha-huygens-check","A single idea explains reflection, refraction and even diffraction","Huygens' wavelet idea is not a special rule invented just for refraction. Apply exactly the same construction to a wavefront hitting a mirror and it correctly predicts ordinary reflection; apply it to a wavefront squeezing through a narrow gap and it predicts **diffraction**, light spreading slightly around obstacles and through slits — a wave behaviour you will meet properly in Extend. One idea, three completely different-looking behaviours, all falling out of the same simple construction.",{"id":312,"type":47,"variant":313,"title":314,"markdown":315},"careful-huygens-limit","model_limit","Huygens' picture predates knowing what actually waves","Huygens proposed this construction over a century before anyone understood that light is an **electromagnetic** wave (that had to wait for James Clerk Maxwell in the 1860s), and about two centuries before anyone suspected light also behaves like a stream of particles (Einstein, 1905, in Extend). Huygens' geometrical construction still works perfectly today for predicting *where* wavefronts go — it says nothing at all about *what* is physically waving, which is a separate and much harder question.",{"id":317,"type":318,"itemId":319,"prompt":320,"check":321,"hints":337,"feedback":340},"prac-huygens","practice","light.deepen-huygens-check","In Huygens' wavelet picture, when a tilted wavefront crosses from air into water at a slant, which part of the wavefront's own tiny wavelets slow down first?",{"kind":322,"options":323,"correct":336},"choice",[324,327,330,333],{"id":325,"label":326},"a","The part that reaches the water first",{"id":328,"label":329},"b","The part that is still furthest from the water",{"id":331,"label":332},"c","All parts slow down at exactly the same instant",{"id":334,"label":335},"d","None of it slows down until the whole wavefront has crossed",[325],[338,339],"Think about which part of a tilted line touches the boundary first.","Only wavelets already inside water can be travelling at water's slower speed.",{"correct":341,"incorrect":342},"Right — the leading edge of the tilted wavefront reaches the boundary first and immediately starts producing slower wavelets, while the trailing edge is still in air and still fast, which is exactly what tilts the wavefront (and the ray) towards the normal.","The edge of the wavefront closest to the water crosses the boundary first, so it is the first part to slow down.",{"id":344,"type":53,"title":345,"eyebrow":346,"navLabel":347},"d-ch4","Fixing your own eyesight with a second lens","Chapter 04","4 Correcting the eye",{"id":349,"type":43,"markdown":350},"vision-setup","A normally sighted eye can focus on anything from about 25 cm away (the **near point**) out to the far horizon (the **far point**, effectively infinity). Two very common departures from this are treated with an extra lens placed in front of the eye's own lens system — spectacles or contact lenses — precisely so that the combined power lands back on 25 cm and infinity.",{"id":352,"type":47,"variant":353,"title":354,"markdown":355},"def-myopia-hyperopia","definition","Myopia and hyperopia","**Myopia** (short-sightedness, or near-sightedness): the eyeball focuses images slightly in front of the retina, usually because the eyeball is a little too long or the lens system a little too strong. Distant objects look blurred; the far point is closer than infinity. Corrected with a **diverging** (concave) lens, which spreads the rays out a little before they enter the eye.\n\n**Hyperopia** (long-sightedness, or far-sightedness): images focus slightly behind the retina. Close objects look blurred; the near point is farther than the usual 25 cm. Corrected with a **converging** (convex) lens, which does some of the extra converging the eye's own lens cannot manage.",{"id":357,"type":205,"title":358,"problem":359,"steps":360,"help":365},"we-myopia","How strong a lens does short sight need?","A student cannot focus clearly on anything beyond 2 m (their far point). What power of corrective lens do they need, and is it converging or diverging?",[361,362,363,364],"For myopia, the corrective lens must take light from an object at infinity and make it *appear* to come from the person's own far point instead — right where their eye can already focus it.","A lens's power in dioptres is 1 ÷ focal length in metres. For an object at infinity imaged at the far point, the required focal length equals minus the far point distance (negative because the image must be