Visual walkthrough — Polarization — Malus's law, Brewster's angle derivation

2.5.17 · D2 · Physics › Optics › Polarization — Malus's law, Brewster's angle derivation

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Step 1 — "Glass se reflect hona" actually dikhta kaise hai

KYA HAI. Ek ray of light upar se hawaon mein neeche aati hai aur ek flat sheet of glass se takraati hai. Ek saath do cheezein hoti hain: uska ek hissa wapas bounce hota hai hawaon mein (reflected ray), aur ek hissa *bend hokar glass ke andar jaata hai (refracted ray). Dekho Reflection and Refraction at Interfaces.

PEHLE YEH KYO DRAW KAREIN. Jo bhi symbols hum use karenge, wo sab is picture pe measure kiye gaye angles hain. Agar picture fix nahi ki, toh symbols ka matlab hi nahi.

PICTURE. Dashed vertical line hai normal — wo line jo surface ke perpendicular hai us point pe jahan light strike karti hai. Optics mein har angle is normal se measure hota hai, surface se kabhi nahi.

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Step 2 — Bounced light pehle se partly polarized kyun hoti hai

KYA HAI. Incoming sunlight unpolarized hoti hai — uska electric field $\vec{E}$ ray ke across har direction mein wiggle karta hai (dekho Unpolarized vs Polarized Light). Hum har wiggle ko do independent pieces mein tod dete hain:

• s-piece: $\vec{E}$ wiggle karta hai page ke plane ke perpendicular (paper se bahar, dots ⊙).
• p-piece: $\vec{E}$ wiggle karta hai page ke plane ke andar (us plane mein arrows ↕ jo rays aur normal ko contain karta hai — plane of incidence).

IS TARAH KYO TODEIN. Dono pieces alag-alag rules se reflect hote hain. Nature "plane mein wiggle" aur "plane se bahar wiggle" ko surface pe alag treat karti hai — isliye hume inhe alag-alag track karna hoga. Plane of incidence hamara natural reference hai.

PICTURE. Reflected ray pe ⊙ (s) strongly survive karta hai; ↕ (p) weak hai. Isliye paani se glare pehle se partly polarized hoti hai. Hamara pura goal: wo special angle dhundho jahan p-piece exactly zero ho jaye.

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Step 3 — Re-radiation idea: electrons chhote antennas hain

KYA HAI. Light literally "bounce" nahi karti. Refracted ray, glass ke andar travel karti hui, electrons ko pakad ke unhe apne khud ke $\vec{E}$ ki direction mein aage-peeche hilati hai. Ek hilta hua electric charge re-radiate karta hai light ko har direction mein — aur woh re-radiated light hi reflected ray hai (dekho Electromagnetic Waves).

YEH KYO MATTER KARTA HAI. Ek hilte hue charge ka ek iron rule hai:

PICTURE. Plum arrows radiation ka doughnut dikhate hain: sides pe bright, axis ke along dark. Woh dark direction hi wo loophole hai jo hum exploit karenge.

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Step 4 — Magic condition: reflected ⟂ refracted

KYA HAI. Glass ke andar p-piece electrons, refracted ray ki p-direction mein wiggle karte hain. Reflected p-light ko exactly unhi electrons dwara radiate hona padega. Antenna rule ke hisaab se, woh reflected p-light disappear ho jaati hai tab aur sirf tab jab reflected ray electrons ki oscillation axis ke along point kare.

ISKA MATLAB RIGHT ANGLE KYO HOTA HAI. Electrons refracted ray ke perpendicular oscillate karte hain (EM wave ka $\vec{E}$ uski travel direction ke perpendicular hota hai). Toh "reflected ray oscillation ke along hai" ka matlab hai reflected ray refracted ray ke perpendicular hai:

$\underset{\begin{array}{c}\text{special incidence}\\ \text{angle}\end{array}}{\underbrace{{\theta }_B}}+\underset{\begin{array}{c}\text{refraction}\\ \text{angle}\end{array}}{\underbrace{{\theta }_r}}=90^{\circ }$

Humne incidence angle ${\theta }_i\to {\theta }_B$ (Brewster's angle) rename kar diya kyunki ab yeh woh ek special value hai jo yeh hone deti hai.

PICTURE. Teal reflected ray aur burnt-orange refracted ray ek clean $90^{\circ }$ par milte hain. Sirf is incidence angle par hi wo right angle banaate hain.

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Step 5 — Geometry ko angles ke baare mein equation mein badlo

KYA HAI. Right-angle condition se, ${\theta }_r$ ke liye solve karo:

${\theta }_r=90^{\circ }-{\theta }_B$

KYO. Hum baad mein ${\theta }_r$ ko eliminate karna chahte hain. Snell's law ${\theta }_B$ aur ${\theta }_r$ ko connect karegi; agar hum ${\theta }_r$ ko purely ${\theta }_B$ ke terms mein rewrite kar sakein, toh hume ek unknown mein ek clean equation milegi.

PICTURE. Dono angles ${\theta }_B$ aur ${\theta }_r$, rays ke beech right-angle "wedge" ke do non-right angles hain, isliye unhe $90^{\circ }$ mein add hona chahiye. Figure us wedge ko shade karta hai.

