Total internal reflection — critical angle derivation

2.5.5 · Physics › Optics

Hum kya explain kar rahe hain

First principles se Derivation

KAHAN se shuru karein — Snell's law. Yeh ek hi physics input hai jo humein chahiye: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$

YEH law kyun? Light denser medium mein slow ho jaati hai ( $v = c / n$ ). Boundary par wavefronts continuous rehne chahiye, isliye ray ko bend karna padta hai. Snell's law exactly usi continuity ki bookkeeping hai (yeh Fermat's least-time principle / phase matching se aati hai).

HOW — critical angle kaise find karein — step by step.

Step 1. Light ko medium 1 (denser, $n_{1}$ ) se medium 2 (rarer, $n_{2}$ ) mein bhejo, toh $n_{1} > n_{2}$ . Yeh step kyun? TIR rarer→denser jaate waqt possible nahi hai, isliye hume dense→rare setup karna padta hai.

Step 2. $θ_{1}$ badhao. Kyunki $n_{1} > n_{2}$ hai, Snell deta hai $sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1} > sin θ_{1}$ , toh $θ_{2} > θ_{1}$ : ray normal se door bend karti hai. Yeh step kyun? Yeh dikhata hai ki $θ_{2}$ , $θ_{1}$ se aage bhaagta hai aur pehle $90°$ hit karega.

Step 3. Critical angle ko woh $θ_{1}$ define karo jis par refracted ray abhi surface ke saath graze kare: $θ_{2} = 90°$ . Yeh step kyun? $90°$ aakhri possible refraction angle hai. Iske baad koi real refracted ray exist nahi karti.

Step 4. $θ_{1} = θ_{c}$ aur $θ_{2} = 90°$ ko Snell mein substitute karo: $n_{1} sin θ_{c} = n_{2} sin 90° = n_{2} \cdot 1$

Step 5. Solve karo:

Step 6 (punchline). Jab $θ_{1} > θ_{c}$ ho, toh Snell demand karega $sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1} > 1$ — impossible. Toh koi refracted ray exist nahi karti, aur 100% energy reflect ho jaati hai: total internal reflection.

Figure — Total internal reflection — critical angle derivation

Worked examples

Common mistakes

Flashcards

TIR ke liye kaun si do conditions zaroori hain?

Light denser→rarer (

n_{1} > n_{2}

) jaaye AUR incidence angle critical angle se zyada ho (

θ_{1} > θ_{c}

Snell's law se critical angle condition derive karo.

n_{1} sin θ_{1} = n_{2} sin θ_{2}

mein

θ_{2} = 90°

set karo →

n_{1} sin θ_{c} = n_{2}

→

sin θ_{c} = n_{2} / n_{1}

Critical angle ka formula?

sin θ_{c} = n_{2} / n_{1}

(rarer index over denser index).

Glass (

n = 1.5

) ka air mein

θ_{c}

kya hai?

sin^{- 1} (1/1.5) \approx 41.8°

θ_{c}

ke baad refraction impossible kyun ho jaata hai?

Snell require karta

sin θ_{2} > 1

, jiska koi real solution nahi, toh koi refracted ray exist nahi karti aur saari light reflect ho jaati hai.

Zyada refractive index bada critical angle deta hai ya chhota?

Chhota — isliye diamond (

θ_{c} \approx 24°

) light trap karta hai aur sparkle karta hai.

Measured critical angle (air ke against) se

n

kaise nikalte hain?

n = 1/ sin θ_{c}

Recall Feynman: 12 saal ke bacche ko samjhao

Socho tum paani ke andar ho aur upar dekh rahe ho. Light hawa mein nikal sakti hai, lekin jitna tirchhaa dekho, utna zyada nikalne wali light bend hoti hai. Ek khaas tirchhepan par (critical angle), light aur nahi nikal sakti — woh paani ki surface ke saath flat ho jaati hai. Thoda aur tirchhaa karo aur paani ek perfect mirror ki tarah kaam karta hai: light seedha neeche bounce ho jaati hai. Yahi mirror-trick total internal reflection hai, aur woh khaas tirchhapan jahan se yeh shuru hota hai woh critical angle hai.

Connections

Snell's Law — woh parent law jisse TIR derive hoti hai.
Refraction of Light — TIR refraction ka limiting case hai.
Refractive Index — $θ_{c}$ ki size control karta hai.
Optical Fibres — TIR light ko fibre core ke andar trapped rakhta hai.
Mirage and Atmospheric Refraction — warm-air layers se gradual TIR.
Brewster's Angle — contrast: polarization angle, refraction-ceasing angle nahi.
Prisms and Total Internal Reflection — periscopes/binoculars mein glass mein $45° > θ_{c}$ use hota hai.