Visual walkthrough — Poynting vector — energy flux in EM waves

1.8.36 · D2 · Physics › Electromagnetism › Poynting vector — energy flux in EM waves

Hum ek plain-English sentence ko pakad ke use equation mein badalne ki koshish kar rahe hain:

"Ek choti si box of space mein stored energy tab hi badal sakti hai jab energy uski walls ke across flow kare, ya fields andar wale charges par kaam karein."

Bas itna hi. Yahi sentence, carefully draw kiya hua, hi Poynting's theorem hai.

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Step 0 — Teenon cheezein jo humein pehle se picture karni chahiye

Symbols se pehle, teen pictures. Agar koi shaky lagta hai, toh neeeche wale arrows wahi fix karte hain.

Hum sab kuch Maxwell's equations (woh laws jo batate hain ki $\vec{E}$ aur $\vec{B}$ kaise change hote hain) aur energy density $u$ se build karte hain.

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Step 1 — Choti box draw karo aur bookkeeping karo

KYA. Hum space ki ek tiny cubic box choose karte hain, fixed jagah par, aur energy andar-bahar count karte hain — bilkul jaise till mein paisa count karte hain.

KYUN. Energy conserved hoti hai. Agar box ke andar joules kam ho jaayein, toh woh vanish nahi hue — ya toh kisi wall se leak huey ya andar wale charges par kaam karne mein kharch hue. Uss balance ko likhna hi poora game hai; Poynting vector uss "wall se leak" ko ek naam dene wala hai.

PICTURE. Neeeche wali box dekho. Green arrows = energy walls cross kar rahi hai (flux). Andar lightning-bolt symbol = fields charges ko push kar rahi hain (kaam). Coins ka stack = stored energy $u$.

Shabdon mein, hamari target equation hai:

$$\underset{\begin{array}{c}\text{kitni tezi se stored}\\ \text{energy change ho}\\ {\text{rahi hai, per m}}^3\end{array}}{\underbrace{\frac{\partial u}{\partial t}}}=-\underset{\text{ise hum }\nabla \cdot \vec{S}\text{ kahenge}}{\underbrace{({\text{energy leaking out per m}}^3)}}-\underset{\vec{J}\cdot \vec{E}}{\underbrace{({\text{charges par kaam per m}}^3)}}$$

Yahan har term per unit volume per unit time hai — yahi $\partial u/\partial t$ humpar impose karta hai, toh baaki do terms ko match karna hoga. Abhi tak pata nahi $\vec{S}$ kya hai. Hum usse discover karne wale hain.

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Step 2 — Stored energy likho, phir dekhte hain woh kaise change hoti hai

KYA. $u$ ko do field energies se likho, phir uska time derivative $\partial u/\partial t$ lo.

KYUN. Hum bookkeeping equation ka left-hand side chahte hain — woh rate jis par coins change hote hain. Symbol $\partial /\partial t$ ka matlab hai "yeh kitni tezi se change hota hai jab time tick karta hai, position fixed rakhte hue." Hum ise use karte hain (ordinary $d/dt$ nahi) kyunki $u$ dono jagah aur time par depend karta hai, aur yahan hum still baithe hain aur sirf time flow karne dete hain.

Ab differentiate karo. Rule $\frac{d}{dt}(E^2)=2E\frac{dE}{dt}$ (vector version) deta hai:

$$\frac{\partial u}{\partial t}={\epsilon }_0\underset{\begin{array}{c}\text{E-arrow kitni tezi se}\\ \text{badh raha hai,}\\ \text{E ke along measure kiya}\end{array}}{\underbrace{\vec{E}\cdot \frac{\partial \vec{E}}{\partial t}}}+\frac{1}{{\mu }_0}\underset{\begin{array}{c}\text{B-arrow ke liye}\\ \text{same}\end{array}}{\underbrace{\vec{B}\cdot \frac{\partial \vec{B}}{\partial t}}}$$

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Step 3 — Maxwell se poochho ki fields actually kaise change hoti hain

KYA. $\partial \vec{E}/\partial t$ aur $\partial \vec{B}/\partial t$ ko Maxwell's equations mein se do se replace karo.

