Visual walkthrough — Faraday's law — EMF = −dΦ - dt

1.8.26 · D2 · Physics › Electromagnetism › Faraday's law — EMF = −dΦ - dt

Hum kuch bhi assume nahi karte, sirf yeh: ek magnet ka field hota hai, aur ek charge ek aisi cheez hai jo pushes feel karta hai. Neeche har symbol pehle banaya jaata hai, phir use kiya jaata hai.

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Step 1 — Magnetic field ek picture ke roop mein kya hota hai?

KYA. Ek magnet apne aas-paas ke space ko ek invisible influence se bharta hai. Hum us influence ko arrows ke roop mein draw karte hain. Jahan arrows dense hain, wahan influence strong hai. Hum ise ek naam dete hain: $\vec{B}$ (B Faraday ka letter hai iske liye). Upar chhota sa arrow, $\stackrel{⃗}{}$, matlab hai yeh kisi direction mein point karta hai — sirf ek size nahi, direction bhi hai.

WHY arrows ka field, sirf ek number kyun nahi? Kyunki ek charge ko jo push milti hai woh depend karti hai ki influence kis taraf point kar raha hai, us charge ki motion ke relative. Ek plain number kabhi direction nahi bata sakta. Isliye humein arrows chahiye.

PICTURE. Neeche $\vec{B}$ seedha page ke bahar point kar raha hai — dots se draw kiya (arrow-tips tumhari taraf aa rahe hain). Hum ise uniform rakhenge (har jagah same) — simple aur honest rehne ke liye.

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Step 2 — Woh ek force jo sab kuch shuru karta hai

KYA. Us field mein ek single positive charge $q$ rakho aur use velocity $\vec{v}$ se move karo ($v$ = speed, arrow = travel ki direction). Nature use ek shove deta hai: $\vec{F}=q\vec{v}\times \vec{B}$

Har symbol ko wahin padhte hain jahan woh baitha hai:

• $q$ — kitna charge hai (zyada charge, zyada shove).
• $\vec{v}$ — kitna fast aur kis direction mein move kar raha hai. Motion nahi, force nahi ($v=0\Rightarrow F=0$).
• $\vec{B}$ — Step 1 ka field.
• $\times$ — cross product. Yeh ordinary multiplication nahi hai.

WHY cross product $\times$ aur plain $\cdot$ kyun nahi? Magnetic force strange hai: yeh sideways point karta hai — motion aur field dono ke perpendicular. Cross product exactly woh mathematical tool hai jiska output hota hai "do given directions ke perpendicular direction," size $qvB\sin \varphi$ ke saath ($\varphi$ = $\vec{v}$ aur $\vec{B}$ ke beech ka angle). Hum ise precisely isliye choose karte hain kyunki experiments dikhate hain ki force sideways hai. Jab $\vec{v}$ aur $\vec{B}$ perpendicular hain ($\varphi =90^{\circ }$, $\sin 90^{\circ }=1$), size simply $qvB$ hoti hai.

PICTURE. $\vec{v}$ right point karta hai, $\vec{B}$ page ke bahar, toh $\vec{F}=q\vec{v}\times \vec{B}$ page ke upar point karta hai. Woh "upar" hi hamara current drive karega.

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Step 3 — Charges se bhara ek rod battery ban jaata hai

KYA. Akele charge ki jagah ek metal rod le lo length $L$ ka jo speed $v$ se slide kar raha hai. Rod free charges se packed hai; har ek ko woh upward $qvB$ shove milta hai. Woh ek end ki taraf pile ho jaate hain — rod ab ek plus end aur ek minus end rakhta hai. Yahi exactly battery hai: ek device jo charge ko separate karta hai.

WHY ise EMF bolte hain aur work-per-charge compute karte hain? Battery ki strength ek charge par force nahi hai — yeh woh energy hai jo woh har unit charge ko deta hai jab woh charge usse cross karta hai. Yeh energy-per-charge EMF kehlaata hai (symbol $\mathcal{E}$, units volts). Toh hum poori length $L$ mein ek charge ko push karne mein kiya gaya work compute karte hain:

$\mathcal{E}=\frac{W}{q}=\frac{(\stackrel{\text{force}}{\overbrace{qvB}})\stackrel{\text{distance}}{\overbrace{L}}}{q}=BvL$

Padho ise: force $qvB$ times distance $L$ deta hai work $qvBL$; $q$ se divide karo aur charge cancel ho jaata hai, bachta hai $\mathcal{E}=BvL$. Charge cancel hota hai kyunki har charge same push feel karta hai — EMF rod ki property hai, kisi ek particle ki nahi.

PICTURE. Rod as a battery: $+$ top par collect, $-$ bottom par, EMF $BvL$ uske across.

