Visual walkthrough — Self-inductance L, mutual inductance M

1.8.28 · D2 · Physics › Electromagnetism › Self-inductance L, mutual inductance M

Yeh deep-dive parent topic ko expand karta hai. Jo tools hum borrow karenge — ek per step — woh hain Ampère's law, Magnetic flux, aur Faraday's law of induction, sign ke liye Lenz's law ke saath.

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Step 0 — Object se Milo aur Uske Har Symbol ko Pehchano

KYA HAI. Ek solenoid bas ek wire hai jo tight helix mein wind ki gayi hai — circular loops ka ek stack. Hum char cheezein label karte hain aur kuch nahi:

• $\ell$ — coil ki length, metres mein (stack kitni door tak jaata hai).
• $N$ — turns ki ginti (kitne loops wind kiye), ek pure count.
• $A$ — ek loop ka cross-sectional area, ${\text{m}}^2$ mein (woh flat disc jiske through field jaati hai).
• $I$ — wire mein flow hone wala current, amperes mein.

KYUN. Yeh charon sirf woh geometric facts hain jo hum control karte hain. Baaki sab — field, flux, inductance — inhi se compute hoga. Inhi se hum ek convenience symbol bhi banate hain:

PICTURE. Green loops hain turns; blue axis hai length $\ell$; yellow disc hai area $A$.

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Step 1 — Current ek Magnetic Field ($B$) Banata Hai

KYA HAI. Wire mein current $I$ daalo. Har loop ek tiny magnet ban jaata hai. Tight stack hone par, unke fields andar add up hote hain aur bahar cancel ho jaate hain, axis ke along seedha point karta hua ek uniform field bachta hai. Uski strength ko $B$ kaho (unit: tesla).

KYUN yeh tool — Ampère's law? Hume "current" ko "field" mein convert karna hai, aur Ampère's law exactly wahi machine hai jo yeh karta hai: yeh kehta hai ki ek chosen loop ke around circulate karne wala field, us loop se thread hone wale current ka ${\mu }_0$ guna hota hai. Hum ek rectangular path ("Amperian loop") choose karte hain jiska ek lamba side solenoid ke andar ho aur ek bahar jahan $B=0$ ho. Is choice se algebra ek single term mein collapse ho jaata hai.

$\underset{\text{andar ka field}}{\underbrace{B}}\underset{\text{andar ki side ki length}}{\underbrace{L_{\text{path}}}}={\mu }_0\underset{\text{turns jo path enclose karta hai}}{\underbrace{(n{L}_{\text{path}})}}\underset{\text{har turn mein current}}{\underbrace{I}}$

Path length $L_{\text{path}}$ dono sides par aata hai aur cancel ho jaata hai — isliye rectangle trick kaam karti hai:

PICTURE. Dashed red rectangle hai Amperian loop; blue arrows hain andar ka uniform axial field; bahar, field zero hai.

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Step 2 — Ek Loop se Guzarta Field Flux ($\Phi$) Ban Jaata Hai

KYA HAI. "Flux" $\Phi$ count karta hai ki kitna field ek loop ki disc se guzarta hai. Field lines ko baarish samjho aur loop ko ek hoop: flux hai ki kitni raindrops hoop se guzarti hain.

KYUN yeh tool — Magnetic flux? Faraday's law (Step 4 mein aane wala) flux ki parwah karta hai, field ki nahi. Toh hume $B$ (field per area) ko $\Phi$ (puri disc se total field) mein convert karna hoga Faraday ke kaam karne se pehle. General rule hai $\Phi =\int \vec{B}\cdot d\vec{A}$, lekin yahan $\vec{B}$ uniform hai aur seedha disc se guzar raha hai, toh integral ek seedhi multiplication ban jaati hai:

$\Phi =\underset{\text{field strength}}{\underbrace{B}}\times \underset{\text{disc area}}{\underbrace{A}}={\mu }_0\frac{N}{\ell }IA$

PICTURE. Yellow disc hai loop; blue arrows seedhe us se guzar rahe hain; shaded yellow region hai flux $\Phi =BA$.

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Step 3 — Saare $N$ Loops Flux Share Karte Hain: Linkage $N\Phi$

KYA HAI. Wahi field line jo pehle loop se guzarti hai har loop se guzarti hai, kyunki loops ek hi axis par stack hain. Toh field "$N$ baar linked" hai. Flux linkage total hai, $N\Phi$.

KYUN. Faraday's law poore circuit mein se flux count karta hai, aur circuit yahan ek wire hai jo $N$ baar loop karta hai. Har loop ek turnstile ki tarah hai jisse usi field line ko guzarna hai; $N$ turnstiles ke saath, field line $N$ baar count hoti hai.

$N\Phi =\underset{\text{linked loops}}{\underbrace{N}}\times \underset{\Phi \text{ from Step 2}}{\underbrace{{\mu }_0\frac{N}{\ell }AI}}={\mu }_0\frac{N^2}{\ell }AI$

Dhyan do: loops count karne wala pehla $N$, $\Phi$ ke andar chupta doosra $N$ se milta hai (kyunki Step 1 mein zyada loops ne zyada strong field banaya). Do $N$ multiply hote hain → $N^2$. Yahi hai famous square ka birthplace.

