2.5.5 · D3Total internal reflection — critical angle derivation
Yeh page critical-angle idea ki drill floor hai. Parent note ne humein woh ek formula diya jo is poore page pe kaam aayega:
Drill se pehle, chalte hain aur har tarah ke questions ki list banate hain jinme yeh ek formula dress up ho sakta hai. Agar hum har row mein ek example solve kar lein, toh exam mein koi cheez humein surprise nahi kar sakti.
Har critical-angle problem actually sin θ c = n 2 / n 1 edge cases bhi hain jahan formula thoda alag behave karta hai. Yeh raha poora map:
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Case class
Kya diya hai
Kya poochha hai
Kaun sa cell
1
Standard forward
n 1 , n 2 θ c forward, do real media
2
Air ke against
n 1 θ c forward, n 2 = 1
3
Inverse
θ c n 1 index ke liye solve karo
4
Two-media inverse
θ c doosra index
index ke liye solve karo
5
Degenerate: equal indices
n 1 = n 2 θ c limiting value θ c = 90°
6
Impossible: rarer→denser
n 1 < n 2 θ c no solution case
7
Critical se aage
n 1 , n 2 , θ 1 > θ c kya hoga?
sign/regime test
8
Word problem
fibre / underwater
θ c real-world
9
Exam twist
prism / 45°
kya TIR hoga?
compare-to-45°
Ab hum har cell ko hit karte hain.
Worked example Example 1 — Cell 1: standard forward, do real mediaLight ek dense plastic (n 1 = 1.60 n 2 = 1.40 θ c
Forecast: dono indices kaafi close hain, toh ray muskil se bend hogi — guess karo: kya θ c 90° 30°
Direction check karo: n 1 = 1.60 > n 2 = 1.40 Yeh step kyun? Agar light doosri taraf jaati, toh koi critical angle exist hi nahi karta (Cell 6). Direction hamesha pehle verify karo.
Formula apply karo: sin θ c = n 1 n 2 = 1.60 1.40 = 0.875 Yeh step kyun? Yahi definition hai — refracted angle ko 90° Snell's Law mein hai.
Sine ko invert karo: θ c = sin − 1 ( 0.875 ) = 61.0° Yeh step kyun? Humein angle ka sine pata hai; sin − 1
Verify karo: indices close hain (ratio near 1), toh θ c 90° 61° sin 61.0° = 0.875 = 1.40/1.60
Worked example Example 2 — Cell 2: forward against airDiamond ka n 1 = 2.42 n 2 = 1 θ c
Forecast: diamond ka index bahut bada hai. Bada index → kya θ c
sin θ c = n 1 n 2 = 2.42 1 = 0.4132 Yeh step kyun? Air ke against, n 2 = 1 sin θ c = 1/ n 1 θ c = sin − 1 ( 0.4132 ) = 24.4° Yeh step kyun? Angle read karne ke liye sine ko undo karo.
Verify karo: ek chhota critical angle matlab hai ki andar ka almost koi bhi ray trap ho jaata hai — bilkul isi wajah se cut diamonds itne sparkle karte hain (light bahar nikalne se pehle kai baar bounce karti hai). Sanity check: sin 24.4° = 0.413
Worked example Example 3 — Cell 3: inverse,θ c θ c = 45.0° n
Forecast: 45° n 1 2
Air formula likho: sin θ c = n 1 Yeh step kyun? Air ke against n 2 = 1 n 1 = n
n n = sin θ c 1 = sin 45° 1 = 0.7071 1 Yeh step kyun? Humein n n = 1.414 Yeh step kyun? Yeh 2
Verify karo: n = 1.414 1.33 1.5 sin θ c = 1/1.414 = 0.707 = sin 45°
Worked example Example 4 — Cell 4: two-media inverseLight glass (n 1 = 1.50 θ c = 60.0° n 2
Forecast: TIR ke liye coating glass se rarer honi chahiye, toh guess karo: kya n 2 1.50
sin θ c = n 1 n 2 Yeh step kyun? Dono media real hain (koi bhi air nahi hai), toh poora do-index form rakhte hain.n 2 n 2 = n 1 sin θ c = 1.50 × sin 60° = 1.50 × 0.8660 Yeh step kyun? Unknown index ko isolate karne ke liye dono sides ko n 1 n 2 = 1.299 ≈ 1.30 Yeh step kyun? Arithmetic.