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Step 4 — Wahi rod, swept area ke roop mein dekha

KYA. Rod ko do rails par rakho taaki woh ek loop close kare. Jab yeh thodi si time $dt$ mein right ki taraf thodi si distance $vdt$ slide karta hai, toh woh naya area sweep out karta hai. Jo strip sweep hoti hai woh ek rectangle hai: height $L$, width $vdt$.

$dA=L\cdot (vdt)⟹\frac{dA}{dt}=Lv$

Padho ise: $dA$ woh tiny naya area hai, $L$ uski height, $vdt$ uski width. Tiny time $dt$ se divide karo aur milta hai area grow hone ki rate: $Lv$.

WHY area rate ke baare mein baat karte hain? Kyunki ab hum bilkul alag sawaal pooch sakte hain — "charges ko kya force feel hoti hai?" nahi, balki "loop mein kitna field thread ho raha hai, aur woh amount kitni fast badh raha hai?" Hum abhi dikhane wale hain ki in dono sawaalon ka ek hi jawaab hai.

PICTURE. Rod ki purani aur nayi position ke beech grey swept strip, uski dimensions labelled ke saath.

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Step 5 — Flux: loop mein se field-lines count karna

KYA. Magnetic flux ${\Phi }_B$ define karo = "loop mein kitna field thread karta hai." Ek flat loop of area $A$ ke liye uniform field mein, field loop ke perpendicular ho toh: ${\Phi }_B=BA$

Zyada generally field loop ko slant par hit kar sakta hai. Hum us slant ko $\theta$ se measure karte hain, $\vec{B}$ aur loop ki normal $\hat{n}$ (ek unit arrow jo loop ke face se seedha bahar nikaalta hai) ke beech ka angle. Toh: ${\Phi }_B=BA\cos \theta$

Har symbol padho:

• $B$ — field strength.
• $A$ — loop area.
• $\hat{n}$ — outward-pointing normal arrow (isi se $\theta$ measure hota hai).
• $\theta$ — $\vec{B}$ aur $\hat{n}$ ke beech tilt.
• $\cos \theta$ — sirf $\vec{B}$ ka woh part rakhta hai jo actually loop ke through poketa hai.

WHY $\cos \theta$ aur $\sin \theta$ kyun nahi? Sirf $\vec{B}$ ka component normal ke along surface ko thread karta hai. Field jo sideways face ke along slide kare, woh kuch nahi thread karta. Jab $\vec{B}$ $\hat{n}$ ke parallel ho ($\theta =0$, $\cos 0=1$) toh poora thread hota hai — maximum flux. Jab $\vec{B}$ loop ke plane mein flat ho ($\theta =90^{\circ }$, $\cos 90^{\circ }=0$) toh kuch bhi thread nahi hota — zero flux. $\cos$ exactly woh function hai jo $0^{\circ }$ par 1 aur $90^{\circ }$ par 0 hota hai: yeh perpendicular part extract karta hai.

PICTURE. Same loop teen tilts par — $\theta =0$ (full), $\theta =60^{\circ }$ (partial), $\theta =90^{\circ }$ (empty) — threading lines drawn ke saath.

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Step 6 — Do pictures takraati hain: $BvL=\frac{d{\Phi }_B}{dt}$

KYA. Hamare rail loop mein $\vec{B}\perp$ loop, toh $\theta =0$, $\cos \theta =1$, aur ${\Phi }_B=BA$. Field $B$ fixed hai; sirf area badh raha hai. Differentiate karo (flux ke rate of change ko measure karo): $\frac{d{\Phi }_B}{dt}=B\frac{dA}{dt}=B(Lv)=BLv$

Ab Step 3 se compare karo, jahan force picture ne diya tha $\mathcal{E}=BvL$. Woh same number hai: $\boxed{\mathcal{E}=\frac{d{\Phi }_B}{dt}=BLv}$

WHY yeh poora miracle hai? Humne ek quantity ko do tareekon se compute kiya jo unrelated lagte the — charges ko push karna (Step 3) versus threading lines count karna (Step 6) — aur identical answers mile. Faraday ki leap: yeh equality sliding rods ka coincidence nahi hai. Yeh tab bhi hold karta hai jab kuch move nahi karta — agar $B$ khud time mein change kare, flux change hota hai aur EMF appear hota hai bina kisi moving charge ke jo push kare. Toh hum result ko sirf flux ke baare mein ek law mein promote karte hain.

PICTURE. Do side-by-side panels — "force story" aur "flux story" — same box $\mathcal{E}=BLv$ par milte hue.

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Step 7 — Minus sign kahan se aata hai

KYA. EMF jo current drive karta hai woh apna khud ka chhota sa field banata hai. Woh kis taraf point karta hai? Suppose karo flux page ke bahar se increasing hai (rod right move kar raha hai, area badh raha hai). Induced current is tarah flow karta hai ki uska khud ka field loop ke andar page ke andar point kare — increase ko fight karte hue.

$\mathcal{E}=-\frac{d{\Phi }_B}{dt}$

Minus sign is opposition ka bookkeeper hai.

