2.5.5 · D1Total internal reflection — critical angle derivation
Light jo ek "slow" medium se "fast" medium mein jaati hai, woh surface ke normal se door bend karti hai, aur ek hard limit hoti hai ki woh kitna bend kar sakti hai — jab woh flat letne se bhi zyada bend karna chahti hai, toh woh simply kar hi nahi sakti , isliye sab kuch andar waapis reflect ho jaata hai. Woh single tipping point hi critical angle hai, aur parent page par jo bhi hai, woh bas usi limit ki bookkeeping hai.
Isse pehle ki tum critical angle derive karo, tumhe har ek letter khud se pata hona chahiye aur parent note jo picture dikhata hai woh clearly visualize honi chahiye. Yeh page har ek cheez ko zero se build karta hai, usi order mein jisme woh ek doosre par depend karti hain.
Poori kahani ek flat surface par hoti hai jahaan do see-through materials milti hain — jaise glass ke upar air, ya paani ke upar air.
Ek medium koi bhi transparent cheez hai jisme se light travel karti hai: glass, water, air, diamond. Hum har medium ko ek number label dete hain. Medium 1 woh jagah hai jahaan se light shuru hoti hai; Medium 2 woh jagah hai jahaan woh jaane ki koshish karti hai.
Figure dekho: horizontal chalk line boundary hai (ise interface bhi kehte hain). Medium 1 neeche bhara hai, Medium 2 upar. Light ki ek single ray upar climb karti hai aur ek point par boundary se takraati hai.
flat boundary?
Ek flat surface par har jagah ek hi "seedha-upar" direction hoti hai. Isse hum angles clearly measure kar sakte hain. Curved surfaces (jaise lenses) bas bahut saari tiny flat pieces hain — pehle flat case master karo.
Tum "ray ka angle" nahi keh sakte bina yeh bataye ki kisse measure kiya . Optics mein hum angles surface se kabhi nahi measure karte. Hum ek special line se measure karte hain jise normal kehte hain.
Normal woh line hai jo boundary par perpendicular (bilkul 90°
Figure mein normal vertical dashed chalk line hai. Woh little square corner symbol notice karo — woh normal aur surface ke beech 90°
Common mistake Normal ki jagah surface se angle measure karnaKyun natural lagta hai: surface hi obvious "flat cheez" hai jisse compare karo.
Fix: hamesha normal se measure karo. Tab "0° 0°
Ab jab hamare paas ek reference line hai, hum angles ko name de sakte hain.
Definition Do rays aur do angles
Incident ray Medium 1 mein aane wali light hai.Refracted ray woh light hai jo Medium 2 mein cross karti hai aur aage jaati hai (bent hokar).θ 1 angle of incidence : incident ray aur normal ke beech ka angle.θ 2 angle of refraction : refracted ray aur normal ke beech ka angle.
θ θ x θ 1 = 30° "incidence angle thirty degrees hai" hai. Isse zyada mysterious kuch nahi.
Yeh figure dhyaan se dekho — ek board par refraction ki poori geometry hai:
Incident ray (chalk blue) neeche se upar aati hai, surface se takraati hai.θ 1 Refracted ray (chalk pink) Medium 2 mein jaari rehti hai, bent hokar.θ 2 uss ray aur normal ke beech hai.
Recall Picture par quick self-checkAgar ek ray normal ke along seedhi upar jaaye, toh θ 1 θ 2 0° normal par hi hai, isliye ray aur normal ke beech koi gap nahi, aur woh bina kisi bending ke seedhi nikal jaati hai.
Light bend kyun hoti hai? Kyunki iski speed change hoti hai. Har medium ko ek number milta hai jo batata hai ki woh light ko kitna slow karta hai.
Definition Refractive indexKisi medium ka refractive index n slower travel karti hai:
n = v c c v
n n n n = 1.5 n = 2.42 n n ≈ 1 1 v c n hamesha ≥ 1 hota hai.
Do labels jo hum baar baar use karenge:
Definition Denser vs rarer
Denser medium = zyada n Rarer medium = kam n
"Optically denser" index n weight ke baare mein nahi — air ki warm layer optically rarer ho sakti hai cold layer se.
n Refractive Index dekho; yahan hume bas "bada n
Snell's Law likhne se pehle humein woh ek piece of mathematics se milna hoga jis par woh depend karti hai: sine of an angle. Tumhe koi bhi aisa symbol use nahi karna chahiye jise tum picture nahi kar sakte, isliye hum sin θ pehle isse kisi law mein dekhne se.
Definition Sine of an angleEk right triangle (ek 90° θ sin θ = length of the longest side (hypotenuse) length of the side opposite θ .
Figure mein ek right triangle hai jisme angle θ θ opposite hai; lambi slanted side hypotenuse hai. Jaise θ 1
Intuition Kyun sine 1 par cap hoti hai — aur kyun woh cap IS poora topicOpposite side kabhi bhi hypotenuse se lambi nahi ho sakti (hypotenuse hamesha right triangle ki sabse lambi side hoti hai). Isliye sin θ kabhi 1 se zyada nahi ho sakta . Iski sabse badi possible value exactly 1 θ = 90°
Yahi single fact total internal reflection ka engine hai (woh phenomenon jahaan light denser medium ke andar poori tarah bounce back karti hai — Section 8 mein fully explain kiya aur wahan naam diya). Snell's law, jo hum aage milenge, kahegi sin θ 2 = n 2 n 1 sin θ 1 n 2 n 1 > 1 1 se badi value demand kar sakta hai — jo geometrically impossible hai. Jab maths impossible cheez maangta hai, refraction ruk jaata hai aur light reflect ho jaati hai. Parent page par jo kuch bhi hai woh isi sentence ko spell out kar raha hai.
