Foundations — Faraday's law — EMF = −dΦ - dt

1.8.26 · D1 · Physics › Electromagnetism › Faraday's law — EMF = −dΦ - dt

Yeh Faraday's law ka toolbox page hai. Parent note mein $\vec{B}$, $\hat{n}$, $\cos \theta$, dot products, derivatives, aur $\mathcal{E}$ aur ${\Phi }_B$ jaise letters ko aise use kiya gaya hai jaise aap pehle se jaante ho. Yahan hum ek-ek symbol earn karte hain, ek aisi order mein jahan har symbol sirf unhi symbols par rely karta hai jo pehle aa chuke hain.

1. Arrows jo ek direction carry karte hain: vectors

Arrows ki zaroorat kyun hai? Kyunki magnetic field sirf "kitna strong" nahi hai — uski ek "kis taraf" bhi hoti hai. Seedha loop ke through point karne wala field kuch alag karta hai, aur sideways phisalta hua field kuch alag. Ek akela number dono mein farak nahi bata sakta; arrow bata sakta hai.

2. Magnetic field arrow: $\vec{B}$

Hum $\vec{B}$ ko track karte hain kyunki yeh poori kahani mein cause hai. Faraday's law jo kuch bhi karta hai, woh $\vec{B}$ (aur loop) ko change hote dekh ke karta hai.

3. Loop aur uski area: $A$ aur normal $\hat{n}$

Lekin 3-D space mein baithi ek flat surface ki ek facing direction bhi hoti hai — woh "dekh" kis taraf rahi hai? Woh direction ek special chhote arrow se capture hoti hai.

$\hat{n}$ kyun banate hain? Kyunki yeh poochhne ke liye ki "kya field loop ke through poke kar rahi hai ya along slide kar rahi hai?" hume $\vec{B}$ se kuch compare karna hoga. Normal "through-ness" ka honest representative hai: agar $\vec{B}$ $\hat{n}$ ke saath line up kare, field seedha through poke karti hai; agar $\vec{B}$ $\hat{n}$ se right angles par ho, field surface ko skim karti hai aur kuch thread nahi karti.

4. Angle $\theta$ aur $\cos$ kyun aata hai

Ab humhe ek aisa tool chahiye jo is angle ko ek number mein badal sake jo bataaye ki field ka kitna hissa seedha through jaata hai. Woh tool cosine hai.

Cosine kyun, kuch aur kyun nahi? Hum ek projection question pooch rahe hain: "is tilted field mein se, kitna through-direction $\hat{n}$ ke along lie karta hai?" Kisi direction par projection exactly wahi hai jo cosine compute karta hai. Sine iska ulta poochhega ("kitna sideways lie karta hai"), jo kuch thread nahi karta — toh cosine sahi tool hai.

Har case dekho taaki baad mein kuch surprise na kare:

Woh negative case matter karta hai: isi se loop bata paata hai "field front se enter ho rahi hai" ya "field back se enter ho rahi hai," jo baad mein induced current ki direction ban jaata hai.

5. Dot product: $\vec{B}\cdot d\vec{A}$

Parent flux ko ${\int }_S\vec{B}\cdot d\vec{A}$ likhta hai. Woh dot dot product hai, aur yeh sirf §4 ka projection idea hai jo do arrows ke liye package kiya gaya hai.

Yahan $\vec{A}$ ka matlab hai "area vector": ek arrow jiska length area $A$ hai aur jiska direction normal $\hat{n}$ hai. Toh $\vec{B}\cdot \vec{A}$ literally hai "field strength $\times$ area $\times$ (kitna aligned hain)" — exactly wahi jo hum flux ka matlab chahte hain. ${\int }_S$ (ek integral) ka matlab sirf hai "poori surface par in chhote contributions ko add karo"; ek uniform field mein flat loop ke liye yeh ek saaf product, $BA\cos \theta$, mein collapse ho jaata hai.

6. Magnetic flux: ${\Phi }_B$

Ab hum star quantity assemble kar sakte hain.

$BA\cos \theta$ mein har symbol ab kuch aisa hai jo aapne build kiya hai: $B$ field ki strength (§2), $A$ loop ka area (§3), $\cos \theta$ alignment fraction (§4). Yeh woh quantity hai jise Faraday's law watch karta hai. Deep-dive: Magnetic Flux.

7. Time ke saath change: derivative $\frac{d{\Phi }_B}{dt}$

Faraday's law flux ke baare mein nahi hai — yeh flux ke changing ke baare mein hai. Toh humhe "rate of change" ke liye ek symbol chahiye.

Derivative kyun, sirf ek difference kyun nahi? Ek magnet ko smoothly push kiya ja sakta hai, speed up aur slow down hote hue. Hum har moment par instantaneous rate chahte hain, sirf "total kitna change hua" nahi. Derivative woh tool hai jo ek single instant par exact slope deta hai. Chhote $d$ ka matlab hai "infinitely tiny change" — toh $d{\Phi }_B/dt$ ek tiny flux change divided by tiny time jo usme laga.

8. Output: EMF, $\mathcal{E}$ likha jaata hai

Pieces ko jod kar, parent ka law $\mathcal{E}=-\frac{d{\Phi }_B}{dt}$ ab plain words mein padhta hai: loop jo push-per-charge produce karta hai woh equal hai spaghetti-count kitni fast change ho rahi hai, minus sign ke saath. Minus sign Lenz's law hai — loop change ko resist karne ke liye push back karta hai — aur iske liye Lenz's Law par apna deep dive hai.

9. Prerequisite map

Equipment checklist

Khud se test karo — jab aap in mein se har ek ka jawab bina dekhein de sakein tab aap parent note ke liye ready hain.