Visual walkthrough — Snell's law — derivation from Fermat's principle

2.5.4 · D2 · Physics › Optics › Snell's law — derivation from Fermat's principle

Hum ek hi sawaal ka jawaab de rahe hain: ek beam of light point $A$ se nikalta hai, point $B$ par pahunchna zaroori hai, aur raaste mein woh ek "fast" material se "slow" material mein cross karta hai. Woh kaunsa rasta leta hai?

---

Step 1 — Stage draw karo aur har distance ka naam rakho

KYA. Hum point $A$ ko top material mein aur point $B$ ko bottom material mein rakhte hain. Ek flat horizontal line — boundary — unhe alag karti hai. Light ko $A$ se shuru karke $B$ par khatam karna hai; uske paas sirf yeh choose karne ka option hai ki boundary par kahan cross kare. Us crossing point ko $P$ kaho, aur uski horizontal position ko $x$ kaho.

KYUN. Jab tak hum "best path" ki baat nahi kar sakte jab tak hum koi bhi path ko ek single number se describe na kar sakein. Woh number hai $x$. $P$ ko left aur right slide karo aur tum har seedha-andar, tedha-bahar wala path generate karte ho jo light le sakti hai. Toh $x$ ek free dial hai.

PICTURE. Figure dekho. $A$ se boundary tak vertical drop $a$ hai; boundary se $B$ tak vertical drop $b$ hai; $A$ se $B$ tak total horizontal gap $w$ hai. Yeh teen ($a$, $b$, $w$) fixed hain — yeh is baat se set hain ki $A$ aur $B$ kahan baithe hain. Sirf $x$ (coral mein marked) slide karne ke liye free hai.

---

Step 2 — Har leg ki length ek right triangle se measure karo

KYA. Path ke do seedhe tukde hain: top medium mein $A\to P$, aur bottom medium mein $P\to B$. Hume unki lengths chahiye.

KYUN. Light ka travel time is baat par depend karta hai ki woh har material mein kitni door jaati hai (aur wahan kitni tez chalti hai). Toh hume geometry ko do honest distances mein convert karna hoga.

PICTURE. $A$ se boundary tak ek vertical line giraa do. Uski height $a$ hai. $P$ tak horizontal run $x$ hai. Yeh do sides right angle par hain, toh $A$, vertical ka foot, aur $P$ milke ek right triangle banate hain. Tedha path $A\to P$ uska hypotenuse hai.

${\ell }_1=\underset{\text{length }A\to P}{\underbrace{\sqrt{\underset{\text{vertical}}{\underbrace{a^2}}+\underset{\text{horizontal}}{\underbrace{x^2}}}}}$

Bottom leg bhi isi idea par hai. Uska vertical drop $b$ hai; uska horizontal run woh hai jo $x$ ke baad bacha hua hai, yani $w-x$:

${\ell }_2=\underset{\text{length }P\to B}{\underbrace{\sqrt{\underset{\text{vertical}}{\underbrace{b^2}}+\underset{\text{leftover horizontal}}{\underbrace{(w-x{)}^2}}}}}$

---

Step 3 — Lengths ko travel time mein badlo

KYA. Hum har distance ko time mein convert karte hain jo light wahan spend karti hai.

KYUN. Fermat's principle (dekho Fermat's Principle) kehta hai light least time ka rasta leti hai, least distance ka nahi. Toh time hi woh cheez hai jo hume banana hai aur phir jitna ho sake chhota karna hai.

PICTURE. Kisi medium mein speed $v=c/n$ hai, jahan $c$ light ki vacuum mein speed hai aur $n$ refractive index hai (bada $n$ = slower medium). Time = distance ÷ speed:

$t=\frac{d}{v}=\frac{d}{c/n}=\frac{nd}{c}.$

Woh product $nd$ optical path length hai. Dono legs jodte hain:

$T(x)=\underset{\text{top medium mein time}}{\underbrace{\frac{n_1}{c}{\ell }_1}}+\underset{\text{bottom medium mein time}}{\underbrace{\frac{n_2}{c}{\ell }_2}}=\frac{n_1}{c}\sqrt{a^2+x^2}+\frac{n_2}{c}\sqrt{b^2+(w-x{)}^2}.$

• $n_1,n_2$ ::: dono media ke indices (har ek light ko kitna slow karta hai).
• $c$ ::: vacuum light speed — ek constant, dono terms mein same.
• $T(x)$ ::: total travel time, hamare ek dial $x$ ka function likha hua.