virtual, on the same side as the object).","Power = −1 ÷ 2 = **-0.5 D**.","The negative sign means a **diverging** lens — exactly the concave spectacle lenses you can see are thinner in the middle if you look at a short-sighted friend's glasses edge-on.",{"simplerExplanation":366},"A shorter far point needs a stronger (more negative) lens. Power = −1 ÷ far point in metres.",{"id":368,"type":205,"title":369,"problem":370,"steps":371,"help":376},"we-hyperopia","How strong a lens does long sight need?","An adult's near point has drifted out to 1 m, instead of the usual 25 cm. What power of lens brings a book held at the normal 25 cm back into focus?",[372,373,374,375],"For hyperopia, the lens must take an object at the normal near point (0.25 m) and make it *appear* to come from the person's own actual near point instead, where their eye can focus it.","Using the lens power formula for this case: power = 1 ÷ 0.25 − 1 ÷ (actual near point in metres).","Power = 1 ÷ 0.25 − 1 ÷ 1 = 4 − 1 = **+3 D**.","The positive sign means a **converging** lens — the thicker-in-the-middle spectacle lenses common in reading glasses, which is exactly what this condition is often nicknamed for.",{"anotherExample":377},"This is the same arithmetic — just rearranged — as the shaving-mirror and lens-case calculations from earlier layers. Optical-power problems almost always reduce to 1\u002Fv ± 1\u002Fu = 1\u002Ff in some form.",{"id":379,"type":47,"variant":380,"title":381,"markdown":382},"careful-eyesight","careful","This is physics, not medical advice","Real prescriptions come from a qualified eye examination, account for astigmatism and other complications this simple model ignores, and are given in a form (like \"−0.50 D\") a person should discuss with an optometrist, not calculate for themselves from a formula. The value of the physics here is understanding *why* the two conditions need opposite kinds of lens, not self-diagnosing.",{"id":384,"type":129,"component":130,"componentVersion":5,"config":385,"objective":419,"textAlternative":420,"help":421},"lab-vision-sort",{"prompt":386,"bins":387,"items":394,"seconds":168},"Does this description point to myopia (short sight, needs a diverging lens) or hyperopia (long sight, needs a converging lens)?",[388,391],{"id":389,"label":390},"myopia","Myopia — diverging lens",{"id":392,"label":393},"hyperopia","Hyperopia — converging lens",[395,399,403,407,411,415],{"id":396,"label":397,"bin":389,"why":398},"far-blur","Distant road signs look blurred, but a book held close is sharp","Trouble with distant objects and a far point closer than infinity is the definition of myopia.",{"id":400,"label":401,"bin":392,"why":402},"near-blur","A book held at the usual reading distance looks blurred, but distant objects are sharp","Trouble focusing up close, with a near point farther than the usual 25 cm, is hyperopia.",{"id":404,"label":405,"bin":389,"why":406},"diverge-fix","Corrected with spectacles that are thinner in the middle than at the edges","A diverging (concave) lens is thinner in the middle — the fix for myopia.",{"id":408,"label":409,"bin":392,"why":410},"converge-fix","Corrected with spectacles that are thicker in the middle than at the edges","A converging (convex) lens is thicker in the middle — the fix for hyperopia, and the classic look of reading glasses.",{"id":412,"label":413,"bin":389,"why":414},"long-eyeball","Often linked to an eyeball that is slightly too long, focusing images just in front of the retina","A too-long eyeball (or too-strong lens system) is a common cause of myopia.",{"id":416,"label":417,"bin":389,"why":418},"negative-power","Needs a lens with a negative power, measured in dioptres","A negative dioptre value always means a diverging lens, used for myopia.","Sort six descriptions into myopia (needs a diverging lens) or hyperopia (needs a converging lens).","Six clues about symptoms, causes and corrective lenses, sorted into myopia and hyperopia. The underlying rule for every card: myopia struggles with far things and is fixed with a diverging (thinner-in-the-middle, negative-power) lens; hyperopia struggles with near things and is fixed with a converging (thicker-in-the-middle, positive-power) lens.",{"hints":422},[423,424],"A lens's shape (thin or thick in the middle) tells you converging or diverging at a glance.","Negative dioptres always mean diverging; positive always mean converging.",{"id":426,"type":53,"title":427,"eyebrow":428,"navLabel":429},"d-ch5","Total internal reflection at planetary scale","Chapter 05","5 TIR at