Faraday's law — EMF = −dΦ - dt: Question bank

1.8.26 · D5 · Physics › Electromagnetism › Faraday's law — EMF = −dΦ - dt

Poore raaste hum do tools pe lean karte hain, unhe yaad kar lo:

Flux $Φ_{B} = B A cos θ$ — dekho Magnetic Flux. Yahan $θ$ woh angle hai $B$ aur loop ki normal $\overset{n}{^}$ ke beech (woh arrow jo loop ki face se seedha bahar nikalti hai), na ki loop ki flat surface se.
The law $E = - \frac{d Φ _{B}}{d t}$ — EMF change ki rate hai, aur minus sign Lenz's Law hai jo energy conservation ka bookkeeping karta hai.

True or false — justify karo

A constant magnetic field through a stationary loop induces an EMF.

False. EMF depend karta hai

d Φ/ d t

pe; agar

B

A

, aur

θ

sab fixed hain toh

Φ

fixed hai aur uski rate of change zero hai — chahe

B

kitna bhi enormous kyun na ho.

Agar kisi loop se flux momentarily zero ho, toh us instant pe induced EMF bhi zero honi chahiye.

False. EMF

Φ (t)

ki slope track karta hai, uski value nahi. Ek sinusoidal flux

Φ = Φ_{0} sin ω t

exactly tab zero se guzarta hai jab woh sabse tezi se change ho raha hota hai, isliye wahan EMF apne peak pe hoti hai.

Turns ki number

N

double karne se same flux change ke liye induced EMF double ho jaati hai.

True. Har turn same flux link karta hai, aur unki EMFs series mein add hoti hain:

E = - N d Φ/ d t

. Yahi wajah hai ki real coils mein bahut saare turns hote hain — dekho Electric Generators and AC.

Ek loop jo flat lie kar raha ho jab

B

uske plane ke along (surface ko graze karte hue) point kare, uska maximum flux hota hai.

False. "Plane ke along" matlab

B ⊥ \overset{n}{^}

, isliye

θ = 9 0^{\circ}

aur

cos 9 0^{\circ} = 0

— flux zero hai. Maximum flux tab hota hai jab

B ∥ \overset{n}{^}

(

θ = 0

Induced EMF nonzero ho sakti hai chahe na

B

change ho, na loop change ho.

False.

Φ = B A cos θ

sirf unhi teen quantities pe depend karta hai. Agar

B

A

θ

mein se koi bhi time ke saath nahi badla, toh

d Φ/ d t = 0

. Koi fourth hidden knob nahi hai.

Loop ki winding ka direction reverse karne se induced EMF ki sign flip ho jaati hai.

True (lekin yeh ek convention flip hai). Winding reverse karne se tumhara chosen

\overset{n}{^}

reverse ho jaata hai, jo

Φ

ki sign flip karta hai aur isliye

E

ki bhi. Loop ke relative physical current direction mein koi badlaav nahi — tumne bas "positive" ko relabel kiya.

Agar do loops same changing flux enclose karein, toh unhe unki shapes ki parwah kiye bina same EMF milti hai.

True. Faraday's law sirf loop ko thread karne wale flux ki parwah karta hai, loop ki geometry ki nahi. Square, circle, ya wiggly — same

d Φ/ d t

se same

E

milti hai.

Motional EMF aur "changing-

B

" EMF do alag laws hain jo similar dikhti hain.

False. Yeh ek hi law

E = - d Φ/ d t

ke do faces hain. Motional EMF charges pe force

q v \times B

se aata hai (dekho Motional EMF); flux picture dono ko unify karta hai.

Error dhundho

"Rod tezi se slide karta hai, isliye flux bada hai, isliye EMF badi hai."

Error hai "flux is bigger." Speed, flux ki value nahi badalta — woh rate badalta hai jis pe area sweep hoti hai,

d A / d t = Lv

. EMF isliye badhti hai kyunki

d Φ/ d t

badhta hai, na ki isliye kyunki

Φ

large hai.

θ

woh angle hai field aur loop ki surface ke beech."

θ

normal

\overset{n}{^}

se measure hota hai, surface se nahi. Surface mein lie karta field

θ = 9 0^{\circ}

deta hai aur

Φ = 0

; yeh

9 0^{\circ}

ka error har generator formula ke maxima aur minima ko flip kar deta hai.

"Kyunki hum sirf magnitude chahte hain, hum minus sign hamesha ke liye safely delete kar sakte hain."

Magnitude-only numeric answer ke liye theek hai, lekin sign is Lenz's Law hai. Ise delete karo aur induced current ki direction kho doge, isliye braking force bhi kho doge aur yeh bhi ki generator motion ko resist karta hai ya assist karta hai.

"Generator EMF

E = N B A ω sin ω t

peak karta hai jab coil field ko head-on face kare."

Ulta hai. Head-on matlab

θ = 0

, jahan

Φ

maximum hai lekin momentarily change nahi ho raha — EMF wahan zero hai. Peak EMF tab hoti hai jab coil edge-on ho (

θ = 9 0^{\circ}

), jahan flux sabse tezi se change hota hai.

"Induced current hamesha is tarah flow karta hai ki loop se flux badhe."

