Visual walkthrough — Maxwell's equations — integral form, all four

1.8.31 · D2 · Physics › Electromagnetism › Maxwell's equations — integral form, all four

Neeche sab kuch assume kiya gaya hai ki tum fresh start kar rahe ho. Har arrow, loop, aur symbol ko use karne se pehle define kiya jayega.

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Step 1 — "Loop" aur uski "cap surface" hoti kya hain

KYA HAI. Ek flat wire imagine karo jisme steady current beh rahi hai, aur uske around ek closed loop draw karo — ek rubber band $C$ jo space mein float kar rahi hai, kisi cheez ko touch nahi kar rahi. Ab us loop ke across ek soap film stretch karo. Woh film ek surface $S$ hai jiska edge bilkul wahi loop $C$ hai.

KYUN. Poora chautha equation ek loop aur us loop ki border karne wali surface ke baare mein ek statement hai. Koi bhi symbol likhne se pehle hume bilkul clear hona chahiye: loop boundary hai, surface woh "cap" hai jo tum uske across stretch kar sakte ho — aur ek loop ke liye infinitely many aisi caps hoti hain. Yahi freedom poori story ki jad hai.

PICTURE. Figure s01 dekho: pale-yellow rubber band loop $C$ hai; usse bulge hoti blue film ek valid cap surface $S$ hai. Chhota arrow $d\vec{\ell }$ loop ke saath chalta hai; arrow $d\vec{A}$ film se bahar nikal raha hai.

Right-hand rule dono ko ek saath jodata hai: apni right hand ki ungliyan $d\vec{\ell }$ ke direction mein curl karo, aur thumb $d\vec{A}$ ka direction dega. Step 2 mein iska zaroorat padegi.

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Step 2 — "$\vec{B}$ ki circulation" kya measure karti hai

KYA HAI. Loop $C$ ke around magnetic field $\vec{B}$ ho sakta hai (har point par ek arrow jo batata hai ki compass needle kidhar mudegi). Hum step by step jod rahe hain ki kitna $\vec{B}$ hamare marching direction ke saath align hai. Yeh running total hi circulation hai.

KYUN. Ampère ne notice kiya ki current $\vec{B}$ ko uske circles mein wrap kar deta hai. Toh "kya is loop se koi current thread ho rahi hai?" detect karne ka natural tarika yeh hai ki poocho "kya $\vec{B}$ loop ke saath swirl karta hai?" — yahi precisely circulation hai. Hum dot product $\vec{B}\cdot d\vec{\ell }$ use karte hain, sirf $|\vec{B}|$ nahi, kyunki sirf march ke direction mein $\vec{B}$ ki component count hoti hai; jo $\vec{B}$ loop ko sideways cross kare woh koi swirling nahi karta.

PICTURE. s02 mein blue field arrows current wire ke around wrap ho rahe hain. Har red step $d\vec{\ell }$ par hum sirf $\vec{B}$ ka woh hissa rakhte hain jo step ke saath lie karta hai (pink mein projection dikhaya gaya hai).

Recall Sirf plain multiplication kyun nahi, dot product kyun?

Dot product $\vec{B}\cdot d\vec{\ell }$ equals $|\vec{B}||d\vec{\ell }|\cos \theta$ hai ::: yeh automatically $\vec{B}$ ka sideways part throw away kar deta hai (jab $\theta =90^{\circ }$, $\cos \theta =0$), sirf loop ke saath swirl bachta hai.

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Step 3 — Purana Ampère law, aur flat cap jo "kaam karta hai"

KYA HAI. Ek straight wire mein current $I$ ke liye, loop $C$ ko wire ke around radius $r$ ka circle choose karo aur cap $S$ ko woh flat disc banao jise wire pierce karta hai. Purana Ampère kehta hai: $\vec{B}$ ki circulation ${\mu }_0$ times us current ke barabar hai jo cap ko stab karta hai.

KYUN. Yeh sirf Step 2 ko ek law mein convert karna hai. Current flat disc ko ek baar pierce karta hai, toh "current through $S$", likhte hain $I_{enc}$, equals $I$ hai. Sab consistent hai — is cap ke liye.

PICTURE. s03: flat blue disc yellow current arrow $I$ se ek baar stab hua hai. Fingers $C$ ke saath curl karti hain, thumb $I$ se agree karta hai.

Woh phrase yaad rakho — "chosen cap". Yeh abhi humein kaatne wala hai.

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Step 4 — Contradiction: cap ko capacitor gap mein bulge karo

KYA HAI. Wire ko kaato aur ek capacitor lagao — do parallel plates jiske beech ek chhota vacuum gap hai (ek capacitor). Current phir bhi wire mein flow karta hai, plates ko charge karta hai. Same loop $C$ wire ke around rakho, lekin ab ek bulging cap $S^{\prime }$ choose karo jo flat disc ki jagah plates ke beech slip kare.

KYUN. Yahi crisis hai. Flat cap ke liye, current $I$ stab karta hai → purana Ampère deta hai ${\oint }_C\vec{B}\cdot d\vec{\ell }={\mu }_{0}I$. Bulging cap ke liye jo gap mein dip karta hai, koi charge gap cross nahi karta — toh $I_{enc}=0$ → purana Ampère deta hai ${\oint }_C\vec{B}\cdot d\vec{\ell }=0$. Same loop, do alag answers! Ek law ek physical cheez ke liye do answer nahi de sakta.