scale",{"id":431,"type":43,"markdown":432},"fibre-deep","An optical fibre keeps light bouncing down a hair-thin glass core by total internal reflection at the boundary with a surrounding layer of slightly lower refractive index (the **cladding**), at an angle always kept steeper than the critical angle. Almost the entire internet's long-distance traffic — video calls, this very page, financial trades between cities — travels this way today, not as electricity in a copper wire.",{"id":434,"type":62,"caption":435,"columns":436,"rows":441},"table-fibre-latency","Time for a signal to cross two real-world fibre routes (light in fibre travels at c ÷ 1.468 ≈ 2.04 × 10⁸ m\u002Fs)",[437,438,439,440],"Route","Cable distance","One-way time","Compare with a vacuum at c",[442,447],[443,444,445,446],"Chennai to Delhi","about 2,200 km","10.8 ms","7.34 ms in a vacuum — fibre adds about 3.5 ms",[448,449,450,451],"London to New York","about 5,900 km (submarine cable)","28.9 ms","19.68 ms in a vacuum — fibre adds about 9.2 ms",{"id":453,"type":47,"variant":454,"title":455,"markdown":456},"nuance-fibre-round-trip","nuance","Why a video call has a bigger delay than the cable alone","The pure travel time for light in fibre between London and New York is only about 29 ms each way — barely noticeable. A round trip (your question there, the answer back) is about 58 ms. Yet real video calls across the Atlantic often feel far laggier than that. The extra delay comes almost entirely from **equipment**, not light's speed: routers, switches, encoding and decoding video, and the receiving computer's own processing, all add up far more than the light's actual travel time ever does.",{"id":458,"type":318,"itemId":459,"prompt":460,"check":461,"hints":466,"feedback":469},"prac-fibre","light.deepen-fibre-distance","A fibre link takes 15 ms to send a signal one way. Using the fibre speed of about 2.04 × 10⁸ m\u002Fs, roughly how far away is the other end, in kilometres (to the nearest 100 km)?",{"kind":462,"answer":463,"tolerance":464,"unit":465},"number",3100,150,"km",[467,468],"Distance = speed × time.","Convert 15 ms to seconds first: 0.015 s.",{"correct":470,"incorrect":471},"Right: 2.04 × 10⁸ m\u002Fs × 0.015 s ≈ 3063 km.","Multiply the fibre speed (in m\u002Fs) by the time in seconds (0.015 s), then convert metres to kilometres.",{"id":473,"type":47,"variant":474,"title":475,"markdown":476},"misc-fibre-instant","misconception","\"The internet is instant\"","It only feels instant for small amounts of data over short distances. Every single signal, however it travels, is limited by the speed of light in its medium — fibre-optic signals at roughly two-thirds of light's vacuum speed, electrical signals in copper at a broadly similar fraction. A satellite in geostationary orbit, 36,000 km up, adds a very noticeable quarter-second of delay each way — which is exactly why satellite phone calls have that unmistakable, slightly awkward pause.",{"id":478,"type":53,"title":479,"eyebrow":480,"navLabel":481},"d-ch6","Descartes works out the rainbow with a bowl of water","Chapter 06","6 Descartes' bowl",{"id":483,"type":43,"markdown":484},"descartes-setup","The 42° figure is not a modern discovery. In 1637, the French philosopher and mathematician **René Descartes** worked it out using nothing but Snell's law (which he had a hand in formulating) and a large glass sphere filled with water standing in for a giant raindrop, decades before Newton split light with a prism and nearly three centuries before anyone had a wave or particle theory of light to explain *why* Snell's law holds.",{"id":486,"type":487,"title":488,"items":489},"timeline-rainbow-history","timeline","From folklore to physics: understanding the rainbow",[490,494,498,502],{"time":491,"title":492,"text":493},"c. 1200","Theodoric of Freiberg","Working in Germany, argued that a rainbow forms from light refracting and reflecting inside individual raindrops, rather than in a mist or cloud as a whole — remarkably close to the modern picture, using only geometry and glass spheres as stand-ins for raindrops.",{"time":495,"title":496,"text":497},"1637","Descartes' calculation","Using Snell's law and a water-filled glass sphere, calculated the precise 42° angle for the primary bow and about 51° for the secondary — matching observation closely, though he had no explanation for *colour* within the bow.",{"time":499,"title":500,"text":501},"1666–1672","Newton's prism experiments","Showed that white light is a mixture of colours that a prism merely separates, explaining why a rainbow shows a