KYUN. Abhi tak $\partial u/\partial t$ fields time mein kaise change hoti hain ke terms mein likhi hai — lekin uska koi alag handle nahi hai hamare paas. Maxwell's laws wahi laws hain jo batate hain ki fields kaise evolve hoti hain, aur woh time-changes ko fields ki spatial twisting (curl) ke terms mein express karte hain. Unhein swap karna woh hai jo secretly geometry introduce karta hai — aur geometry mein hi ek flow direction chhup sakti hai.

Hum symbol $\nabla \times$ use karte hain (padho "curl"). Ise ek tiny paddlewheel ki tarah picture karo: $\nabla \times \vec{B}$ measure karta hai ki $\vec{B}$ ek point ke around kitna circulate karta hai — woh wheel ko kitna tezi se spin karega.

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Step 4 — Substitute karo aur dekho kya appear hota hai

KYA. Step 2 ke $\partial u/\partial t$ expression mein do Maxwell replacements daalo.

KYUN. Yeh pure substitution hai — lekin yeh "rate of stored energy" ko $\vec{E}$, $\vec{B}$, unke curls, aur $\vec{J}$ se built expression mein convert kar deta hai. $\vec{J}$ term work-on-charges term banega; curl terms flow term ka beej hain.

$$\frac{\partial u}{\partial t}=\vec{E}\cdot \left(\underset{={\epsilon }_0\partial \vec{E}/\partial t}{\underbrace{\frac{1}{{\mu }_0}\nabla \times \vec{B}-\vec{J}}}\right)+\frac{1}{{\mu }_0}\vec{B}\cdot \left(\underset{=\partial \vec{B}/\partial t}{\underbrace{-\nabla \times \vec{E}}}\right)$$

Expand aur group karo:

$$\frac{\partial u}{\partial t}=\underset{\text{do curl terms — mystery}}{\underbrace{\frac{1}{{\mu }_0}[\vec{E}\cdot (\nabla \times \vec{B})-\vec{B}\cdot (\nabla \times \vec{E})]}}-\underset{\begin{array}{c}\text{charges ko deliver ki gayi}\\ {\text{power, per m}}^3\end{array}}{\underbrace{\vec{J}\cdot \vec{E}}}$$

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Step 5 — Woh identity jo do curls ko ek flow mein package karti hai

KYA. Vector calculus ki ek line use karo: $\nabla \cdot (\vec{E}\times \vec{B})=\vec{B}\cdot (\nabla \times \vec{E})-\vec{E}\cdot (\nabla \times \vec{B})$

KYUN. Gaur karo right side hamare Step 4 ke bracket ka negative hai. Yeh identity woh ek tool hai jo do curl terms ko ek single vector ke divergence mein fold kar deta hai. Aur "divergence" — symbol $\nabla \cdot$ — literally matlab hai net outflow per unit volume: exactly woh leak jo humein chahiye thi.

Identity ko hamare bracket par apply karte hain:

$$\frac{1}{{\mu }_0}[\vec{E}\cdot (\nabla \times \vec{B})-\vec{B}\cdot (\nabla \times \vec{E})]=-\frac{1}{{\mu }_0}\nabla \cdot (\vec{E}\times \vec{B})$$

Us vector ko define karo jiska divergence abhi appear hua:

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Step 6 — Poynting's theorem padho

KYA. Packaged term ko wapas substitute karo. Bracket $-\nabla \cdot \vec{S}$ ban gaya.

KYUN. Ab har piece Step 1 ki bookkeeping sentence se, symbol for symbol, match karta hai.

Plain English mein rearrange karein: stored energy kam hoti hai ($-\partial u/\partial t$) exactly utni jitni flow away hoti hai ($\nabla \cdot \vec{S}$) aur utni jitni charges ko di jaati hai ($\vec{J}\cdot \vec{E}$). Kuch create ya destroy nahi hota.

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Step 7 — Edge case A: empty space, koi charges nahi

KYA. $\vec{J}=0$ set karo (vacuum — push karne ke liye koi charges nahi).

