2.5.17 · D1Polarization — Malus's law, Brewster's angle derivation
Light ek aisi "wiggle" hai jiska ek direction hota hai, aur dono — ek polarizer aur ek shiny surface — us direction ke hisaab se light ko filter kar sakte hain. Is topic mein sab kuch sirf itna hai ki kitni wiggle bachti hai — yeh ek vector ko ek axis par project karke measure karte hain (Malus), ya Snell's law ke saath geometry ko align karke (Brewster).
Parent note padhne se pehle, tumhe har ek symbol ko samajhna padega. Neeche, har cheez pichli cheez se build hoti hai: plain words → picture → kyun topic ko yeh chahiye. Koi bhi cheez define hone se pehle use nahi ki jaati.
Definition Transverse waveEk wave jisme jo cheez wiggles karti hai woh wave ke travel direction ke perpendicular (right angle par) move karti hai. Ek jump-rope imagine karo: wave rope ke saath chalti hai, lekin rope ka har hissa upar-neeche move karta hai, yani us direction ke sideways.
Kyun topic ko yeh chahiye. Polarization sirf transverse waves ke liye meaningful hai. Agar wiggle travel direction ke along hoti (jaise sound mein), toh koi "sideways choice" filter karne ke liye hoti hi nahi — koi up-down vs left-right nahi. Dekhein Wave Nature of Light aur Electromagnetic Waves — kyun light qualify karti hai.
Definition Electric fieldE E E length batata hai ki woh push kitni strong hai. Plain letter E length (magnitude) ko denote karta hai, jo ek positive number hoti hai.
Intuition Arrow vs. numberE E direction hi polarization hai.
Kyun topic ko yeh chahiye. E hi polarization hai. Malus's law poochta hai "filter ke baad E Electromagnetic Waves se, light ek E
E 0 0 E 0 sabse badi length hai jo arrow E
Rope imagine karo: rope middle line ko kai baar cross karti hai, lekin E 0 is maximum swing par karta hai.
Kyun topic ko yeh chahiye. Malus's law incoming amplitude E 0
Yeh Malus's law ka dil hai, isliye hum ise dheere-dheere build karte hain.
θ θ angle ke liye use hota hai — do directions ke beech tilt ki miqdar, degrees (∘ 36 0 ∘ E
Ab: jab E 0 θ projection kehlati hai.
cos θ E 0 hypotenuse ) ho, aur axis par seedhi line drop karo. Woh piece jo axis ke saath lie karta hai woh adjacent side hai. Tab:
cos θ = hypotenuse adjacent = E 0 length along axis . E 0 cos θ
Intuition Cosine kyun, aur kuch kyun nahi?Humein ek aisa tool chahiye jo answer kare "ek tilted arrow ka kitna fraction ek chosen line ke along hai?" Cosine exactly wahi fraction hai — yeh 1 θ = 0 0 θ = 9 0 ∘
Kyun topic ko yeh chahiye. Polarizer se guzarne ke baad bachne wali amplitude E 0 cos θ
Worked example Extremes check karo
θ = 0 ∘ cos 0 = 1 E 0 θ = 9 0 ∘ cos 9 0 ∘ = 0 θ = 3 0 ∘ cos 3 0 ∘ = 2 3 ≈ 0.866
I I brightness hai jo ek detector actually read karta hai — light jo energy per second per area deliver karti hai. Tumhari aankh aur ek light meter I nahi .
E E 2 Electromagnetic Waves se — energy density field squared ke saath jaati hai, bilkul jaise ek spring energy store karti hai displacement squared ki tarah). Toh jab bachne wali amplitude E 0 cos θ brightness hai ( E 0 cos θ ) 2 = E 0 2 cos 2 θ cos 2 θ square kyunki brightness amplitude squared hai.
Kyun topic ko yeh chahiye. Malus's law intensity mein likha jaata hai (I = I 0 cos 2 θ E I
cos 2 θ ( cos θ ) 0 1 2 1
Intuition Average exactly2 1 Unpolarized vs Polarized Light ) sab tilt angles ka ek equal mix hai. Uska bachne wala fraction nikalne ke liye hum cos 2 θ cos 2 θ 2 1 2 1
Kyun topic ko yeh chahiye. Unpolarized light ke liye 2 1
Definition Refractive indexn slow aur bend karta hai. Vacuum/air: n ≈ 1 n = 1.33 n = 1.5 n kaunsa medium: n 1 n 2
Kyun topic ko yeh chahiye. Brewster's angle poori tarah ratio n 2 / n 1 Reflection and Refraction at Interfaces .
Definition Normal, aur sinesin θ Normal woh dashed line hai jo surface par jahan light hit karti hai wahan perpendicular draw ki jaati hai — yeh saare angles usi se measure hote hain, surface se nahi.
Sine , sin θ = hypotenuse opposite saamne wali side slanted side se divide. Yeh cosine ka natural partner hai aur yahi bending govern karta hai.
Kyun topic ko yeh chahiye. Brewster's derivation hi Snell's law hai jisme ek extra geometric fact plug in ki gayi hai. Full details Snell's Law and Refraction mein.
tan θ tan θ = c o s θ s i n θ = adjacent opposite steepness measure karta hai — tilted arrow ka rise over run.
arctan Question-undoer : arctan ( x ) "kis angle ka tangent x Agar tan θ B = 1.5 θ B = arctan ( 1.5 ) = 56. 3 ∘
Intuition Tangent Brewster's law mein kyun aata haiBrewster derivation mein ek Snell equation ko cos θ B sin / cos tan "ratio n 2 / n 1 — aur arctan us ratio ko actual angle mein wapas convert karta hai.
Angle theta and cosine projection
Malus amplitude E0 cos theta
Intensity goes as E squared
Average of cos2 is one half
Unpolarized halves at first filter
Brewster tan theta B equals n2 over n1
Parent par wapas jao: Polarization — Malus's law, Brewster's angle derivation .
Right side cover karo aur khud ko test karo — jab har line instantly aaye tab tum ready ho.
Ek wave "transverse" kyun hoti hai? Wiggle travel direction ke perpendicular hoti hai.
E E E wiggle arrow hai (direction + strength);
E sirf uski length hai.
E 0 Peak amplitude — arrow ki sabse badi swing.
cos θ Adjacent over hypotenuse — ek tilted arrow ka woh fraction jo axis ke along hai.
Polarizer se guzarne ke baad bachne wali amplitude E 0 cos θ Yeh tilted field ka transmission axis par projection hai.
Intensity I E Woh brightness jo ek detector read karta hai; I ∝ E 2
Malus's law mein cos cos 2 Amplitude ke projection se cosine aata hai, phir square isliye kyunki I ∝ E 2
cos 2 θ 2 1 Yeh ek puri turn mein 2 1
Refractive index n Koi medium light ko kitna slow aur bend karta hai; n = 1 1.33 1.5
Snell's law state karo aur angles kis se measure hote hain. n 1 sin θ 1 = n 2 sin θ 2
tan θ sin θ / cos θ
arctan ( x ) "Kis angle ka tangent x
