Total internal reflection — critical angle derivation: Question bank

2.5.5 · D5 · Physics › Optics › Total internal reflection — critical angle derivation

Shuru karne se pehle, ek one-line reminder notation ka jo poore bank mein use hoti hai taaki kuch bhi unexplained na rahe:

$n_{1}$ = us medium ka index jisme light start karti hai (incident side).
$n_{2}$ = us medium ka index jisme light jaane ki koshish kar rahi hai.
$θ_{1}$ = angle of incidence, $θ_{2}$ = angle of refraction, dono normal se measure kiye jaate hain (surface ke perpendicular line se).
$θ_{c}$ = critical angle, woh special $θ_{1}$ jis par $θ_{2} = 90°$ hota hai.

Ek picture ka reminder ki kyun $sin θ_{c} = n_{2} / n_{1}$

Traps tackle karne se pehle, boundary khud dekho — kyunki neeche ke almost har trap mein actually is ek picture ki galat reading hai.

Figure — Total internal reflection — critical angle derivation

Figure ko left se right padhо. Sabse left panel mein incidence angle $θ_{1}$ chhota hai: light rarer medium mein cross karti hai aur normal se door bend hoti hai, isliye $θ_{2} > θ_{1}$ . Jaise-jaise tum $θ_{1}$ upar slide karte ho (middle panel), refracted ray surface ki taraf aur aur tip karti jaati hai. Right panel mein $θ_{1}$ exactly us value par pahunch gaya hai jahan refracted ray boundary ke saath flat lie karti hai — yahi $θ_{2} = 90°$ hai, aur yahi incidence angle $θ_{c}$ hai.

Formula kahan se aata hai — ek wavefront argument, koi memorised rule nahi. Snell's law $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ sirf yeh statement hai ki wavefronts boundary ke along continuous rehti hain (dekho Snell's Law). Grazing condition $θ_{2} = 90°$ plug karo, jiske liye $sin 90° = 1$ : $n_{1} sin θ_{c} = n_{2} \cdot 1 ⟹ sin θ_{c} = \frac{n _{2}}{n _{1}}$ Toh formula bas itna hi hai ki "refracted ray ke paas bend karne ki jagah khatam ho gayi." Kyunki hamein $n_{1} > n_{2}$ chahiye (dense→rare), right side hamesha 1 se kam hoti hai, aur yahi reason hai ki ek valid angle exist karta hai. Yeh panel dimaag mein rakho — "flat refracted ray" picture neeche har item ka anchor hai.

True or false — justify karo

TIR tab ho sakta hai jab light rarer medium se denser medium mein travel kare.

False —

n_{1} < n_{2}

ke saath

sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1} < 1

har

θ_{1}

ke liye hota hai, isliye ek real refracted ray hamesha exist karti hai aur refraction kabhi cease nahi hoti.

Exactly

θ_{1} = θ_{c}

par, saari light already totally internally reflect ho chuki hoti hai.

False — exactly

θ_{c}

par refracted ray abhi bhi exist karti hai lekin surface ke saath

θ_{2} = 90°

par graze karti hai; TIR (100% reflection) sirf tab shuru hota hai jab

θ_{1}

θ_{c}

se strictly greater ho.

Denser medium ka bada refractive index ek bada critical angle deta hai.

False —

sin θ_{c} = n_{2} / n_{1}

, isliye

n_{1}

badhane se fraction chhota hota hai aur isliye

θ_{c}

bhi; yahi reason hai ki diamond (

n = 2.42

θ_{c} \approx 24°

) glass se kahin zyada aasaani se light trap karta hai.

Critical angle sirf denser medium ke index par depend karta hai.

False — yeh ratio

n_{2} / n_{1}

par depend karta hai; same glass ka air ke against ek

θ_{c}

hota hai aur water ke against bilkul alag.

Critical angle se neeche, boundary par koi bhi light reflect nahi hoti.

False — kuch light hamesha reflect hoti hai (partial reflection har boundary par hota hai);

θ_{c}

se neeche zyaadatar energy transmit hoti hai, aur reflected fraction sirf badhta hai jaise-jaise

θ_{1}

θ_{c}

ke paas pahunchta hai.

Agar do media ke refractive indices equal hain, toh koi critical angle nahi hoga.

True —

n_{1} = n_{2}

ke saath,

sin θ_{c} = 1

isliye

θ_{c} = 90°

; light kabhi bend nahi hoti aur kabhi trap nahi ho sakti, isliye TIR effectively occur nahi kar sakta.

TIR sirf kuch light reflect karta hai; kuch hamesha leak hoti rehti hai.