PICTURE. s04: ek yellow loop $C$, do caps — flat blue disc jise current pierce karta hai, aur pink balloon cap jo empty gap mein slide karta hai, kisi cheez se pierce nahi hota.

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Step 5 — Gap mein actually kya thread ho raha hai: ek changing electric field

KYA HAI. Plates ke beech koi charge flow nahi hai — lekin wahan electric field $\vec{E}$ zaroor hai, $+$ plate se $-$ plate ki taraf point karta hua, aur jaise jaise charge piles up hota hai yeh grow kar raha hai. Bulging cap is growing $\vec{E}$ se pierce ho raha hai.

KYUN. Agar kuch "current-jaisa" bulging cap ko thread nahi karta, toh Ampère doomed hai. Maxwell ki leap: cap ke through electric flux measure karo — $\vec{E}$ ki total amount jo poke through kare — aur dekho woh time mein kaise change hota hai. Woh rate of change hi missing ingredient hai.

PICTURE. s05: pink $\vec{E}$ arrows gap fill kar rahe hain, aur bulging cap ke through flux ${\Phi }_E$ grow karta hai jaise plates par charge $Q$ grow karta hai.

Note karo ki plate area $A$ cancel ho gaya: ${\Phi }_E=Q/{\epsilon }_0$ sirf charge par depend karta hai, geometry par nahi. Yeh clean cancellation hint hai ki yeh term exactly wahi hai jo humein chahiye.

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Step 6 — Maxwell ki fix: displacement current

KYA HAI. Us flux ka time-derivative lo aur ${\epsilon }_0$ se multiply karo. Result ko displacement current $I_d$ kaho. $I_d$ ko Ampère's law mein add karna dono caps ko agree karwa deta hai.

KYUN. Humein ek aisa term chahiye jo bulging cap par $I$ ke barabar ho (jahan real current $0$ hai) aur flat cap par $0$ ho (jahan $\vec{E}$ negligible hai, real current kaam karta hai). Electric flux ke change ki rate precisely wahi term hai — algebra dekho: kyunki ${\Phi }_E=Q/{\epsilon }_0$, $I_d={\epsilon }_0\frac{d{\Phi }_E}{dt}={\epsilon }_0\cdot \frac{1}{{\epsilon }_0}\frac{dQ}{dt}=\frac{dQ}{dt}=I.$ Gap ke through changing $\vec{E}$ wire current ke exactly same number carry karta hai. Current "conserved" hai — yeh seamlessly real charge flow se field-change ko hand off karta hai.

PICTURE. s06: wire current $I$ (yellow) plate mein flow in karta hai; gap ke andar pink "displacement current" $I_d$ baton uthata hai identical value $dQ/dt$ ke saath, toh dono caps ke across total threading continuous hai.

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Step 7 — Edge & degenerate cases (kabhi gap mat chhodna)

KYA HAI. Woh corners check karo jahan cheezein toot sakti hain.

KYUN. Ek law ko har scenario mein survive karna chahiye, sirf friendly wale mein nahi.

PICTURE. s07 teen mini-cases side by side stack karta hai.

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Ek-picture summary

s08 pura walkthrough compress karta hai: ek loop, do caps, wire current gap ke across displacement current ko hand off karta hua, aur resulting swirl of $\vec{B}$ jo same hai chahe koi bhi cap choose karo.

Recall Feynman: pura walkthrough plain words mein

Ek rubber band imagine karo jo ek wire ke around float kar rahi hai, aur uske across ek soap film stretched hai — tum film ko jis tarah chaaho bulge kar sakte ho. Purana rule kehta tha: "film ko stab karne wali current count karo, woh batayegi ki magnetic field band ke around kitna swirl karta hai." Flat film ke liye theek hai. Lekin wire kaato aur do metal plates daalo ek tiny gap ke saath. Ab agar main apni film gap mein bulge karun, koi current use stab nahi karta — phir bhi band ke around magnetic field swirl kar raha hai! Do films, do alag answers — yeh ho nahi sakta. Maxwell ne notice kiya ki gap empty nahi hai: jaise plates charge hoti hain, wahan ek electric field grow kar raha hai. Usne measure kiya ki us electric field ka "poke-through" kitni tezi se grow karta hai, sahi constants se multiply kiya, aur — magic — woh exactly wire current ke barabar nikla. Toh growing electric field gap ke across baton uthata hai ek invisible current ki tarah. Woh term add karo aur har film agree karta hai. Aur bonus prize: truly empty space mein, sirf ek changing electric field ek magnetic field swirl kar sakta hai, jo ek electric field swirl karta hai, jo... aur woh wave ban jaati hai. Woh wave light hai.

Recall Quick self-test

Ek charging capacitor mein bulging cap par real current $I_{enc}$ kya hai? ::: Zero — koi charge gap cross nahi karta. Law abhi bhi ${\mu }_{0}I$ de sake, uske liye kya replace karta hai? ::: Displacement term ${\mu }_0{\epsilon }_{0}d{\Phi }_E/dt$, jo ${\mu }_{0}I$ ke barabar hai. Jab capacitor fully charged ho jata hai, $I_d$ ka kya hota hai? ::: Woh zero ho jata hai kyunki $dQ/dt=0$; law purane Ampère tak reduce ho jaata hai.