Snell's law — derivation from Fermat's principle: Foundations

2.5.4 · D1 · Physics › Optics › Snell's law — derivation from Fermat's principle

Pehle tum Snell's law ki parent derivation padh sako, usse pehle har symbol jo woh use karta hai pehle kuch aisa mean karna chahiye jo tum dekh sako. Yeh page har ek ko absolute zero se build karta hai, us order mein jis order mein woh ek doosre par depend karte hain.

1. Ek point aur ek straight line (stage)

Sab kuch ek flat drawing par hota hai. Usmein do special points hain:

$A$ — jahan se light shuru hoti hai (fast material mein upar).
$B$ — jahan light khatam hoti hai (slow material mein neeche).

Unke beech ek flat horizontal line hai: boundary (jise interface ya surface bhi kehte hain). Uske upar material 1 hai; neeche material 2 hai.

Humein $A$ , $B$ , aur boundary chahiye kyunki poora sawaal yeh hai: " $A$ se shuru hokar, $B$ par khatam hokar, light line ko kahan cross karni chahiye?"

2. Distance $d$ , aur speed $v$

Picture: do dots ek segment se jude hain jisme length $d$ hai; ek chhota runner ise time $t$ mein cover karta hai.

Humein $v$ isliye chahiye kyunki poori "least time" story is baat ke baare mein hai ki light kuch materials mein fast hoti hai aur kuch mein slow. Alag-alag speeds hi woh reason hai ki woh bend karti hai.

3. Time $t$ , aur rule $t = d / v$

Distance ko speed se kyun divide karte hain? Kyunki agar tum har second $v$ metres cover karte ho, toh $d$ metres cover karne ke liye tumhe bas $d / v$ seconds chahiye.

$t = \frac{d}{v}$

Neeche diya figure dekho. Dashed navy line shortest distance hai ( $A$ se $B$ tak ek straight line). Solid magenta line least time path hai: yeh boundary ko aage cross karti hai taaki light fast top material mein zyada length spend kare aur slow bottom material mein kam. Boundary par kink exactly woh hai jo Snell's law pin down karegi.

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

4. Light ki speed $c$ , aur refractive index $n$ (aur $n_{1}$ , $n_{2}$ )

Glass ya paani ke andar, light $c$ se slower hoti hai. Hum "kitna slower" ek number se measure karte hain:

Kyunki Snell's law do materials compare karti hai, hum index ko ek number ke saath tag karte hain yeh batane ke liye ki hum kaunsa material mean kar rahe hain:

Neeche picture: upar $c$ label ki ek bar (full speed), aur uske neeche ek chhoti bar $v = c / n$ — jitna bada $n$ , utni choti bar. Paani ki bar air ki bar se choti hai; glass ki aur bhi choti hai.

Topic ko $n$ (aur dono $n_{1}$ , $n_{2}$ ) kyun chahiye: yeh awkward speed $v$ ko $c / n$ ki tarah rewrite karne deta hai, toh medium 1 mein time $n_{1} d_{1} / c$ ban jaata hai aur medium 2 mein $n_{2} d_{2} / c$ . Dekho Refractive Index.

5. Optical path length: $n \cdot d$

Kyunki $t = n d / c$ aur $c$ ek fixed constant hai, time minimize karna quantity $n \cdot d$ minimize karne ke barabar hai. Us quantity ka apna ek naam hai:

Picture: $d$ length ka ek real ruler, lekin material ruler ko $n$ factor se stretch karta hai jab tum "cost" count karte ho. Dekho Optical Path Length.

Kyun: yeh constant $c$ ko har calculation se hata deta hai, toh hum paths ko pure lengths aur indices use karke compare karte hain — koi seconds nahi chahiye.

6. Geometry variables: $a$ , $b$ , $w$ , $ℓ$ , aur $x$

Derivation trip ko do straight legs mein slice karti hai aur har measurement ko ek naam deti hai.

Neeche figure dekho paancho ek saath dekhne ke liye: $a$ aur $b$ vertical legs hain, $x$ aur $w - x$ total width $w$ ko split karte hain, aur har slanted magenta segment ek $ℓ$ hai.

7. Right triangle aur uske sides

Angles measure karne ke liye humein ek right triangle chahiye — ek aisa triangle jisme ek square ( $9 0^{\circ}$ ) corner ho.

Neeche figure dekho. Light ka har leg exactly aisa hi triangle banata hai: ek vertical side ( $a$ ya $b$ ), ek horizontal side ( $x$ ya $w - x$ ), aur slanted ray $ℓ$ sabse lambi side ke roop mein.