False —

θ_{c}

se aage 100% energy reflect hoti hai (ideally); koi ordinary refracted ray time-averaged energy carry nahi karta, aur yahi "total" ka matlab hai.

Critical angle se aage electromagnetic field boundary ke dusri taraf exactly zero hota hai.

False — surface ke baad ek non-propagating evanescent field exist karta hai, jo wavelength ke fraction par exponentially decay karta hai; yeh boundary ke across koi net energy carry nahi karta, isliye reflection abhi bhi total hai (edge-case section dekho).

Total internal reflection aur metal mirror se reflection ek hi phenomenon hai.

False — ek metal mirror har bounce par kuch percent absorb karta hai, jabki TIR essentially lossless hai; yahi reason hai ki fibre optics aur prisms TIR use karte hain na ki metallic coatings.

Error dhundho

sin θ_{c} = n_{1} / n_{2}

, kyunki incident medium pehle likha jaata hai."

Ratio flip hai.

n_{1} sin θ_{c} = n_{2} sin 90°

mein

θ_{2} = 90°

set karne par

sin θ_{c} = n_{2} / n_{1}

milta hai — rarer over denser, jo correctly 1 se kam hai.

"Air se glass mein jaane wali light TIR undergo kar sakti hai agar angle kaafi steep ho."

Direction ulta hai. Air→glass rarer→denser hai, isliye us side koi critical angle exist nahi karta; TIR ke liye light ko denser medium ke andar hona chahiye boundary hit karte waqt.

"Critical angle woh hai jahan reflected ray exactly

90°

incident ray se hoti hai."

Yeh Brewster's condition describe karta hai —

θ_{B}

par reflected aur refracted rays perpendicular hoti hain (

θ_{B} + θ_{2} = 90°

), jo reflection ko polarize karta hai. Critical angle alag hai: isse refracted ray ke

θ_{2} = 90°

par grazing se define kiya jaata hai, yeh refraction cease hone ka statement hai, polarization ka nahi (dekho Brewster's Angle).

"Kyunki

sin θ_{c} = n_{2} / n_{1}

, ek fibre core jiska

n = 1.0

hai air ke against light perfectly trap karega."

n_{1} = 1.0 = n_{2}

ke saath fraction 1 hai aur

θ_{c} = 90°

; kuch bhi trap nahi hota. Ek working fibre ko ek core chahiye jo uski cladding se denser ho (dekho Optical Fibres).

θ_{c} = sin^{- 1} (1.4/1.2)

light ke liye jo

n_{1} = 1.2

n_{2} = 1.4

mein jaati hai."

Argument

1.4/1.2 > 1

ka koi valid arcsine nahi hai — yeh sign hai ki yeh direction (rarer→denser) koi critical angle allow hi nahi karta, isliye setup invalid hai.

θ_{c}

ke aage bhi refraction hoti rehti hai, bas

90°

se zyada bend hoti hai."

Transmitted ray ke liye "

90°

se zyada" bend jaisi koi cheez nahi hoti — woh use incident side par wapas rakh deta. Snell demand karta hai

sin θ_{2} > 1

, jo impossible hai, isliye refraction genuinely ruk jaati hai (dekho Refraction of Light).

Why questions

Kyun refracted ray

90°

par pahunch jaati hai pehle incident ray se, jaise-jaise

θ_{1}

badhta hai?

Kyunki dense→rare mein

sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1}

hota hai jisme

n_{1} / n_{2} > 1

, isliye

θ_{2} > θ_{1}

hamesha;

θ_{2}

aage bhaagta hai aur

90°

ki ceiling ko hit karta hai jabki

θ_{1}

abhi bhi chhota hota hai.

Kyun

θ_{2} = 90°

θ_{c}

ke liye defining condition choose kiya jaata hai?

Kyunki

90°

sabse bada possible refraction angle hai — ray surface ke saath flat lie karti hai (figure ka rightmost panel). Kisi bhi bade

θ_{2}

ki demand geometrically impossible hai, isliye yahi exact tipping point hai jahan refraction exist nahi kar sakti.

Kyun diamonds same cut ke glass gems se zyada sparkle karte hain?

Diamond ka high index (

2.42

) ek chhota

θ_{c} \approx 24°

deta hai, isliye zyaadatar internal rays ise exceed karti hain aur escape se pehle andar kai baar bounce karti hain, light ko concentrate aur delay karti hain.

Kyun

sin θ_{c}

ek valid critical angle ke liye hamesha 1 se kam aana chahiye?

Kyunki

sin θ_{c} = n_{2} / n_{1}

aur TIR ke liye

n_{1} > n_{2}

chahiye (denser to rarer);

\geq 1

ki value signal karti hai ki ya toh ratio flip ho gaya ya galat direction choose kiya (dekho Refractive Index).