spectrum rather than a single colour at each angle.",{"time":503,"title":504,"text":505},"1803","Young's wave evidence","Thomas Young's interference experiments supported treating light as a wave, eventually enabling a fuller explanation of the finer bands sometimes seen inside a rainbow (supernumerary bows), which Descartes' simple ray picture cannot predict at all.",{"id":507,"type":47,"variant":92,"title":508,"markdown":509},"aha-descartes-no-colour","Descartes got the angle right decades before anyone could explain the colours","Descartes' geometry correctly predicted *where* the rainbow would appear using only Snell's law and trigonometry — no knowledge of wavelengths, no idea that different colours refract by even slightly different amounts. He treated all light as behaving identically. It took Newton's later prism experiments to show that red and violet actually take very slightly different paths through a raindrop (different refractive indices), which is what spreads the single bright angle into a full band of colour rather than a plain white arc.",{"id":511,"type":205,"title":512,"problem":513,"steps":514,"help":520},"we-descartes-trace","Tracing one ray, Descartes' way","Trace a single ray entering a spherical raindrop at 59° from the drop's centre line, using Snell's law with water's refractive index of 1.333, to see why this particular angle turns out to matter so much.",[515,516,517,518,519],"On entry: sin(refraction angle) = sin(59°) ÷ 1.333, giving a refraction angle inside the drop of about 40.2°.","That refracted ray crosses the drop and strikes the inside of the far surface, where it partially reflects (the rest escapes and is lost from this particular path).","It crosses back to the near surface and refracts a second time on the way out, by the same Snell's law relationship in reverse, bending away from the normal back into air.","Add up all the bending at both refractions and the one reflection, and the ray's total change of direction from where it started comes to about 138° — which, measured back from the direction the light originally came from, is the **42°** you see the rainbow at.","Repeat this whole trace for entry angles a few degrees either side of 59° (as the table above already did numerically) and the total deviation barely changes at all — Descartes had, without calculus, found a minimum.",{"simplerExplanation":521},"Following one specific ray step by step through a raindrop, using Snell's law twice and one reflection, reproduces the 42° by direct calculation rather than by simply trusting the answer.",{"id":523,"type":53,"title":524,"eyebrow":525,"navLabel":526},"d-ch7","The rainbow, minimised","Chapter 07","7 Rainbow minimum",{"id":528,"type":43,"markdown":529},"rainbow-deep","Investigate told you the primary bow sits at about 42° because deviation reaches a **minimum** near one particular entry angle, and rays bunch up around that minimum. Here is the actual shape of that minimum, worked out point by point rather than asserted.",{"id":531,"type":62,"caption":532,"columns":533,"rows":536},"table-bow-deviation","Total deviation of light through a raindrop (refract, reflect once, refract), by entry angle, for water (n = 1.333)",[534,535],"Entry angle","Total deviation",[537,540,543,546,549,552],[538,539],"10°","170.1°",[541,542],"30°","151.9°",[544,545],"50°","139.7°",[547,548],"59°","137.9°",[550,551],"70°","140.7°",[553,554],"89°","163.6°",{"id":556,"type":47,"variant":92,"title":557,"markdown":558},"aha-bow-dip","A real dip, not a straight decline","Notice the pattern in the table: deviation falls from 170.1° at a grazing 10° entry down to a minimum of about 137.9° near 59°, then climbs back up to 163.6° at an almost edge-on 89°. Every entry angle *except* the small range right around the minimum sends its ray off to a distinctly different final deviation — meaning the light from those angles gets thinly spread across a wide range of directions, too dilute to see. Only the narrow bundle of angles near the minimum, roughly 55°–63°, all land within about half a degree of 42.1°, concentrating enough light to be visible as a distinct, bright arc. This mathematical bunching at a minimum (or maximum) is called a **caustic**, and it is the same basic idea that makes a coffee cup show a bright curved line of light on the surface when sunlight catches it.",{"id":560,"type":205,"title":561,"problem":562,"steps":563,"help":567},"we-secondary-bow","Why the secondary bow is fainter and reversed","The secondary rainbow comes from light that reflects **twice** inside