Total internal reflection — critical angle derivation: Visual walkthrough

2.5.5 · D2 · Physics › Optics › Total internal reflection — critical angle derivation

Step 0 — Woh chaar cheezein jo hume pehle draw karni hain

Kisi bhi physics se pehle, aao hum characters ki cast par agree kar lein. Yahan abhi koi formula nahi hai — bas labelled pictures hain.

Figure dekho: peeli dotted line normal hai. Neeli ray boundary ke neeche se aati hai aur woh normal ke against jo angle banati hai woh $θ_{1}$ hai. Refracted ray (green) upar jaati hai aur $θ_{2}$ angle banati hai. Angles normal ko hug karte hain, surface ko nahi — yeh ek aadat aadhi sign mistakes khatam kar deti hai.

Step 1 — Ek hi physics jo hum import karte hain: Snell's Law

KYA. Hum Snell's Law se ek equation import karte hain: $medium 1 ka index n_{1} incident ray ka angle sin θ_{1} = medium 2 ka index n_{2} refracted ray ka angle sin θ_{2}$

Ise term by term padho: boundary ke dono taraf slow-down number aur angle ka sine ka product same hota hai. Woh product ek conserved bookkeeping quantity hai — jo medium 1 ke paas hai, medium 2 ko match karna padega.

KYun yeh tool aur koi nahi? Hume ek rule chahiye jo dono angles ko boundary ke aacross connect kare. Sirf reflection yeh nahi bata sakta ki refraction kab rukti hai; sirf ek law jo $θ_{1}$ aur $θ_{2}$ ko link kare woh kar sakta hai. Snell's law ek aisa unique link hai. Yeh light ke $v = c / n$ tak slow hone se nikalta hai — jahaan $v$ medium ke andar light ki speed hai aur $c$ speed of light in vacuum hai (ek fixed universal number, lagbhag $3 \times 1 0^{8} m/s$ , har jagah same) — saath mein wavefronts ka boundary ke aacross continuous rehna; dekho Refraction of Light.

KYun sine aur tan ya cosine nahi? Sine ray ka sideways lean normal se measure karta hai — exactly woh quantity jo tab matched rehni chahiye jab wavefronts boundary ke along glide karti hain. Cosine forward part measure karega (unmatched); tan dono ko mix karta hai. Sine yahan honest bookkeeper hai.

Figure mein do shaded rectangles $n_{1} sin θ_{1}$ aur $n_{2} sin θ_{2}$ ko equal area ke saath dikhaya gaya hai — woh equality hi Snell's law ko visible roop mein dikhana hai.

Step 2 — Woh direction choose karo jo trapping possible banata hai

KYA. Hum light ko denser medium se rarer medium mein bhejtein hain: $n_{1} > n_{2}$ . (Denser = bada index = slow light.)

KYun. Snell ko rearrange karo outgoing angle isolate karne ke liye: $sin θ_{2} = bigger > 1 \frac{n _{1}}{n _{2}} sin θ_{1}$

Kyunki $n_{1} > n_{2}$ hai, fraction $\frac{n _{1}}{n _{2}}$ 1 se bada hai. Yeh $sin θ_{1}$ ko amplify karta hai. Trapping ka poora seed yehi hai: outgoing angle upar ki taraf push ho raha hai, aur $sin$ sirf $1$ tak hi pahunch sakta hai uske baad jagah khatam ho jaati hai.

Figure mein red arrow: $θ_{2}$ (green ray) $θ_{1}$ (blue ray) se wider draw ki gayi hai. Ray normal se door bend karti hai.

Step 3 — Dial ghuma do aur dekho $θ_{2}$ aage bhaagte huye

KYA. $θ_{1}$ ko $0°$ se dhire dhire badhao aur $θ_{2}$ ko track karo.

KYun. Step 2 se, $sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1}$ hamesha $sin θ_{1}$ ki ek stretched copy hai. Toh jaise jaise $θ_{1}$ badhta hai, $θ_{2}$ zyaada tezi se badhta hai aur top ( $90°$ ) par pehle pahunch jaata hai. Yeh " $θ_{2}$ pehle pahunchta hai" wala fact sabse pivotal hai.

Figure mein teen incident rays (blue, badhti hui) aur unke refracted partners (green, zyaada tezi se badhte aur surface ki taraf flat hote hue) dikhaye hain. Notice karo ki green ray almost flat lie kar rahi hai jabki blue ray abhi bhi lean kar rahi hai — $θ_{2}$ race jeet raha hai.

Step 4 — Tipping point: critical angle define karo

KYA. Critical angle $θ_{c}$ woh special value hai $θ_{1}$ ki, jis par refracted ray just surface ko graze karti hai, yaani $θ_{2} = 90°$ .

KYun $90°$ ? Kyunki $90°$ sabse bada possible refraction angle hai — refracted ray boundary ke saath flat lie karti hai. Ek real ray ke liye escape karne ke liye koi " $91°$ " nahi hai. Toh $θ_{2} = 90°$ woh last frame hai jisse pehle refraction khatam hoti hai.