Kyun binoculars aur periscopes ke andar silvered mirrors ki jagah TIR prefer kiya jaata hai?

TIR essentially 100% light reflect karta hai bina kisi metallic absorption ke, isliye images bright rehti hain; ek

45°

prism angle comfortably glass ke

θ_{c} \approx 42°

se exceed karta hai (dekho Prisms and Total Internal Reflection).

Kyun garmi ke din ek road door se geeli, reflective surface jaisi dikhti hai?

Garmi ki road ke paas ki hawa ek continuous index gradient banati hai (surface par sabse rarer); har thin layer sky-light ko thoda aur refract karti hai jab tak ek ray wapas upar bend nahi ho jaati. Yeh ek single sharp interface nahi hai, isliye strictly yeh gradient refraction hai, textbook TIR nahi — though trapping effect similar lagta hai (dekho Mirage and Atmospheric Refraction).

Edge cases

θ_{c}

kya hoga jab do media identical hain (

n_{1} = n_{2}

sin θ_{c} = 1

, isliye

θ_{c} = 90°

; light seedhi nikal jaati hai bina kisi bending ke aur kabhi trap nahi ho sakti — TIR impossible hai.

Jaise

n_{2} \to n_{1}

neeche se, critical angle ka kya hoga?

n_{2} / n_{1} \to 1

, isliye

θ_{c} \to 90°

; "trapping window"

(θ_{c}, 90°)

kuch nahi tak sir jaati hai, matlab jaise indices ek doosre ke paas aate hain TIR achieve karna mushkil hota jaata hai.

Jaise

n_{1}

n_{2}

ki tulna mein bahut bada hota jaata hai,

θ_{c}

kya approach karta hai?

n_{2} / n_{1} \to 0

, isliye

θ_{c} \to 0°

; incidence ka almost koi bhi angle light ko trap kar leta hai, jo lossless light-piping ke liye ideal limit hai.

Exactly

θ_{1} = θ_{c}

par kya hota hai — boundary mirror hai ya window?

Na poora ek, na poora doosra — refracted ray exist karti hai lekin surface ke saath skim karti hai (

θ_{2} = 90°

), ek knife-edge case; true mirror behaviour sirf is angle se thoda aage switch on hota hai.

Agar light dense→rare setup mein

θ_{1} = 0°

(seedha) par boundary hit kare, toh kya TIR hoga?

Nahi — normally-incident ray seedhi nikal jaati hai

θ_{2} = 0°

ke saath; kyunki

0° < θ_{c}

, refraction freely hoti hai aur koi trapping nahi hoti.

Kya TIR ho sakta hai agar incident medium vacuum ho?

Nahi — vacuum ka sabse kam possible index hai (

n = 1

), isliye koi rarer medium nahi hai jisme light jaaye; TIR ke liye destination se kisi denser cheez mein shuru karna zaroori hai.

θ_{c}

ke aage, kya koi bhi energy rarer medium mein cross kar sakti hai?

Flat boundary akele se nahi — wahan field evanescent hai, penetration depth

\sim λ / (2 π)

ke saath exponentially decay karti hai aur across koi net energy carry nahi karti. Lekin agar ek doosra dense medium us decay distance ke andar laaya jaaye, toh light gap ke across "tunnel" kar sakti hai (frustrated TIR) — quantum tunnelling ka wave equivalent.

θ_{1} > θ_{c}

par evanescent field surface ke aage kitni deep tak jaata hai?

Sirf wavelength ke fraction tak — uski amplitude roughly

λ / (2 π)

boundary ke andar

1/ e

tak fall ho jaati hai, yahi reason hai ki "total" reflection genuinely total hai jab tak surface ke bilkul saath koi aur medium nahi rakh diya jaata.

Connections

Snell's Law — woh ek law jis par upar ke har trap ka trace wapas jaata hai.
Refraction of Light — TIR refraction ki failure limit hai.
Refractive Index — ratio ke through $θ_{c}$ ki size set karta hai.
Optical Fibres — "identical index" edge case explain karta hai kyun cladding rarer honi chahiye.
Brewster's Angle — Spot-the-error mein orthogonal-rays condition clear ki gayi.
Mirage and Atmospheric Refraction — gradient-refraction edge case.
Prisms and Total Internal Reflection — lossless-mirror "why" question.

Ek picture ka reminder ki kyun sinθc​=n2​/n1​

True or false — justify karo

Error dhundho

Why questions

Edge cases

Connections

Ek picture ka reminder ki kyun $sin θ_{c} = n_{2} / n_{1}$