a raindrop instead of once. Explain, without heavy calculation, why it is both fainter and colour-reversed compared with the primary bow.",[564,565,566],"Fainter: each time light reflects off the inside of a raindrop's surface, a little of it escapes rather than reflecting — the surface is not a perfect mirror. Two reflections means two chances for light to leak out, so less survives to reach your eye than after only one reflection. The secondary bow is reliably dimmer.","Reversed: with one internal reflection, the ray that finally reaches you has effectively been \"flipped\" once by that reflection. With two reflections, it has been flipped twice, restoring the original left-right sense but shifting *which* colour ends up on the inside versus the outside of the arc.","The result: the primary bow (one reflection) shows red on the outside, violet on the inside, at about 42°. The secondary bow (two reflections) shows the colours swapped — violet outside, red inside — at about 51°.",{"simplerExplanation":568},"One bounce inside the drop gives one arrangement of colours; two bounces both dims the light and flips the colour order.",{"id":570,"type":53,"title":571,"eyebrow":572,"navLabel":573},"d-ch8","Colour, precisely: scattering, addition, subtraction","Chapter 08","8 Colour, precisely",{"id":575,"type":43,"markdown":576},"rayleigh-full","Rayleigh's scattering law states that the intensity scattered by particles much smaller than a wavelength is proportional to **1 ÷ λ⁴**, where λ is the wavelength. That fourth power is what makes the sky's blue so decisive rather than a weak trend: red light (700 nm) is scattered about 5.9 times less than blue (450 nm), and about 9.4 times less than violet (400 nm) — because (700 ÷ 450)⁴ ≈ 5.9 and (700 ÷ 400)⁴ ≈ 9.4.",{"id":578,"type":579,"items":580},"f-rayleigh","formulas",[581,584,587],{"expression":582,"caption":583},"scattered intensity ∝ 1 ÷ λ⁴","Rayleigh scattering law for particles much smaller than the light's wavelength, such as nitrogen and oxygen molecules in air.",{"expression":585,"caption":586},"(700 ÷ 450)⁴ ≈ 5.9","Blue light scatters this many times more strongly than red light of wavelength 700 nm.",{"expression":588,"caption":589},"(700 ÷ 400)⁴ ≈ 9.4","Violet scatters even more strongly still, though there is less of it in sunlight and our eyes are less sensitive to it.",{"id":591,"type":43,"markdown":592},"colour-mix-precise","Additive and subtractive colour mixing are opposite operations on the same underlying spectrum. **Additive** mixing (coloured lights) adds wavelengths together: shine red and green light on the same white screen and both sets of wavelengths reach your eye at once, which your brain reads as yellow — no yellow wavelength is actually present. **Subtractive** mixing (paints, inks, filters) works by removing wavelengths: a cyan pigment absorbs red light and reflects blue and green; a yellow pigment absorbs blue and reflects red and green; mix cyan and yellow paint and each removes what the other would have reflected, leaving mostly green.",{"id":594,"type":62,"caption":595,"columns":596,"rows":600},"table-colour-precise","What survives when subtractive primaries overlap (each pigment absorbs its opposite colour)",[597,598,599,66],"Pigments mixed","Each removes","What is left to reflect",[601,605,609,613],[602,603,604,604],"Cyan + Yellow","Cyan removes red; yellow removes blue","Green",[606,607,608,608],"Cyan + Magenta","Cyan removes red; magenta removes green","Blue",[610,611,612,612],"Magenta + Yellow","Magenta removes green; yellow removes blue","Red",[614,615,616,617],"Cyan + Magenta + Yellow","Red, green and blue all removed","Almost nothing","Near-black (real ink adds a true black, \"K\", because pigment mixing is never perfect)",{"id":619,"type":47,"variant":454,"title":620,"markdown":621},"nuance-cmyk","Why printers add a fourth, black ink","In principle, mixing cyan, magenta and yellow ink should produce black by removing red, green and blue entirely. In practice, real inks are not perfectly pure, so the mixture comes out as a muddy dark brown rather than a true black — and using three coloured inks to make black also wastes ink and money. Printers add a fourth, pure black ink instead, giving the CMYK system its name (K stands for \"key\", the printing trade's term for the black plate).",{"id":623,"type":129,"component":624,"componentVersion":5,"config":625,"objective":631,"textAlternative":632,"help":633},"lab-scatter-deepen","prism-lab",{"modes":626,"rounds":630},[627,628,629],"scattering","filters","mixing",8,"Test Rayleigh scattering's wavelength dependence directly, then compare it with how filters and mixed lights change colour.","Three modes on one virtual bench: scattering (particles deflecting different colours by different amounts, standing in for air molecules and sunlight), filters (a coloured filter absorbing every wavelength except its own), and mixing (coloured lights adding together).