Figure mein green curve $T$ ko $x$ ke against plot kiya gaya hai. Notice karo iska ek lowest point hai — ek valley. Light us valley ke bottom par rehti hai.

---

Step 4 — Valley ka bottom dhundo

KYA. Hum woh $x$ locate karte hain jo $T(x)$ ko sabse chhota banata hai.

KYUN yeh tool. Tum ek smooth valley ka bottom aankhon se andaaze lagaye bina kaise dhoondoge? Bilkul bottom par curve momentarily flat hoti hai — uska slope zero hota hai. Derivative $\frac{dT}{dx}$ wahi slope hai. Toh "least time" kehne ka machine-precise tarika hai:

$\frac{dT}{dx}=0.$

Yahi idea hai Calculus — minimization and stationary points mein: minimum wahan hota hai jahan tangent line horizontal ho jaati hai.

PICTURE. Figure mein time-curve par tangent line teen jagah draw ki gayi hai. Left slope par woh neeche jhukti hai, right par upar, aur bottom par bilkul level hai — woh flat tangent hi winning $x$ mark karti hai.

---

Step 5 — Actually derivative lo

KYA. Hum $\frac{dT}{dx}$ term by term compute karte hain.

KYUN chain rule. Har leg "(x ke saath kuch)" ka square root hai. Ek function-inside-a-function differentiate karne ke liye tum use layers mein peel karte ho — yahi chain rule hai. $\sqrt{a^2+x^2}$ ke liye outer layer  ​ hai, inner layer $a^2+x^2$ hai:

$\frac{d}{dx}\sqrt{a^2+x^2}=\underset{\text{derivative of }\sqrt{}}{\underbrace{\frac{1}{2\sqrt{a^2+x^2}}}}\cdot \underset{\text{derivative of inside}}{\underbrace{2x}}=\frac{x}{\sqrt{a^2+x^2}}.$

Bottom leg mein andar $w-x$ hai; inside ka derivative $-1$ hai, jo saamne ek minus sign kheench laata hai:

$\frac{d}{dx}\sqrt{b^2+(w-x{)}^2}=\frac{-(w-x)}{\sqrt{b^2+(w-x{)}^2}}.$

Dono ko $dT/dx=0$ mein daalte hain:

$\frac{dT}{dx}=\underset{\text{top-leg time kaise badlata hai}}{\underbrace{\frac{n_1}{c}\cdot \frac{x}{\sqrt{a^2+x^2}}}}+\underset{\text{bottom-leg time kaise badlata hai}}{\underbrace{\frac{n_2}{c}\cdot \frac{-(w-x)}{\sqrt{b^2+(w-x{)}^2}}}}=0.$

Figure do competing "urges" dikhata hai: $x$ ko right nudge karna top leg mein time add karta hai lekin bottom leg mein time subtract karta hai. Balance wahan hai jahan yeh dono cancel ho jaate hain.

---

Step 6 — Fractions mein chhupe sines pehchaano

KYA. Woh do ugly fractions secretly dono angles ke sines hain.

KYUN. Step 2 ke har right triangle mein, normal se angle define karo (boundary ke perpendicular dashed vertical line). Trigonometry kehta hai:

$\sin (\text{angle})=\frac{\text{opposite side}}{\text{hypotenuse}}.$

Top triangle mein normal-se-angle ke opposite wala side horizontal run $x$ hai, aur hypotenuse ${\ell }_1$ hai. Toh:

$\sin {\theta }_1=\frac{x}{\sqrt{a^2+x^2}},\sin {\theta }_2=\frac{w-x}{\sqrt{b^2+(w-x{)}^2}}.$

• ${\theta }_1$ ::: incoming ray aur normal ke beech ka angle (top medium).
• ${\theta }_2$ ::: outgoing ray aur normal ke beech ka angle (bottom medium).

PICTURE. Figure har triangle mein opposite side (coral) over hypotenuse (lavender) highlight karta hai — woh ratio hi sine hai. Yahi exact fractions Step 5 se hain.

---

Step 7 — Substitute karo, cancel karo, aur law padho

KYA. Fractions ki jagah sines daal do, phir clean up karo.