Verify karo: 1.30 < 1.50 hai hi. ✅
Worked example Example 5 — Cell 5: degenerate, equal indicesDo media dono ka n = 1.50 θ c
Forecast: agar media identical hain, toh kya koi aisi boundary bhi hai jo light ko bend kare? Kaunsa angle light ko "trap" karta?
sin θ c = n 1 n 2 = 1.50 1.50 = 1 Yeh step kyun? Equal indices plug karo — ratio exactly 1 θ c = sin − 1 ( 1 ) = 90° Yeh step kyun? Sirf wahi angle jiska sine 1 90°
Verify karo: θ c = 90° exactly flat chale, yaani effectively kabhi nahi . Yeh sahi hai: jab koi index difference nahi hota toh koi bending nahi hoti (dekho Refraction of Light ), toh TIR occur nahi ho sakta. Formula gracefully limiting value 90°
Worked example Example 6 — Cell 6: impossible direction (rarer → denser)Light air (n 1 = 1.00 n 2 = 1.50
Forecast: light denser medium mein ja rahi hai. Kya TIR yahan sense bhi banta hai?
Blindly formula apply karo: sin θ c = n 1 n 2 = 1.00 1.50 = 1.50 Yeh step kyun? Contradiction expose karne ke liye, formula ko bolne dete hain.
Poochho: kya sin θ c = 1.50 Nahi. Kisi bhi real angle ka sine [ − 1 , 1 ] Yeh step kyun? sin θ c = 1.5 koi real solution nahi hai — geometrically, aisa koi angle exist nahi karta.
Conclude karo: rarer→denser jaate waqt koi critical angle nahi hota . TIR is direction mein impossible hai.
Yeh step kyun? Denser medium mein refraction hamesha succeed karta hai (ray normal ki taraf bend hoti hai), toh light kabhi "trap" nahi hoti.
Verify karo: yeh parent note ki pehli condition se match karta hai — TIR ke liye n 1 > n 2 n 1 < n 2
Worked example Example 7 — Cell 7: critical angle se aageWater ke andar (n 1 = 1.33 n 2 = 1 θ 1 = 60° θ c
Forecast: kya 60°
Critical angle: sin θ c = 1.33 1 = 0.7519 ⇒ θ c = 48.8° Yeh step kyun? Compare karne se pehle threshold chahiye.
Compare karo: θ 1 = 60° > 48.8° = θ c Yeh step kyun? Regime decide hota hai is baat se ki hum θ c
Kyunki θ 1 > θ c sin θ 2 = n 2 n 1 sin θ 1 = 1.33 × sin 60° = 1.33 × 0.8660 = 1.152 > 1 Yeh step kyun? Snell ka demand check karne se pata chalta hai ki woh kuch impossible maang raha hai.
Koi refracted ray nahi → total internal reflection : 100% light wapas water mein bounce kar jaati hai.
Yeh step kyun? Energy ko kahin jaana hai; refraction forbidden hone par, reflection sab le leta hai.
Verify karo: sin θ 2 = 1.152 > 1 60° > 48.8°
Worked example Example 8 — Cell 8: real-world word problem (optical fibre)Ek optical fibre core ka n 1 = 1.50 n 2 = 1.45 θ c
Forecast: dono indices bahut close hain. Kya trapping angle 90° 0°
Direction check: 1.50 > 1.45 Yeh step kyun? Fibres deliberately core mein denser glass rakhte hain taaki light trap ho sake.
sin θ c = 1.50 1.45 = 0.9667 Yeh step kyun? Rarer (cladding) over denser (core).θ c = sin − 1 ( 0.9667 ) = 75.2° Yeh step kyun? Sine ko undo karo.Trapped range: wall se θ 1 75.2° 90° Yeh step kyun? θ c upar ke (grazing tak) saare incidence angles totally reflect hote hain.
Verify karo: close indices ek bada θ c 90°
Worked example Example 9 — Cell 9: exam twist (45° prism)Ek right-angled glass prism (n = 1.50 45° periscope prism trick hai.)
Forecast: incidence 45° θ c 45°
Glass–air ke liye θ c sin θ c = 1.50 1 = 0.6667 ⇒ θ c = 41.8° Yeh step kyun? Threshold woh hai jo 45°
Compare karo: 45° > 41.8° = θ c Yeh step kyun? TIR tab hota hai jab incidence critical angle se zyada ho.
Conclusion: haan — light totally internally reflect hoti hai, toh prism bina silvering ke perfect mirror ki tarah kaam karta hai.
Yeh step kyun? Kyunki 45°
Verify karo: 45° − 41.8° = 3.2° barely . Agar glass low-index hota (n < 1/ sin 45° = 1.414 θ c 45° n = 1.50 > 1.414
Recall Rapid-fire self-test (answers cover karo)Glass n = 1.5 θ c sin − 1 ( 1/1.5 ) ≈ 41.8° n = 2.42 θ c sin − 1 ( 1/2.42 ) ≈ 24.4° θ c = 45° n 1/ sin 45° = 1.414 n 1 = n 2 θ c 90° θ c sin θ c = 1.5 > 1 1.50 1.45 θ c sin − 1 ( 1.45/1.50 ) ≈ 75.2°
Mnemonic Sab kuch ek sentence mein"Pehle direction, phir formula, phir threshold se compare." Dense→rare confirm karo, sin θ c = n rarer / n denser θ c