Topic ko yeh names isliye chahiye kyunki angle ka sine unse bana hai, aur poori derivation in triangles ko read karke khatam hoti hai.

8. Normal, aur angle $θ$

Picture: flagpole (normal) khada hai; ray us se angle $θ$ se door jhukti hai. Pole ke paas ray ka chhota $θ$ hota hai; paani ke saath almost flat ray ka $θ$ $9 0^{\circ}$ ke paas hota hai.

9. Sine, $sin θ$

Ab jab hamare paas ek triangle aur normal se angle hai, hum woh tool define kar sakte hain jo unhe connect karta hai.

Derivation mein run $x$ (ya $w - x$ ) hai aur hypotenuse path length $ℓ$ hai, isliye $sin θ_{1} = \frac{x}{a ^{2} + x ^{2}} = \frac{run}{ray length} .$

Section 7 ka figure triangle par yeh ratio draw karke dikhata hai: horizontal run $x$ slanted ray $ℓ$ ke upar.

10. Square root, aur $a^{2} + x^{2}$

Picture: $A$ se crossing point tak ki ray ek right triangle ki slanted hypotenuse hai jisme vertical leg $a$ hai ( $A$ ki line ke upar height) aur horizontal leg $x$ hai (kitna sideways). Iska length $a^{2} + x^{2} = ℓ_{1}$ hai.

Topic ko yeh isliye chahiye: yeh har straight path length $ℓ$ ko us ek cheez ke terms mein likhta hai jo light choose kar sakti hai — crossing point $x$ .

11. Total time $T$

Ab hum do legs ko ek single quantity mein add karte hain jise hum minimize kar sakein.

Picture: $x$ ke against $T$ plot ki gayi ek U-shaped curve; actual light path bilkul bottom par hoti hai, jahan curve flat hoti hai.

12. Minimize karna, aur derivative $d T / d x$

Yeh tool kyun? Kyunki "least time" matlab "U ka bottom" hai, aur "bottom, jahan curve flat ho" ka mathematical naam woh jagah hai jahan derivative zero ho. Derivative exactly woh spot find karne ki machine hai, bina guess kiye. Yeh poori derivation ka ek calculus step hai — aur yeh total time $T$ par act karta hai, kisi ek leg par nahi.

Prerequisite map

$T$ mein $c$ aur optical path length dono ke do arrows note karo: $T = OPL / c$ , toh constant $c$ definition se kabhi drop nahi hota — sirf baad mein, jab $d T / d x = 0$ ke dono sides ek $1/ c$ share karte hain jo cancel ho jaata hai.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle answer do.

Refractive index

n

tumhe physically kya bataata hai?

Light ek material mein vacuum se kitni guna slower travel karti hai,

n = c / v

n_{1}

aur

n_{2}

mein subscripts ka kya matlab hai?

Woh do media label karte hain:

n_{1}

boundary ke upar (medium 1) ka index hai,

n_{2}

neeche (medium 2) ka.

Time ko distance aur speed ke terms mein likho.

t = d / v

Time minimize karna

n \cdot d

minimize karne ke barabar kyun hai?

Kyunki

t = n d / c

aur

c

ek fixed constant hai.

Setup mein

a

b

aur

w

kya hain?

a

= boundary ke upar

A

ki height,

b

= uske neeche

B

ki depth,

w

A

B

tak total horizontal span.

ℓ

kya hai aur pehli leg kitni lambi hai?

ℓ

ek straight leg ki length hai;

ℓ_{1} = a^{2} + x^{2}

Ek right triangle par

sin θ

define karo.

sin θ = opposite / hypotenuse

Angle

θ

kis line se measure hota hai, aur iska range kya hai?

Normal se;

0^{\circ} \leq θ < 9 0^{\circ}

Normal incidence

θ = 0^{\circ}

par kya hota hai?

sin θ = 0

, toh ray bina bend kiye seedhi nikal jaati hai.

Legs

a

aur

x

wali ray ki length kya hai?

a^{2} + x^{2}

(Pythagoras).

Crossing point

x

ka domain kya hai?

0 \leq x \leq w

Total time

T (x)

likho.

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

\frac{d T}{d x} = 0

geometrically kya mean karta hai?

Total-time curve ka slope zero hai — valley ka bottom, yaani least time.

sin 9 0^{\circ}

calculate karo aur batao yeh kaunsi ray describe karta hai.

sin 9 0^{\circ} = 1

; ek ray jo surface ke saath flat lie karti hai (grazing incidence).

Connections

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