\n\nRunning all three back to back makes the distinction concrete: scattering *redirects* light sideways without absorbing it, a filter *absorbs* the wavelengths it does not pass, and mixing *adds* different wavelengths together to the eye — three different mechanisms that a young learner might otherwise lump together as \"changing the colour\".",{"hints":634},[635,636],"Scattering favours short wavelengths (blue, violet) far more than long ones (red).","A filter's colour is decided by what it lets through, not what it reflects.",{"id":638,"type":318,"itemId":639,"prompt":640,"check":641,"hints":644,"feedback":647},"prac-rayleigh","light.deepen-rayleigh-ratio","Using scattering ∝ 1 ÷ λ⁴, roughly how many times more strongly is light of wavelength 450 nm (blue) scattered than light of wavelength 600 nm (orange)? Round to one decimal place.",{"kind":462,"answer":642,"tolerance":643},3.2,0.2,[645,646],"Compute the ratio of wavelengths first: 600 ÷ 450.","Then raise that ratio to the fourth power.",{"correct":648,"incorrect":649},"Right: (600 ÷ 450)⁴ ≈ 3.2.","Divide 600 by 450, then raise the result to the power of 4.",{"id":651,"type":53,"title":652,"eyebrow":653,"navLabel":654},"d-ch9","Mix-ups worth clearing up","Chapter 09","9 Common mix-ups",{"id":656,"type":47,"variant":474,"title":657,"markdown":658},"mix-mirror-formula","\"1\u002Fv = 1\u002Ff − 1\u002Fu means bigger f always gives a bigger image\"","A longer focal length does change the image, but not in one fixed direction: for a real, distant object it gives a *larger* real image (that is why long telephoto camera lenses have long focal lengths), but for an object closer than the focus it changes the *virtual* magnification too, and the relationship is never simply \"longer f, bigger image\" in every situation. Always work through the formula rather than guessing from the focal length alone.",{"id":660,"type":47,"variant":474,"title":661,"markdown":662},"mix-critical-material","\"The critical angle is a property of light, not of the material\"","The critical angle depends entirely on the refractive index of the material light is trying to leave — water (48.8°), glass (41.8°) and diamond (24.4°) each have their own, and the same beam of light has a different critical angle depending which material it is inside. Light itself has no fixed critical angle.",{"id":664,"type":47,"variant":474,"title":665,"markdown":666},"mix-rainbow-object","\"A rainbow is a real object hanging in the sky\"","It has no location independent of the observer: it is a direction and an angle (about 42° from the point opposite the Sun), reconstructed fresh by each viewer's own eyes from a completely different set of raindrops. Fly towards a rainbow in an aeroplane and, under the right conditions, you can see it as a complete **circle** — evidence that it was never a fixed arc-shaped object in the first place, only ever a cone of directions centred on your own eye.",{"id":668,"type":47,"variant":454,"title":669,"markdown":670},"mix-color-absorbed","A red object is not \"making\" red light","A red apple looks red because its skin's pigments absorb most of the green and blue wavelengths in white light and reflect mostly the red ones back to your eye. The apple does not produce, add or create any light itself; it only edits what falls on it. Take away the light source entirely, and the apple, like everything non-luminous, is not red, or any other colour: it is simply dark.",{"id":672,"type":53,"title":673,"eyebrow":674,"navLabel":675},"d-ch10","Pulling it together","Chapter 10","10 Pulling it together",{"id":677,"type":678,"title":679,"terms":680},"gloss-deepen","glossary","Terms from this lesson",[681,685,688,691,695,699],{"term":682,"meaning":683,"example":684},"Dioptre (D)","Unit of lens or mirror power, equal to 1 divided by the focal length in metres.","A −0.5 D lens corrects a far point of 2 m.",{"term":686,"meaning":687},"Myopia","Short-sightedness: the far point is closer than infinity; corrected with a diverging lens.",{"term":689,"meaning":690},"Hyperopia","Long-sightedness: the near point is farther than the usual 25 cm; corrected with a converging