WHY oppose karna zaroori hai (aur help nahi)? Energy conservation. Agar induced current flux ko help kare badhne mein, toh woh stronger flux aur zyada current drive karta, hamesha ke liye — free infinite energy. Impossible. Toh nature hamesha push back karta hai: yahi hai Lenz's Law. Minus sign encode karta hai "induced effect apne khud ke cause ko resist karta hai."

PICTURE. Growing out-of-page flux (orange), induced current (teal loop) circulate karta hai page-into-page field banane ke liye jo ise oppose kare.

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Step 8 — Teen tareekon se flux change ho sakta hai (har case cover karta hai)

KYA. Flux ek product hai ${\Phi }_B=BA\cos \theta$. Iske teen ingredients mein se koi bhi move kar sakta hai. Calculus ka product rule rate of change ko teen alag causes mein split karta hai: $\frac{d{\Phi }_B}{dt}=\underset{\text{field changes}}{\underbrace{\frac{dB}{dt}A\cos \theta }}+\underset{\text{area changes}}{\underbrace{B\frac{dA}{dt}\cos \theta }}+\underset{\text{loop rotates}}{\underbrace{-BA\sin \theta \frac{d\theta }{dt}}}$

WHY product rule? Jab teen cheezein multiply hoti hain aur sab vary kar sakti hain, product ki rate hoti hai "har ek change ho rahi hai baaki fixed rahate hue, add up karke." Yahi exactly product rule hai — sahi tool kyunki $\Phi$ ek product hai.

Teen cases padho:

• Field changes ($\frac{dB}{dt}\ne 0$): ek fixed loop ek strengthening field mein. Yeh transformer / Inductance idea hai.
• Area changes ($\frac{dA}{dt}\ne 0$): hamara sliding rod, Steps 3–6.
• Rotation ($\frac{d\theta }{dt}\ne 0$): loop spin karta hai, $\theta =\omega t$, deta hai $\sin \omega t$ jo har AC generator mein hota hai — Electric Generators and AC.

Degenerate case — sab constant. Agar $B$, $A$, aur $\theta$ sab fixed hain, toh teeno terms zero hain: $\mathcal{E}=0$. Ek bahut bada constant field kuch bhi induce nahi karta. Value matter nahi karta; sirf change matter karta hai.

PICTURE. Teen mini-panels — ek per knob — har ek apne change ke arrow ke saath.

Recall Khud check karo

Ek loop perfectly still baitha hai ek huge, unchanging field mein. EMF? ::: Zero — $d\Phi /dt$ ka har term vanish ho jaata hai; sirf change EMF banata hai. Kaun sa term power-station generator chalata hai? ::: Rotation term $-BA\sin \theta \frac{d\theta }{dt}$.

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Ek-picture summary

Is page ki sab cheez ek single flow mein collapse hoti hai: ek moving charge $q\vec{v}\times \vec{B}$ feel karta hai → charges pile up hote hain → rod ek battery ban jaata hai EMF $BvL$ ka → woh equal hai us rate ke jis par flux growing loop ko thread karta hai → Faraday ise kisi bhi flux change tak promote karta hai → Lenz's minus sign ise oppose karta hai.

Recall Feynman retelling — poora walkthrough plain words mein

Socho invisible field-arrows page se bahar nikal rahe hain. Unse ek metal rod slide karo. Rod mein har free charge move kar raha hai, aur field mein moving charges sideways shove paate hain — toh charges rod ke ends par pile up ho jaate hain aur rod ek chhoti si battery ban jaati hai. Compute karo kitni energy har charge ko rod cross karte waqt milti hai: aata hai $B$ times speed times length, $BvL$. Ab charges bhool jao aur sirf woh area dekho jo rod slide karte waqt sweep karta hai — bada area matlab zyada field-arrows loop ko thread karte hain, aur woh "threading amount" flux kehlata hai. Flux ke badhne ki rate exactly $BvL$ bhi nikalta hai. Same number, do bilkul alag stories! Faraday ne realize kiya ki flux story zyada deep hai — yeh tab bhi kaam karta hai jab rod ko still rakho aur bajaye iske field khud badhao ya ghataao, ya loop ko tilt karo. Finally: loop grumpy hai. Tum uske flux ke saath jo bhi karo, woh jo current banata hai woh fight back karta hai cheezein same rakhne ke liye — wahi minus sign hai, aur yeh sirf energy conservation hai jo tumhe free electricity nahi dene deta.

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Connections

• Motional EMF — Steps 2–3, force origin
• Magnetic Flux — Step 5, woh quantity jo differentiate hoti hai
• Lenz's Law — Step 7, minus sign
• Electric Generators and AC — Step 8, rotation term
• Inductance — Step 8, self-induced EMF from $dB/dt$
• Maxwell's Equations — Faraday's law as $\nabla \times \vec{E}=-\partial \vec{B}/\partial t$
• Hinglish version