Is exact incidence par (yellow ray), green refracted ray surface par flat lie karti hai — grazing, gayab hone wali hai. Yeh woh frame hai jisse hum formula banate hain.

Step 5 — Substitute karo aur solve karo (term by term)

KYA. $θ_{1} = θ_{c}$ aur $θ_{2} = 90°$ Snell's law mein daalo.

$n_{1} sin θ_{c} = n_{2} = 1 sin 90°$

Yahan $sin 90° = 1$ isliye hai kyunki — Step 0 mein waapis dekho — surface ke along flat ray, normal ke saath $90°$ angle banati hai, aur $90°$ ka sine iska maximum hai, $1$ . Yehi reason hai ki grazing frame itna clean hai: yeh right side ko sirf $n_{2}$ tak collapse kar deta hai.

Solve karo dono sides ko $n_{1}$ se divide karke:

Figure ek right triangle hai jiska hypotenuse $n_{1}$ hai aur vertical side $n_{2}$ hai: $sin θ_{c} = \frac{opposite}{hyp} = \frac{n _{2}}{n _{1}}$ — formula ek shape ke roop mein draw kiya gaya hai.

Step 6 — $θ_{c}$ se aage jao: kyun light trap hoti hai

KYA. $θ_{1} > θ_{c}$ set karo aur Snell se pucho ki $θ_{2}$ kya hona chahiye.

$sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1} > \frac{n _{1}}{n _{2}} sin θ_{c} = \frac{n _{1}}{n _{2}} \cdot \frac{n _{2}}{n _{1}} = 1$

Toh Snell demand karta hai ki $sin θ_{2} > 1$ ho.

KYun yeh impossible hai. Kisi bhi real angle ka sine $- 1$ aur $+ 1$ ke beech rehta hai. Koi real angle nahi hai jiska sine $1$ se zyaada ho. Toh koi refracted ray exist nahi kar sakti. Kahin escape karne ki jagah nahi hone se, 100% light energy denser medium mein wapas reflect hoti hai — yeh hai Total Internal Reflection.

Left panel: $θ_{1} = θ_{c}$ , grazing green ray. Right panel: $θ_{1} > θ_{c}$ — green ray gone hai, ek red reflected ray seedha wapas bounce karti hui ayi hai. Yahi Optical Fibres aur Prisms and Total Internal Reflection mein periscope prisms ko power deta hai.

Step 7 — Degenerate aur edge cases (koi gap mat chhodna)

Recall Forecast-then-verify: do glass media

Predict karo $θ_{c}$ for $n_{1} = 1.6$ into $n_{2} = 1.4$ . Verify: $sin θ_{c} = 1.4/1.6 = 0.875 \Rightarrow θ_{c} = sin^{- 1} (0.875) \approx 61.0°$ . Dono indices matter karte hain — kabhi assume mat karo ki doosra medium air hai.

Ek picture mein poora summary

Yeh single figure poori story stack karta hai: incident rays (blue) ka ek fan boundary ke neeche; har ek escape karti hai (green) door bend hoti hui — jab tak ek $θ_{c}$ (yellow) par flat graze nahi karti — aur isse steep har ray reflect (red) hoti hai. Formula $sin θ_{c} = n_{2} / n_{1}$ tipping ray par print hai.

Recall Feynman ki style mein retelling — poora walkthrough plain words mein

Socho tum ek beam of light ho jo glass mein reh rahi ho, upar air mein escape karne ki koshish kar rahi ho. Air mein light zyaada tezi se chalti hai, toh jaise hi tum boundary cross karte ho, tum seedhi upar wali line se door swing karte ho — aur tum jitne zyaada slanted hote ho, utna hi zyaada violently swing karte ho. Ek catch hai: tum tabhi tak swing kar sakte ho jab tak tum surface ke saath flat nahi ho jaate. Snell's law rulebook hai, aur woh kehta hai tumhara outgoing tilt tumhara incoming tilt ka ek stretched copy hai ( $n_{1} / n_{2}$ se stretch hua, ek number jo 1 se bada hai). Toh tumhara outgoing tilt "flat" pehle hit karta hai tumhare incoming tilt se. Us exact moment ka incoming angle critical angle hai, aur "flat = $90°$ " rulebook mein daalne par $sin θ_{c} = n_{2} / n_{1}$ milta hai. Thoda aur lean karo aur rulebook ek aise tilt ki maang karta hai jo flat se aage ho — jo exist nahi karta — toh tum phans jaate ho, aur seedha andar wapas bounce karte ho. Woh bounce total internal reflection hai.

Connections

Snell's Law — ek hi imported rule (Step 1).
Refraction of Light — normal se door bending (Steps 2–3).
Refractive Index — slow-down number $n$ (Step 0).
Optical Fibres — trapping kaam mein layi gayi (Step 6).
Prisms and Total Internal Reflection — glass mein $45° > θ_{c}$ .
Mirage and Atmospheric Refraction — usi trapping ka gradual version.
Brewster's Angle — ek alag named angle ( $tan θ_{B} = n_{2} / n_{1}$ ), trapping ke baare mein nahi balki polarization ke baare mein.