KYUN. Yahi payoff hai: hamaari balance equation sirf dono angles aur dono media ke baare mein ek statement ban jaati hai.

Step 5 se shuru karke Step 6 ke sines substitute karte hain:

$\frac{n_1}{c}\sin {\theta }_1-\frac{n_2}{c}\sin {\theta }_2=0.$

Har term mein $\frac{1}{c}$ hai. Kyunki $c\ne 0$, $c$ se multiply karo aur woh gayab ho jaata hai:

$n_1\sin {\theta }_1=n_2\sin {\theta }_2.$

---

Step 8 — Degenerate aur edge cases (kuch bhi nahi chhoda)

Har scenario cover hona chahiye. Yeh rahe, har ek seedha law se padha hua.

Case A — Seedha-andar (${\theta }_1=0$). Agar light seedha hit kare, $\sin {\theta }_1=0$, toh $n_2\sin {\theta }_2=0$, jis se ${\theta }_2=0$ forced hota hai. Light seedhi nikal jaati hai, koi bending nahi. (Uski speed phir bhi badlti hai — sirf direction unbent hai.)

Case B — Same medium ($n_1=n_2$). Law collapse ho kar $\sin {\theta }_1=\sin {\theta }_2$ ban jaata hai, toh ${\theta }_2={\theta }_1$: koi boundary effect nahi, ek seedhi line. Yeh sanity check hai.

Case C — Rare → dense ($n_2>n_1$). $\sin {\theta }_2=\frac{n_1}{n_2}\sin {\theta }_1<\sin {\theta }_1$, toh ${\theta }_2<{\theta }_1$ — normal ki taraf jhukta hai.

Case D — Dense → rare ($n_1>n_2$). Ab $\sin {\theta }_2=\frac{n_1}{n_2}\sin {\theta }_1>\sin {\theta }_1$, toh ${\theta }_2>{\theta }_1$ — normal se door jhukta hai.

Case E — Breaking point (Total Internal Reflection). Case D mein, jaise ${\theta }_1$ badhta hai, $\sin {\theta }_2$ apni ceiling $1$ ki taraf chadh jaata hai. Woh angle jahan $\sin {\theta }_2=1$ (yaani ${\theta }_2=90^{\circ }$, refracted ray surface ke along skimming kar rahi hai) critical angle ${\theta }_c$ hai:

$n_1\sin {\theta }_c=n_2\cdot 1\Rightarrow \sin {\theta }_c=\frac{n_2}{n_1}.$

${\theta }_1$ ko ${\theta }_c$ se aage push karo aur law demand karta hai $\sin {\theta }_2>1$, jo koi real angle satisfy nahi karta — refraction impossible hai, toh saari light wapas reflect ho jaati hai: Total Internal Reflection.

---

Ek-picture summary

Upar sab kuch, ek single frame mein compress kiya hua: free crossing point $x$, do right triangles, flat bottom wali time-valley, aur balanced sines jo Snell's law ban jaate hain.

Recall Feynman retelling — poora walk simple shabdon mein

Light $A$ se $B$ tak least time mein pahunchna chahti hai. Uske paas sirf yeh choice hai ki boundary par kahan cross kare — us jagah ko $x$ kaho. $x$ ko idhar-udhar slide karo aur trip alag-alag time leta hai, kyunki fast material aur slow material har crossing point ke liye alag tarah trade off karte hain. Us trip-time ko $x$ ke against plot karo aur tumhe ek smooth valley milti hai. Light valley ke bottom par baihi hai, aur valley ka bottom wahan hai jahan zameen flat ho — jahan slope (derivative) zero ho. Us flatness condition ko work out karo, trip ka har leg ek aisa fraction contribute karta hai jo seedha-upar-se-angle ka sine hai. Dono legs ko balance karne se milta hai $n_1\sin {\theta }_1=n_2\sin {\theta }_2$. Aur woh chhota sa rule sab kuch quietly explain karta hai: seedhi light nahi jhukti, slow glass mein jaana tumhe perpendicular ki taraf moda deta hai, aur glass chhodne ki koshish mein zyada jhuk jaana escape impossible bana deta hai toh light wapas bounce ho jaati hai — total internal reflection.

---

Connections

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

Concept Map