Woh hamesha change ko oppose karta hai. Agar flux badh raha hai, toh current use kam karne ki koshish karta hai; agar flux ghatt raha hai, toh current use banaye rakhne ki koshish karta hai. Yeh kis direction mein hoga woh

d Φ/ d t

ki sign pe depend karta hai, koi fixed direction nahi.

"Ek superconducting loop jisme zero resistance ho woh infinite induced current deta hai, isliye free energy milti hai."

Current infinite nahi hoti — loop ki Inductance limit karti hai ki flux kitni tezi se change ho sakta hai, aur Lenz ki opposition matlab hai ki flux badalne ke liye tum kaam karte ho. Energy conservation kabhi violate nahi hoti.

Why questions

Flux definition mein sirf

B A

ki jagah dot product

B \cdot d A

kyun use hota hai?

Sirf field ka woh component jo surface ke perpendicular hai usse thread karta hai; dot product ka

cos θ

exactly wahi perpendicular part extract karta hai aur sideways-sliding field ko discard karta hai.

Induced current change ko assist karne ki jagah oppose (Lenz) kyun karta hai?

Agar assist karta, toh induced current usi change ko strengthen karta jo use bana raha tha, khud ko feed karta → runaway free energy. Opposition hi energy conservation ke saath consistent ek option hai.

Generator alternating (sinusoidal) EMF kyun produce karta hai steady EMF ki jagah?

Coil rotate karne se

θ = ω t

hota hai, isliye

Φ = B A cos ω t

. Differentiate karne pe

sin ω t

aata hai — ek oscillation jiska sign har half turn pe flip hota hai jab coil face

B

ke relative reverse hoti hai.

Purely time-varying magnetic field kisi bhi moving part ke bina EMF induce kyun kar sakta hai?

Faraday's law ko sirf

d Φ/ d t \neq = 0

chahiye. Ek changing

B

fixed loop se seedha flux change karta hai, koi motion required nahi — yeh Maxwell's

\nabla \times E = - \partial B / \partial t

ka seed hai.

Zyada turns

N

EMF ko multiply kyun karte hain lekin

Φ_{B}

ko nahi?

Φ_{B}

ek loop ke area se flux hai; turns add karne se woh nahi badalta. Lekin turns series mein hote hain, isliye unki individual EMFs stack hoti hain: total

E = N \times (- d Φ/ d t)

Minus sign same physics kyun hai Lenz's law ke?

Minus sign kehta hai EMF current ko us direction mein drive karta hai jiska apna magnetic flux imposed change ko oppose karta hai — yeh precisely Lenz ka statement hai, bas algebraically likha gaya.

Edge cases

Exactly us instant pe jab ek rotating coil flat lie kare field ko face karte hue (

θ = 0

), EMF kya hai?

Zero. Flux apne maximum

Φ = B A

pe hai, isliye uski slope

d Φ/ d t = 0

hai — cosine momentarily apne top pe flat hai. Coil peak flux se "coasting" kar rahi hai.

Agar field change hone se pehle ek loop ko figure-eight mein fold kar diya jaaye toh EMF ka kya hoga?

Dono lobes opposite normals enclose karte hain, isliye unka flux subtract hota hai. Agar lobes equal hain, net

Φ \approx 0

aur

d Φ/ d t \approx 0

— almost koi EMF nahi, chahe strong changing field mein bhi.

Ek loop uniform field mein constant velocity se move kare (fully inside, koi edge bahar ya andar nahi). Kaisi EMF appear hoti hai?

Zero. Loop se guzarta flux nahi badalta jab woh poori tarah immersed ho — same

B

, same

A

, same

θ

. EMF sirf tab appear hoti hai jab koi edge field boundary cross kare.

Agar

B

linearly time ke saath badhta ho,

B = k t

, fixed loop se, toh EMF kya hogi?

Ek constant EMF:

E = - A d B / d t = - A k

, time mein steady. Flux mein linear ramp flat (DC) EMF deta hai — yeh ek time-varying source hai jo constant EMF yield karta hai.

Agar kisi instant pe loop ki area zero ho jaaye, kya wahan EMF zaroor large hogi?

Zaroori nahi. Jo matter karta hai woh hai

d Φ/ d t = B cos θ d A / d t

— area kitni tezi se change ho rahi hai, yeh nahi ki

A

zero ho jaaye. Ek slow collapse small EMF deta hai chahe

A = 0

pe bhi.

Do coaxial coils: inner mein steady current hai. Kya outer coil mein EMF hai?

Nahi, jab current steady ho. Steady current → steady

B

→ steady flux → zero EMF. Sirf inner current ko switch ya vary karne par (mutual Inductance) outer EMF induce hoti hai.

Recall Ek-line survival summary

EMF flux curve ki slope mein rehti hai, kabhi uski height mein nahi; minus sign Lenz hai jo energy ki raksha karta hai; aur $θ$ hamesha normal $\overset{n}{^}$ se measure hoti hai. Teen-word charm ::: "Change Makes EMF; Nature Says No."

Connections

Lenz's Law — minus sign aur current direction ka source
Magnetic Flux — $Φ_{B} = B A cos θ$ jiske around yeh traps orbit karte hain
Motional EMF — same law ka $q v \times B$ face
Maxwell's Equations — "no moving parts" case ke liye field form
Electric Generators and AC — kyun edge-on pe peak EMF milti hai
Inductance — kyun zero resistance ka matlab infinite current nahi