Recall Sine sanity valuessin 0° = ? 0 sin 90° = ? 1 sin 30° = ? 0.5
Ab jab hum sin θ n 1 , n 2 θ 1 , θ 2
sine kyun aata hai?
Humein ek aisa tool chahiye tha jo ek angle (ray kitni slanted hai) ko ek length ratio mein convert kare (wavefront boundary cross karte waqt sideways kitna stretch hota hai). Woh tool sine hai, yahi wajah hai ki humne ise Section 5 mein pehle build kiya.
Derivation ke liye Snell's Law note padho; is page par hum ise apna single given input maante hain, exactly jaise parent karta hai.
Snell's law humein ek sine ke liye ek number deta hai; actual angle report karne ke liye humein sine ulta chalana hoga.
Definition Inverse sine (arcsine)sin − 1 ( x ) "kaunse angle ka sine x Yeh sine ko undo karta hai.
sin ( 30° ) = 0.5 sin − 1 ( 0.5 ) = 30° sin − 1 ratio leta hai aur ek angle bahar deta hai — sin
Intuition Woh kaunsa angle choose karta hai? — principal branchEk catch hai: bahut saare angles ka sine same hota hai (jaise sin 30° = sin 150° = 0.5 sin − 1 ek choose karna hota hai. Universal convention se woh θ ∈ [ − 90° , + 90° ] principal branch kehte hain. Yeh hamare liye perfect hai: physical incidence aur refraction angles 0° 90°
sin − 1 Kyun sahi lagta hai: algebra mein x − 1 1/ x Fix: yahan − 1 inverse function mark karta hai, reciprocal nahi. sin − 1 ( 0.5 ) = 30° 1/ sin ( 0.5° ) "arcsine of..." padho.
Intuition Hume is topic meinsin − 1 number dega. Lekin hum actual angle degrees mein chahte hain ("41.8° sin − 1
Upar ki har cheez ab humein payoff symbol precisely state karne deti hai — including exact conditions jinke under woh exist bhi karta hai.
Definition Critical angleθ c Sirf tab jab light denser medium se rarer medium mein jaaye — yaani, jab n 1 > n 2 θ c θ 1 surface ke along flat lie karti hai, yaani θ 2 = 90° aakhri incidence angle hai jiske liye koi refracted ray exist karti hai.
Agar n 1 ≤ n 2 har angle ke liye ek refracted ray exist karti hai aur koi critical angle nahi hota.
Yahi wajah hai ki domain condition matter karti hai: formula sin θ c = n 2 / n 1 ≤ 1 n 2 < n 1 n 1 < n 2 sin θ c > 1
Yeh samajhne ke baad, parent ki derivation clearly padhti hai: θ 2 = 90° sin 90° = 1 sin θ c = n 2 / n 1 sin − 1
Two media meet at a flat boundary
Normal: perpendicular reference line
Angles theta1 and theta2 measured from normal
Refractive index n = c / v (slowness)
Denser vs rarer media (n1 greater than n2)
Sine = opposite over hypotenuse, capped at 1
Snells law n1 sin theta1 = n2 sin theta2
Critical angle theta c where theta2 = 90 deg, needs n1 greater than n2
Inverse sine (principal branch) turns a ratio into an angle
Total internal reflection: light fully reflected inside
Right side cover karo; kya tum derivation page par jaane se pehle har ek ka jawaab de sakte ho?
"Normal" kya hai aur hum angles surface se nahi balki usse kyun measure karte hain? Boundary par hit-point par perpendicular line; usse measure karne par seedha-through ray 0°
θ 1 θ 2 Angle of incidence aur angle of refraction, dono apni ray aur normal ke beech measure kiye.
Refractive index n n = c / v ≥ 1
Denser ya rarer medium mein se kiska n Denser medium ka (light wahan slower hoti hai).
Triangle ke terms mein sin θ Opposite side over hypotenuse; maximum 1 θ = 90°
Woh maximum-of-1 total internal reflection ki key kyun hai? Denser→rarer jaate waqt, Snell sin θ 2 > 1
Snell's law state karo aur har symbol batao. n 1 sin θ 1 = n 2 sin θ 2 n 1 , n 2 θ 1 , θ 2
sin − 1 ( x ) Woh angle return karta hai jiska sine x [ − 90° , 90° ]
Critical angle θ c Woh incidence angle jis par refracted ray surface ko graze karti hai (θ 2 = 90° sirf tab exist karta hai jab n 1 > n 2
Snell's Law — woh single physics law jisme yeh foundations feed hoti hain.Refractive Index — number n Refraction of Light — woh bending jo yeh angles describe karti hain.Total internal reflection — critical angle derivation — woh parent topic jiske liye yeh page tumhe prepare karta hai. 