lens.",{"term":692,"meaning":693,"example":694},"Caustic","A bright line or curve formed where many light rays bunch together at a minimum or maximum deviation.","The rainbow, and the bright curve inside a sunlit coffee cup.",{"term":696,"meaning":697,"example":698},"Rayleigh scattering","Scattering of light by particles much smaller than its wavelength, with intensity proportional to 1 ÷ wavelength⁴.","Explains the blue sky and red sunsets.",{"term":700,"meaning":701},"CMYK","The four-ink printing system: cyan, magenta, yellow and a separate true black (K).",{"id":703,"type":129,"component":704,"componentVersion":5,"config":705,"objective":722,"textAlternative":723,"help":724},"lab-deepen-match","match-pairs",{"prompt":706,"mode":707,"pairs":708},"Match each deepen-layer term to its meaning.","connect",[709,712,714,716,718,720],{"a":710,"b":711},"Dioptre","Unit of lens power: 1 ÷ focal length in metres",{"a":686,"b":713},"Short sight; far point closer than infinity; corrected with a diverging lens",{"a":689,"b":715},"Long sight; near point farther than 25 cm; corrected with a converging lens",{"a":692,"b":717},"A bright line or curve where many rays bunch at a minimum deviation",{"a":696,"b":719},"Scattering ∝ 1 ÷ wavelength⁴; explains the blue sky",{"a":700,"b":721},"Cyan, magenta, yellow and a separate true black printing ink","Match six deepen-layer terms — dioptre, myopia, hyperopia, caustic, Rayleigh scattering and CMYK — to their meanings.","A connect-the-pairs game with six terms introduced in this layer, matched to plain-language meanings covering optical power, the two common eyesight conditions, the rainbow's bright-line mathematics, and the two colour-mixing systems.",{"hints":725},[726,727],"A negative dioptre value always means a diverging lens.","Rayleigh scattering's fourth-power law is what makes the sky decisively blue rather than only slightly blue.",{"id":729,"type":730,"prompt":731},"reflect-deepen","reflection","The metre is now defined using the speed of light instead of the speed of light being measured using a fixed metre. Why might scientists prefer to build definitions on a constant of nature (like light's speed) rather than on a physical object (like an old metal bar)? Suggest one advantage and one possible drawback.",{"id":733,"type":734,"title":735,"questions":736},"quiz-deepen","quiz","Test what you worked out",[737,750,763,776,789],{"itemId":738,"prompt":739,"options":740,"correct":331,"why":749},"light.deepen-q-michelson","Michelson's 1879 rotating-mirror measurement of light's speed was:",[741,743,745,747],{"id":325,"label":742},"Less accurate than Rømer's",{"id":328,"label":744},"About as accurate as Fizeau's",{"id":331,"label":746},"Far more precise than earlier methods, within about 0.04% of the modern value",{"id":334,"label":748},"Exactly equal to the modern defined value","Michelson's method was a major improvement in precision over both Rømer's astronomical estimate and Fizeau's toothed wheel.",{"itemId":751,"prompt":752,"options":753,"correct":328,"why":762},"light.deepen-q-mirror-derive","The mirror formula 1\u002Fu + 1\u002Fv = 1\u002Ff comes from:",[754,756,758,760],{"id":325,"label":755},"A rule that must simply be memorised, with no underlying reason",{"id":328,"label":757},"Two pairs of similar triangles in the ray diagram, set equal to each other",{"id":331,"label":759},"A direct measurement with no theory behind it",{"id":334,"label":761},"Newton's laws of motion","It is derived exactly as shadow-size and magnification rules are: from similar triangles in a diagram.",{"itemId":764,"prompt":765,"options":766,"correct":328,"why":775},"light.deepen-q-myopia","A corrective lens for myopia (short sight) is:",[767,769,771,773],{"id":325,"label":768},"Converging, because the eye needs more bending power",{"id":328,"label":770},"Diverging, because the eye already bends light too strongly for its length",{"id":331,"label":772},"Always exactly +2.00 D",{"id":334,"label":774},"Not a lens at all, but a mirror","Myopia focuses images in front of the retina; a diverging lens spreads the rays out a little before the eye, moving the focus back onto the retina.",{"itemId":777,"prompt":778,"options":779,"correct":328,"why":788},"light.deepen-q-fibre","Most of the delay you notice on a long-distance video call is caused by:",[780,782,784,786],{"id":325,"label":781},"Light itself travelling too slowly through fibre",{"id":328,"label":783},"Equipment: routers, encoding and processing, not the light's travel time",{"id":331,"label":785},"The video being sent by radio