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Step 6 — Snell's law mein daalo

KYA HAI. Snell's law (Snell's Law and Refraction se) surface ke dono sides ko relate karta hai:

$\underset{\begin{array}{c}\text{index of}\\ \text{air}\end{array}}{\underbrace{n_1}}\sin \underset{\text{incidence}}{\underbrace{{\theta }_B}}=\underset{\begin{array}{c}\text{index of}\\ \text{glass}\end{array}}{\underbrace{n_2}}\sin \underset{\text{refraction}}{\underbrace{{\theta }_r}}$

KYO. Snell ek hi aisa law hai jo medium 1 mein angle ko medium 2 mein angle se jodata hai. Yeh surface ke paas bridge hai. Jitna bada $n$, utna zyada ray normal ki taraf bend hogi.

Ab Step 5 se ${\theta }_r=90^{\circ }-{\theta }_B$ substitute karo:

$n_1\sin {\theta }_B=n_2\sin (90^{\circ }-{\theta }_B)$

Yahan key trick hai — KYO cosine aata hai. Angle $90^{\circ }-{\theta }_B$, ${\theta }_B$ ka complement hai. Ek right triangle mein, ek acute angle ka sine doosre ka cosine hota hai ("opposite" ek ka "adjacent" doosra hai). Toh:

$\sin (90^{\circ }-{\theta }_B)=\cos {\theta }_B$

Jisse milta hai:

$n_1\sin {\theta }_B=n_2\cos {\theta }_B$

PICTURE. Figure ek right triangle dikhata hai jahan same pair of sides ${\theta }_B$ ke liye "opposite" hai lekin $90^{\circ }-{\theta }_B$ ke liye "adjacent" — exactly wahi swap hai jis se $\sin$ $\cos$ ban jaata hai.

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Step 7 — Cosine se divide kyo karein → tangent sahi tool kyo hai

KYA HAI. Hamare paas hai $n_1\sin {\theta }_B=n_2\cos {\theta }_B$. Dono sides ko $\cos {\theta }_B$ se divide karo:

$n_1\frac{\sin {\theta }_B}{\cos {\theta }_B}=n_2⟹n_1\tan {\theta }_B=n_2$

TANGENT KYO, SINE YA COSINE KYO NAHI? Hum same angle ke dono $\sin$ aur $\cos$ ke saath atak gaye hain. Ek hi tool jo "sine over cosine" ko ek clean quantity mein combine karta hai woh hai tangent:

$\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\text{opposite}}{\text{adjacent}}$

Yeh angle ki steepness measure karta hai. Toh $\cos {\theta }_B$ se divide karna koi random algebra move nahi — yeh woh deliberate step hai jo do trig functions ko us ek function mein collapse karta hai jo answer deta hai "is steepness wala angle kaun sa hai?"

$\boxed{\tan {\theta }_B=\frac{n_2}{n_1}}$

Air → medium ke liye ($n_1=1$, $n_2=n$): $\tan {\theta }_B=n$.

PICTURE. Ek right triangle jiska vertical side $n_2$ hai aur horizontal side $n_1$: base pe angle ka $\tan =n_2/n_1$ hai, aur wahi base angle ${\theta }_B$ hai.

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Step 8 — Numbers daalo, aur edge cases check karo

Degenerate / limiting cases — reader ko kabhi surprise nahi hona chahiye:

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Ek picture mein summary

Sab kuch ek canvas par: incident ray ${\theta }_B$ par strike karti hai; reflected (teal, s-only ⊙) aur refracted (orange) perfect $90^{\circ }$ par nikalte hain; Snell + complement identity + tangent sab milke $\tan {\theta }_B=n_2/n_1$ banaate hain.

Recall Feynman retelling — simple shabdon mein poora walkthrough

Light ek glass floor se takraati hai aur split ho jaati hai: kuch bounce hoti hai, kuch andar sink ho jaati hai. Jo hissa sink hota hai woh glass ke electrons ko hilata hai jaise chhoti hands jump-rope par. Hilti hands har direction mein nayi light throw karti hain — except seedha shake ke along, jahan wo kuch throw nahi karti. Woh "kuch nahi" direction hamara loophole hai. Incoming light ko itna tilt karo ki bounce-direction exactly shake ke along line up ho jaye — ab in-plane wiggle simply bounce nahi ho sakti, aur reflection cleanly ek-directional (pure glare) ban jaati hai. Geometrically, "bounce shake ke saath line up hai" ka matlab hai bounced ray aur sunk-in ray perfect right angle banate hain: ${\theta }_B+{\theta }_r=90^{\circ }$. Isse Snell's law mein daalo (dono sides ke beech bridge), notice karo ki $\sin (90^{\circ }-{\theta }_B)$ bas $\cos {\theta }_B$ hai, divide karo, aur do trig pieces ek tangent mein fuse ho jaate hain: $\tan {\theta }_B=n_2/n_1$. Glass ($n=1.5$) daalo → $56.3^{\circ }$. Yahi fisherman-ke-sunglasses wala angle hai.

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Connections

• Snell's Law and Refraction — Step 6 mein Snell hai complement substituted ke saath.
• Reflection and Refraction at Interfaces — Steps 1–2 ray geometry set karte hain.
• Electromagnetic Waves — Step 3 ka antenna rule EM radiation hai.
• Unpolarized vs Polarized Light — kyun raw ray mein s aur p dono pieces hain.
• Wave Nature of Light — transverse wiggle jo sab kuch underpin karti hai.