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

2.5.17 · Physics › Optics

1. Setup: polarizer kya karta hai

Sirf parallel component kyun bachta hai? Field vector ko do perpendicular pieces ke roop mein socho. Polarizer physically ek piece ko pass karta hai aur doosre ko khatam kar deta hai — jaise ek picket fence sirf vertical rope-wiggles ko pass karne deti hai.

Agar incoming polarized light ki amplitude $E_{0}$ hai aur woh transmission axis se $θ$ angle par hai, toh projected (surviving) amplitude hai:

$E_{trans} = E_{0} cos θ$

$cos θ$ kyun? Vector projection: kisi vector ka kisi axis ke saath component uski magnitude aur unke beech ke angle ke cosine ka product hota hai.

2. Malus's Law — scratch se derive karo

Derivation (KAISE):

Intensity field amplitude ke square ke proportional hoti hai: $I \propto E^{2}$ . Kyun? EM wave ki energy $\propto E^{2}$ hoti hai.
Surviving amplitude: $E = E_{0} cos θ$ (projection, upar wala step).
Isliye $I \propto E_{0}^{2} cos^{2} θ$ . $I_{0} \propto E_{0}^{2}$ ko incoming intensity likhne par: $I = I_{0} cos^{2} θ .$

Special case: unpolarized light polarizer se takraati hai

$\frac{1}{2}$ kyun? Unpolarized light sabhi angles $θ$ ka random mix hai. Hum $cos^{2} θ$ ka average poore circle par lete hain: $⟨ cos^{2} θ ⟩ = \frac{1}{2 π} \int_{0}^{2 π} cos^{2} θ d θ = \frac{1}{2} .$ Toh exactly aadhi intensity pass hoti hai — aur output ab axis ke saath polarized ho jaata hai.

3. Worked examples (Malus)

4. Brewster's Angle — scratch se derive karo

Figure — Polarization — Malus's law, Brewster's angle derivation

Derivation (KAISE):

Key geometric condition: reflected ⟂ refracted, yaani $θ_{B} + θ_{r} = 9 0^{\circ} \Rightarrow θ_{r} = 9 0^{\circ} - θ_{B} .$ Yeh condition kyun? Refracted ray electrons ki oscillation drive karti hai. Reflected light unhi electrons dwara radiate hoti hai. Ek charge apni oscillation axis ke saath kuch radiate nahi karta. Reflected ray (in-plane polarization ke liye) tab gaayab ho jaati hai jab woh exactly us oscillation ke saath point karti — jo tab hota hai jab reflected ⟂ refracted.
Snell's law lagao $n_{1} sin θ_{B} = n_{2} sin θ_{r}$ : $n_{1} sin θ_{B} = n_{2} sin (9 0^{\circ} - θ_{B}) = n_{2} cos θ_{B} .$
$cos θ_{B}$ se divide karo: $n_{1} tan θ_{B} = n_{2} .$

5. Common mistakes (steel-manned)

6. Active recall

Recall Quick self-test (answers cover karo)

Malus's law batao aur square explain karo. → $I = I_{0} cos^{2} θ$ ; intensity $\propto E^{2}$ .
Unpolarized light ke liye $\frac{1}{2}$ kyun? → $cos^{2} θ$ ka average sabhi angles par.
Reflected & refracted rays ke beech Brewster condition? → Woh perpendicular hain.
$tan θ_{B} = n$ derive karo. → Snell + $θ_{r} = 90 - θ_{B}$ .

Recall Feynman: ek 12-saal ke bachche ko explain karo

Light ek hilti-dulti jump-rope jaisi hai. Ek polarizer vertical slots wali fence hai: sirf upar-neeche wiggles pass hoti hain, side-to-side wiggles rok di jaati hain. Agar teri wiggle tilted hai, toh uska sirf ek hissa sneaks through hota hai — zyada tilt karo, kam pass hoga. Malus's law exactly measure karti hai ki kitna. Ab, jab light kisi shiny floor ya lake se bounce karti hai, woh bounced light zyaadatar side-to-side wiggling (glare) ban jaati hai. Ek special angle par dekhne par (Brewster's angle), bounce perfectly ek direction ka hota hai — aur isliye machhimaron ke sunglasses paani ki chamak ko ekdum khatam kar dete hain.

7. Connections

Snell's Law and Refraction — Brewster's derivation directly Snell mein plug hoti hai.
Electromagnetic Waves — $I \propto E^{2}$ EM energy density se aata hai.
Reflection and Refraction at Interfaces — Fresnel equations Brewster ke peeche puri reflectance dete hain.
Wave Nature of Light — polarization prove karta hai ki light transverse hai.
Unpolarized vs Polarized Light — woh averaging jo $\frac{1}{2}$ factor deta hai.

Malus's law ka statement

I = I_{0} cos^{2} θ

polarized light ke liye jo transmission axis se

θ

angle par hai.

Intensity

cos^{2} θ

ke proportional kyun hai,

cos θ

ke nahi?

Amplitude

E_{0} cos θ

project hoti hai, aur intensity

\propto E^{2}

hai, isliye squaring karne par

cos^{2} θ

milta hai.

Ek polarizer ke baad unpolarized light ki intensity

\frac{1}{2} I_{0}

(

cos^{2} θ

ka sabhi angles par average = 1/2).

Brewster's law

tan θ_{B} = n_{2} / n_{1}

; reflected light tab completely polarized hoti hai.

Brewster's angle par geometric condition

Reflected aur refracted rays perpendicular hain:

θ_{B} + θ_{r} = 9 0^{\circ}

Snell se Brewster derive karo

n_{1} sin θ_{B} = n_{2} sin (90 - θ_{B}) = n_{2} cos θ_{B} \Rightarrow tan θ_{B} = n_{2} / n_{1}

θ_{B}

par reflected ray ki polarization direction

Plane of incidence ke perpendicular (surface ke parallel).

Glass (n=1.5) ke liye Brewster's angle

arctan (1.5) \approx 56. 3^{\circ}

Reflected light 100% polarized kyun ho sakti hai, physical reason

Ek oscillating charge apni oscillation axis ke saath kuch radiate nahi karta;

θ_{B}

par reflected direction us axis ke saath align ho jaata hai in-plane polarization ke liye.

Teen polarizers (0°,45°,90°), polarized input

I_{0}

, output?

I_{0} /4

— middle polarizer polarization rotate karta hai toh light wapas aa jaati hai.