Foundations — Maxwell's equations — integral form, all four

1.8.31 · D1 · Physics › Electromagnetism › Maxwell's equations — integral form, all four

Is page par assume kiya gaya hai ki aapne kuch bhi nahi dekha. Hum har letter, arrow, aur symbol ko parent note mein use hone se pehle ground up se build karenge, har ek ko earn karke.

Layer 0 — Arrows jo direction carry karte hain: vectors

Ek plain number jaise $5$ sirf kitna batata hai. Lekin hawa, paani ka behna, aur electric push ki bhi ek direction hoti hai. Hum inhe arrow ki tarah draw karte hain: length amount hai, aur jis taraf point kare woh direction hai. Ek arrow-quantity ko vector kehte hain.

Hum ek vector ko upar ek chhote arrow ke saath likhte hain: $\vec{F}$ matlab "vector jiska naam $F$ hai." Parent note mein $\vec{E}$ (electric field), $\vec{B}$ (magnetic field), $\vec{A}$ (area, ek arrow ki tarah — aage explain hoga), aur $\vec{\ell }$ (curve ke saath ek tiny step) use hote hain.

• $\vec{E}$ = electric field — woh arrow jis taraf ek $+1$ test charge ko push feel hogi.
• $\vec{B}$ = magnetic field — woh arrow jis direction mein ek compass needle align hoti hai.

Layer 1 — Unit arrow: $\hat{r}$

Kabhi kabhi hume sirf direction chahiye, size nahi. Hat ka matlab hai "theek 1 length ka arrow us taraf point karta hua." To $\hat{r}$ ("r-hat") woh unit arrow hai jo radially outward point karta hai, kisi central point se seedha door.

Parent mein likha hai $\vec{E}=\frac{q}{4\pi {\epsilon }_0{r}^2}\hat{r}$: woh messy fraction size hai, aur $\hat{r}$ sirf kehta hai "...seedha bahar ki taraf."

Layer 2 — Do arrows ka multiplication: dot product $\vec{F}\cdot d\vec{A}$

$\vec{F}\cdot d\vec{A}$ hume har jagah milta hai. Do vectors ke beech dot ($\cdot$) ek special multiplication hai jo poochta hai: "yeh do arrows kitna ek hi direction mein point kar rahe hain?"

Layer 3 — Surfaces ko bhi arrows milte hain: $d\vec{A}$

Yeh woh trick hai jo flux ko kaam karwaati hai. Ek surface ko tiny flat patches mein kaato. Har patch ko apna arrow $d\vec{A}$ milta hai:

• uski length = patch ki tiny area,
• uski direction = patch se seedha bahar, perpendicular ("normal").

Chhota $d$ matlab "ek infinitesimally small piece." To $d\vec{A}$ = "area ka ek tiny piece, khud se seedha bahar ki taraf point karta hua."

Ek closed surface ke liye (ek sealed bag), $d\vec{A}$ hamesha outward point karta hai. Isliye "bahar jaane waali lines" positive count hoti hain aur "andar aane waali lines" negative — dekhein Gauss's Law.

Layer 4 — Saare pieces ko add karna: $\int$ aur $\oint$

Stretched-S symbol $\int$ ("integral") ka matlab sirf hai "tiny pieces ka ek continuous pile add karo." Kyunki humne surface ko infinitely many patches $d\vec{A}$ mein kaata, hum ordinary $+$ use nahi kar sakte; integral infinitely many infinitesimals ke liye grown-up plus sign hai.

Circle $\oint$ sirf ek reminder hai, kuch aur nahi: "yeh cheez khud par close hoti hai." Closed vs open mein galti karna parent ke mistakes list mein ek classic error hai.

Layer 5 — Flux $\Phi$: "kitna poke through karta hai" wala number

Ab hum pieces assemble karte hain. Har patch par field arrow $\vec{E}$ lo, use us patch ke $d\vec{A}$ se dot karo (us patch se guzarne ki amount milti hai), aur sab ko $\int$ se add karo:

${\Phi }_E$ matlab electric field ka flux; ${\Phi }_B$ matlab magnetic field ka flux. Subscript ka poora matlab yahi hai.

Layer 6 — Loop mein walk karna: $d\vec{\ell }$ aur circulation

"Swirl" laws (Faraday, Ampère) ke liye hum surface ko pierce nahi karte — hum ek closed loop $C$ ke around walk karte hain. Loop ko tiny steps mein kaato; har step ek arrow $d\vec{\ell }$ hai ("d-ell"): length = tiny step size, direction = jis taraf tum chal rahe ho.

$\vec{E}\cdot d\vec{\ell }$ poochta hai "field mujhe mere step ke saath kitna push kar raha hai?" Loop ke around saare steps add karo:

Layer 7 — Rate-of-change tool: $\frac{d}{dt}$

Faraday aur Ampère–Maxwell flux ki value se nahi balki flux kitni tezi se change hoti hai se care karte hain. Symbol $\frac{d\Phi }{dt}$ matlab hai "jaise jaise time $t$ ticks, $\Phi$ kitni rate se change hoti hai" — flux-vs-time graph ka slope.

$-d{\Phi }_B/dt$ mein minus sign Lenz's law hai: nature change ke khilaf push karta hai.

Layer 8 — Named constants

Inका combination speed of light deta hai: $c=1/\sqrt{{\mu }_0{\epsilon }_0}$ — woh punchline jo Electromagnetic Waves ko equations se nikaalti hai.

Parent mein use hone waale doosre letters:

• $q,Q$ = charge ki amount (small/enclosed).
• $I$ = current, charge flowing per second. $I_d$ = displacement current (dekhein Displacement Current) — moving charge nahi, balki ${\epsilon }_{0}d{\Phi }_E/dt$.
• $\lambda$ = charge per unit length; $r$ = center se distance; $s$ = side length; $L$ = ek length.
• $\text{EMF}$ = loop ke around charge per total push, $\oint \vec{E}\cdot d\vec{\ell }$.
• $\vec{E},\vec{B}$ worked example (C) mein ek capacitor ke plates/gaps par rehte hain.

Yeh foundations topic ko kaise feed karte hain

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