Snell's law — derivation from Fermat's principle

2.5.4 · Physics › Optics

WHAT is Fermat's Principle?

Optical path length (OPL): $OPL = n \cdot d$ , jahan $n$ refractive index hai aur $d$ geometric distance hai.
WHY $n$ appear karta hai: kisi medium mein light ki speed $v = c / n$ hoti hai. Distance $d$ cross karne ka time $t = d / v = n d / c$ hota hai. Isliye time minimize karna $\Leftrightarrow$ $n d$ = OPL minimize karna (kyunki $c$ constant hai).

Hum light ko point $A$ se medium 1 (index $n_{1}$ ) se point $B$ medium 2 (index $n_{2}$ ) mein bhejte hain. Boundary ek flat horizontal line hai. Maano ray boundary ko horizontal position $x$ pe cross karti hai.

Figure — Snell's law — derivation from Fermat's principle

Setup (ye variables kyun?): Hum $x$ ko crossing point lete hain — yahi ek free choice hai jo light karti hai. Baaki sab kuch (heights $a$ , $b$ , total width $w$ ) fix hai, depending on kahan $A$ aur $B$ hain.

Medium 1 mein path length: $ℓ_{1} = a^{2} + x^{2}$
Medium 2 mein path length: $ℓ_{2} = b^{2} + (w - x)^{2}$

Total time (Why $n / c$ ? kyunki $t = n d / c$ ): $T (x) = \frac{n _{1}}{c} a^{2} + x^{2} + \frac{n _{2}}{c} b^{2} + (w - x)^{2}$

Minimize: light woh $x$ choose karti hai jisse $T$ stationary ho, isliye $\frac{d T}{d x} = 0$ set karo.

Ye step kyun? Stationary = derivative zero ho jaata hai; yeh "least time" ka mathematical translation hai.

$\frac{d T}{d x} = \frac{n _{1}}{c} \cdot \frac{x}{a ^{2} + x ^{2}} + \frac{n _{2}}{c} \cdot \frac{- ( w - x )}{b ^{2} + ( w - x ) ^{2}} = 0$

Ye step kyun? Har square root chain rule se differentiate hoti hai: $\frac{d}{d x} a^{2} + x^{2} = \frac{x}{a ^{2} + x ^{2}}$ .

Ab geometry padho: $sin θ_{1} = \frac{x}{a ^{2} + x ^{2}}, sin θ_{2} = \frac{w - x}{b ^{2} + ( w - x ) ^{2}}$

Ye step kyun? $θ_{1}$ woh angle hai jo normal (vertical) se banta hai. Opposite side horizontal run $x$ hai; hypotenuse $ℓ_{1}$ hai. Isliye $sin θ_{1} = x / ℓ_{1}$ — pure right-triangle trig.

Substitute karo, $c$ cancel karo: $\frac{n _{1}}{c} sin θ_{1} - \frac{n _{2}}{c} sin θ_{2} = 0$

Worked Examples

Common Mistakes

Recall Feynman: 12-year-old ko explain karo

Light jaldi mein hoti hai par woh thick cheez jaise water ya glass mein fast nahi ja sakti. Isliye jab use fast air se slow glass mein cross karna hota hai, toh woh clever hoti hai: apni trip ka kam hissa slow glass ke andar aur zyada hissa fast air mein spend karti hai. Isse uska path surface pe bend ho jaata hai, bilkul uss lifeguard ki tarah jo slow paani mein koodne se pehle beach pe zyada door daudta hai. Bending ki exact maatra ek simple rule follow karti hai: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ .

Flashcards

Fermat's principle kya kehta hai?

Light us path pe chalti hai jiske liye travel time (optical path length) stationary, usually minimum ho.

Time minimize karna

n \cdot d

minimize karne ke barabar kyun hai?

Kyunki

t = d / v = n d / c

, aur

c

constant hai, isliye time minimize karna ⇔ optical path length

n d

minimize karna.

Snell's law batao.

n_{1} sin θ_{1} = n_{2} sin θ_{2}

, angles normal se measure hote hain.

Derivation mein single free variable kya hai?

Boundary pe horizontal crossing point

x

;

d T / d x = 0

set karne se Snell's law milta hai.

Geometry se

sin θ_{1}

kaise aata hai?

sin θ_{1} = x / a^{2} + x^{2}

= normal ke saath right triangle mein (horizontal run)/(path length).

Dense medium mein enter karte waqt, light normal ki taraf bend karti hai ya door?

Normal ki taraf (bada

n

⇒ chhota

sin θ

⇒ chhota angle).

Critical angle ki condition kya hai?

Jab

θ_{2} = 9 0^{\circ}

sin θ_{c} = n_{2} / n_{1}

(dense se rare mein jaate waqt).

Speeds ke terms mein Snell's law?

\frac{sin θ _{1}}{v _{1}} = \frac{sin θ _{2}}{v _{2}}

, kyunki

n = c / v

Incidence ka angle kis line se measure hota hai?

Normal se (surface ke perpendicular), surface se nahi.

Connections

Fermat's Principle
Refractive Index
Total Internal Reflection
Critical Angle
Reflection — law from Fermat's principle
Optical Path Length
Huygens' Principle and Snell's Law
Calculus — minimization and stationary points

Snell's law — derivation from Fermat's principle

WHAT is Fermat's Principle?

HOW to derive Snell's law

Worked Examples

Common Mistakes

Flashcards

Connections

Concept Map

Deep Dive