Question bank — Snell's law — derivation from Fermat's principle

False. Ek single uniform medium mein leta hai, lekin across media light time minimize karta hai, jo ki OPL $=\sum n_i{d}_i$ ko minimize karna hai, raw length ko nahi — isliye path boundary par kink karta hai.

False. Ye reflection ka law hai. Refraction deta hai $n_1\sin {\theta }_1=n_2\sin {\theta }_2$; angles generally unequal hote hain jab tak $n_1=n_2$ na ho.

Zyada tar sahi hai lekin imprecise hai: ye stationary time ki demand karta hai ($dT/dx=0$). Flat boundary par refraction ke liye ye stationary point ek minimum hai, lekin exact statement hai "stationary," jo maxima aur saddle paths bhi cover karta hai (jaise mirrors). Dekho Fermat's Principle.

False. Denser matlab bada $n$; kyunki $n\sin \theta$ conserved hai, bada $n_2$ force karta hai chhota $\sin {\theta }_2$ — ray normal ki taraf bend karti hai.

False. Frequency source se fixed hoti hai aur boundary ke across continuous rehti hai; jo cheez change hoti hai woh speed aur isliye wavelength hai ($\lambda =v/f$, aur $v=c/n$).

True. Kyunki $n=c/v$, law ban jaata hai $\frac{\sin {\theta }_1}{v_1}=\frac{\sin {\theta }_2}{v_2}$; $c$ cancel ho jaata hai. Dekho Refractive Index.

True. ${\theta }_1=0$ par, $\sin {\theta }_1=0$, toh $\sin {\theta }_2=0$ aur ${\theta }_2=0$ kisi bhi indices ke liye — ray seedhi nikal jaati hai (uski speed phir bhi change hoti hai, sirf direction nahi).

False. TIR ke liye $n_1>n_2$ chahiye (dense to rare). Air→glass jaana rare→dense hai, toh $\sin {\theta }_2=(n_1/n_2)\sin {\theta }_1<\sin {\theta }_1$ ka hamesha ek real solution hota hai — TIR nahi. Dekho Total Internal Reflection.

"Zyada" word galat apply ho raha hai. Conserved quantity $n\sin \theta$ hai; bada $n$ ke saath product ko constant rakhne ke liye chhota $\sin \theta$ chahiye. Denser ⇒ chhota angle.

Snell's law normal se angle use karta hai, surface se nahi. Asli incidence angle hai ${\theta }_1=90^{\circ }-60^{\circ }=30^{\circ }$.

$\sqrt{a^2+x^2}$ ka $x$ ke respect mein chain-rule derivative $\frac{x}{\sqrt{a^2+x^2}}$ hai, na ki $\frac{a}{\sqrt{a^2+x^2}}$ — tum free variable $x$ ke respect mein differentiate karte ho, aur $a$ ek fixed constant hai. Dekho Calculus — minimization and stationary points.

Premise (slow medium mein kam path spend karo) sahi hai, lekin ye bend hone se achieve hota hai, seedha jaane se nahi. Ek straight line time minimize nahi karegi; woh kink jo slow-length ko fast-length se trade karta hai, woh karta hai.

Nahi — ye inverted hai. Dense→rare jaana ${\theta }_2=90^{\circ }$ ke saath: $n_1\sin {\theta }_c=n_2\sin 90^{\circ }$, toh $\sin {\theta }_c=n_2/n_1$ (upar chhota index). Dekho Critical Angle.

Nahi. $dT/dx=0$ solve karne se stationary point milta hai; flat refracting boundary ke liye second derivative positive hai, toh ye minimum time hai. Ye case ke liye "maximum" wording galat hai.

Kyunki time matter karta hai: $t=d/v=nd/c$. Speed $n$ ke factor se slow hoti hai, toh dense medium mein ek distance "cost" karta hai $n$ times zyada time — ise $n\cdot d$ ke roop mein encode karne se hum ek length-like quantity minimize karke time minimize kar sakte hain.

Endpoints $A$ aur $B$ fixed hain, toh unki heights $a,b$ aur separation $w$ locked hain. Light ka ek hi choice hai — boundary par kahan cross karna hai — har path us ek number $x$ se parameterize hota hai.

Derivative $\frac{x}{\sqrt{a^2+x^2}}$ exactly incidence right triangle ka (horizontal run)/(hypotenuse) hai, aur us triangle mein run woh side hai jo normal-se-angle ke opposite hai — toh ye ratio definition se $\sin \theta$ hai.

Snell's law kehta hai $n_1\sin {\theta }_1=n_2\sin {\theta }_2$: ye combination interface ke dono taraf same value rakhta hai. Physically ye wave direction ka horizontal component hai (tangential wave-vector), jise flat boundary change nahi kar sakti.

Lifeguard sand par fast hai, paani mein slow — jaise light rare phir dense medium mein hoti hai. Total time minimize karne ke liye woh fast sand par zyada dauda aur slow swim ko chhota kiya, ek bent path produce kiya boundary par kink ke saath, exactly jaise light boundary par bend karti hai.

Snell's law purely geometric (kinematic) statement hai transmitted ray ki directions ke baare mein. Reflection aur refraction ke beech energy split hone ki amount ek alag dynamical question hai jo Fresnel equations se answer hoti hai, Fermat's least-time condition se nahi.

Dono same physical fact encode karte hain — ki wavefronts continuous rehte hain aur light medium mein $n$ se slow hoti hai. Huygens boundary par wavefront geometry match karta hai, Fermat time minimize karta hai; ye ek hi truth ke do views hain. Dekho Huygens' Principle and Snell's Law.

Tab $\sin {\theta }_1=\sin {\theta }_2$ toh ${\theta }_2={\theta }_1$ — koi bending nahi. Effectively koi boundary nahi hai, aur path ek straight line hai, uniform medium mein least-time ke consistent hai.

Isko $\sin {\theta }_2=(n_1/n_2)\sin {\theta }_1>1$ chahiye hoga, jiska koi real angle nahi hai. Koi transmitted ray nahi hai: saari light internally reflect hoti hai — total internal reflection.

${\theta }_2=90^{\circ }$ tak: refracted ray surface ke along skim karti hai (grazing), woh limiting orientation jiske baad transmission bilkul khatam ho jaati hai.

Ye grazing incidence hai; rare→dense jaate waqt ye sabse bada possible ${\theta }_2$ deta hai, aur symmetry se woh limiting ${\theta }_2$ reverse (dense→rare) trip ke critical angle ke barabar hota hai.

Tab phir se effectively $n_1>n_2$ hai: ray normal se door bend karti hai, aur vacuum→medium direction ke liye TIR ka critical angle exist kar sakta hai — "denser" wala role usi side ko milta hai jis par bada $n$ hota hai, rozana ke materials ke baare mein intuition chahe kuch bhi bole.

Haan, locally. Har point par tum local normal use karte ho curved surface ka; Snell's law ek point condition hai. Overall path sirf har crossing point par alag normal direction use karta hai.