KYUN. Light wave empty space mein travel karti hai. Hume dikhana hoga ki theorem wahan bhi sense deta hai, warna poori picture suspect hai.

$$\frac{\partial u}{\partial t}+\nabla \cdot \vec{S}=0$$

Yeh ek pure continuity equation hai: energy na banti hai na kharch hoti hai, sirf move karti hai. Jo ek box se nikalti hai woh agli mein jaati hai — energy surf karti hai aage. Yeh degenerate case wahi hai jo Earth tak pahunchne wali sunlight par govern karta hai (dekho Intensity and amplitude of waves).

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Step 8 — Edge case B: ek plane wave, aur sign/geometry check

KYA. Ek plane wave mein specialize karo jahan $\vec{E}\perp \vec{B}$ aur dono travel direction ke across point karte hain, $E=cB$ ke saath.

KYUN. Hume geometry ke saare cases check karne hain: jab fields perpendicular hain (wave case) toh $|\vec{S}|$ kya hai, aur jab woh parallel ya zero hain (degenerate cases) toh kya hota hai? Ek formula jo stress-test nahi kar sakte, woh trustworthy nahi hai.

Kyunki $|\vec{E}\times \vec{B}|=EB\sin \theta$ aur yahan $\theta =90^{\circ }$ toh $\sin \theta =1$:

$$S=\frac{EB}{{\mu }_0}=\frac{E(E/c)}{{\mu }_0}=\frac{E^2}{{\mu }_{0}c}={\epsilon }_{0}c{E}^2$$

$c^2=1/({\mu }_0{\epsilon }_0)$ use karte hue, isliye $\frac{1}{{\mu }_{0}c}={\epsilon }_{0}c$.

$S$ ka time-average (ek oscillation ke over) intensity hai, $⟨{\cos }^2⟩=\frac{1}{2}$ use karte hue:

$$I=⟨S⟩=\frac{1}{2}{\epsilon }_{0}c{E}_0^2$$

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Ek-picture summary

Upar sab kuch, compress kiya hua: stored energy ki ek box se shuru karo, Maxwell se poochho ki fields kaise change hoti hain, ek vector identity use karo, aur nikalta hai ek conserved flow $\vec{S}$ jo — ek wave ke liye — wahan point karta hai jahan light jaati hai.

Recall Feynman retelling — poora walkthrough plain words mein

Socho ek tiny glass box space mein float kar rahi hai, jisme electric aur magnetic field-arrows thread ho rahe hain. Box ke andar stored energy hai, jaise till mein coins. Agar coins ki number kam ho, toh paisa kahin gaya — ya toh kisi wall se leak hua, ya kisi andar wale charge par kaam khareedne mein laga. Woh ek honest sentence hi poora theorem hai.

Ise math mein turn karne ke liye humne poochha: stored energy kitni tezi se change hoti hai? Woh $u$ ka derivative hai. Lekin hum akele uska jawaab nahi de sakte, toh humne Maxwell's laws se poochha ki fields khud kaise change hoti hain — aur Maxwell jawaab deta hai ki fields space mein kaise curl karti hain ke terms mein. Humne woh plug in kiya. Teenon pieces nikle: do twisty curl terms, aur ek clean term jo obviously "charges par kiya gaya kaam" hai. Phir vector calculus ki ek magic line ne do twisty terms ko ek nayi arrow ke outflow mein fold kar diya. Humne us arrow ko $\vec{S}$ naam diya — Poynting vector. Ise kabhi assume nahi kiya; woh isliye nikla kyunki energy conserved honi chahiye.

Aakhir mein humne corners check kiye: empty space (energy sirf aage surf karti hai), parallel fields (bilkul koi flow nahi, aur sahi bhi hai), aur ek real light wave (fields right angles par, flow maximal aur exactly wahan point karta hai jahan wave travel karti hai). Apni ungliyan $\vec{E}$ ke along rakho, $\vec{B}$ ki taraf curl karo, aur tumhara thumb wahan point karta hai jahan light — aur uski energy — ja rahi hai.