PICTURE. Ek red field line saare green loops mein se snaking karti hai; ek counter tik karta hai $1,2,\dots ,N$ jab woh har ek ko pierce karta hai.

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Step 4 — Flux-per-amp ko Naam Do: Woh Hai $L$

KYA HAI. Last line dekho: $N\Phi$ bas ek constant times $I$ hai. $I$ ke alawa sab kuch fixed geometry hai. Us constant ko hum ek naam dete hain, $L$.

Naam kyun dena? Taaki hume kabhi Steps 1–3 dobara na chalane padein. Ek baar $L$ pata lag jaaye, toh kisi bhi current ka flux linkage turant $LI$ ho jaata hai. Yeh pure bookkeeping hai — coil par stamped ek "flux per amp" price tag.

PICTURE. Straight-line graph $N\Phi$ versus $I$: uska slope hi $L$ hai.

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Step 5 — Kyun Care Karna: $L$ Back-EMF Ban Jaata Hai

KYA HAI. Ab $I$ ko time mein change hone do. Kyunki $N\Phi =LI$ aur $L$ fixed hai, changing current matlab changing flux linkage — aur Faraday's law of induction use ek induced voltage (EMF), $\epsilon$, mein badal deta hai.

KYUN Faraday's law of induction yahan? Yeh sirf wahi law hai jo kehta hai "changing flux → voltage." Hum $N\Phi =LI$ ko time ke respect mein differentiate karte hain. Kyunki $L$ ek constant hai (geometry move nahi karti), yeh derivative se bahar aa jaata hai:

$\epsilon =-\frac{d(N\Phi )}{dt}=-\frac{d(LI)}{dt}=-L\frac{dI}{dt}$

• $\frac{dI}{dt}$ — current ki rate of change (amps per second), current khud nahi.
• Minus sign hai Lenz's law: induced EMF change se ladta hai. Current badh raha hai? EMF pushback deta hai use slow karne ke liye. Current gir raha hai? EMF aage dhakelta hai use sustain karne ke liye.

PICTURE. Upar: ek current ramp $I(t)$. Neeche: resulting EMF — ramp ke dauran ek flat step, current level hone par zero, current mudne par sign flip.

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Step 6 — Edge & Degenerate Cases (Reader ko Kabhi Stranded Mat Chodo)

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Ek-Picture Summary

Sab kuch ek canvas par: current field banata hai (Ampère), field flux banata hai (integrate karo), flux $N$ baar linked hota hai, slope $L$ hai, aur use change karne se back-EMF banta hai (Faraday–Lenz).

Recall Poore walkthrough ki Feynman-style retelling

Wire ko loops ki ek tube mein wind karo. Current daalo — Ampère kehta hai andar ek seedha field ughta hai, strong agar loops tight packed hain. Woh field har loop ki opening se guzarta hai; ek loop se kitna guzra, yeh count karna flux hai. Lekin wahi field saare loops se guzarta hai, toh hum use $N$ baar count karte hain — yeh hai linkage. Kyunki zyada loops pack karne se field dono strong hota hai aur zyada link hota hai, hume $N$ ke do factors milte hain, isliye $N^2$. Ab, linkage hamesha same constant times current hota hai, aur us constant ko hum $L$ naam dete hain taaki hume counting dobara nahi karni pade. Finally, agar tum current badlane ki koshish karo, toh flux badalti hai, aur Faraday tumhe ek voltage deta hai jo (Lenz ki zimmedaari hai) change se ladta hai — ek bhaari current-flywheel jo speed up ya slow down hone se nafrat karta hai. Yahi hai self-inductance, bilkul kuch nahi se draw kiya gaya.

Recall Quick self-test

Do factors of $N$ kahan se aate hain? ::: Ek $N$ turns ko link karne ki counting se (Step 3), ek field ke $N$ guna zyada strong hone se kyunki $N$ turns use bana rahe hain (Step 1). $I$ final $L$ se kyun gayab ho jaata hai? ::: $L=N\Phi /I$ aur $N\Phi$ already $I$ ke proportional hai, toh $I$ cancel ho jaata hai — $L$ pure geometry hai. $\epsilon =-LdI/dt$ mein minus sign kyun hai? ::: Lenz's law — induced EMF us current ke change ka virodh karta hai jisne use cause kiya.

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Connections

• Ampère's law — Step 1, current ko field mein badalta hai.
• Magnetic flux — Steps 2–3, linked quantity.
• Faraday's law of induction — Step 5, changing flux → EMF.
• Lenz's law — minus sign.
• Energy stored in magnetic field — aage $\frac{1}{2}L{I}^2$ kahan se aata hai.
• Transformers aur RL circuits — jahan $L$ aur $M$ use hote hain.