instead of fibre",{"id":334,"label":787},"Sound always being faster than light","The pure fibre travel time even across an ocean is only tens of milliseconds; encoding, routing and processing typically add far more.",{"itemId":790,"prompt":791,"options":792,"correct":328,"why":801},"light.deepen-q-secondary","Compared with the primary rainbow, the secondary rainbow is:",[793,795,797,799],{"id":325,"label":794},"Brighter and in the same colour order",{"id":328,"label":796},"Fainter, with its colours reversed",{"id":331,"label":798},"Exactly the same in every way",{"id":334,"label":800},"Only visible from an aeroplane","An extra internal reflection loses some light at each bounce (fainter) and flips the colour order (reversed).",{"id":803,"type":804,"title":805,"points":806},"cheat-deepen","summary","Cheat sheet",[807,808,809,810,811,812],"**Speed of light history:** Rømer (1676, astronomical) → Fizeau (1849, toothed wheel) → Michelson (1879, rotating mirror, within 0.04%) → defined exactly since 1983 as 299,792,458 m\u002Fs.","**Mirror\u002Flens formula 1\u002Fu + 1\u002Fv = 1\u002Ff** comes directly from similar triangles in a three-ray diagram (parallel ray, focal ray, centre ray).","**Corrective lenses:** myopia (short sight) needs a diverging lens, power = −1 ÷ far point (m); hyperopia (long sight) needs a converging lens, power = 1\u002F0.25 − 1 ÷ near point (m).","**Optical fibre:** total internal reflection carries light at about c ÷ 1.468. Even London to New York takes only about 29 ms one way — most video-call lag is equipment, not light speed.","**Rainbow:** deviation through a raindrop has a genuine minimum near 59° entry, bunching light into a caustic at about 42°. Two internal reflections give a fainter, colour-reversed secondary bow near 51°.","**Rayleigh scattering ∝ 1 ÷ λ⁴:** blue scatters about 6 times more than red, violet about 9 times more — the reason for a blue sky and red sunsets.",{"id":814,"type":815,"conceptId":816,"relation":817,"explanation":818},"conn-deepen-angles","connection","angles","helps_understand","Every ray-diagram derivation in this lesson leans on properties of similar triangles and angles measured precisely from a normal or an axis.",{"id":820,"type":815,"conceptId":821,"relation":822,"explanation":823},"conn-deepen-eclipses","eclipses","related_to","The same caustic-like bunching that concentrates light into a sharp rainbow also explains why a total solar eclipse's path of totality is a narrow band rather than a smeared-out region.",{"id":825,"type":826,"sourceIds":827},"sources-deepen","sources",[828,829,830,831,832,833,834,835],"light-wikipedia-speed-of-light","light-nist-speed-of-light","light-hyperphysics-mirror","light-hyperphysics-lens","light-hyperphysics-totint","light-hyperphysics-rainbow","light-wikipedia-rayleigh","light-physicsclassroom-refraction",[828,829,830,831,832,833,834,835],"needs_review",{"generatedBy":839,"notes":840},"claude-code","Draft generated locally; every number computed and asserted in scratchpad\u002Flight\u002Fnumbers.py. Pending owner review.","bfb70ffe7a286693092c372c0ce439bb04b8a27058196c88e2f54a0c5de18c06",{"component:sort-game@1":843,"component:light-ray@1":844,"logic:practice":845,"component:prism-lab@1":846,"component:match-pairs@1":847,"source:light-hyperphysics-lens":848,"source:light-hyperphysics-mirror":849,"source:light-hyperphysics-rainbow":850,"source:light-hyperphysics-totint":851,"source:light-nist-speed-of-light":852,"source:light-physicsclassroom-refraction":853,"source:light-wikipedia-rayleigh":854,"source:light-wikipedia-speed-of-light":855},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","d66d44021dc28328ce2ea82e0d6cefc9466b5dc65dbb93c92b05b88161725be6","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","835c8ba7fd707c60ded46494f4b672aa095199ba9a2628409905335d0b3434a9","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","4cdd337c59a170b52016937cc2cc4327db640e3f7c8e883689608119257931d4","cc4f04df90efb4f54357cdc43554d566fcd0bb5efb3303655a1579a15d292259","dce907528f01e833f82d68150b423cc68c60d1c9c88673a96a84eb269bbbd3ce","fbea9871167acaf5df4f2d491a5417bc634467e447ecc5463c5d581b47e11cc6","525bf3f6ba3b6422ad2b7c0fa6254ee5070e1e8cb385320d308ae53d75c4ede4","43e2d25a3db005014fd7d491471e9e47242ce863ec6c79370b74a25bfe96acfb","4c7f726ffcab1b913b628c2df33237f78d9472f867c4ab3682ea2c2f080d0ea0","e6620f59bd5a24800f47530a4ed586390c383a730742e4e6bdf5fda99401c43a",{"state":857,"reviewer":858,"selfReview":272,"reviewedAt":859,"method":860},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598400]