Ampere's circuital law — magnetostatic form: Question bank

1.8.23 · D5 · Physics › Electromagnetism › Ampere's circuital law — magnetostatic form

Yeh page sirf parent law ko assume karta hai: $\oint_C \vec B\cdot d\vec l = \mu_0 I_{\text{enc}}$. Neeche har symbol wahin se liya gaya hai. $B$ = magnetic field, $d l$ = closed loop $C$ ke saath ek chota sa step, $I_{enc}$ = woh current jo actually loop ke across khiche surface ko pierce karti hai.

Definition Is page par use hone wale symbols

Kisi bhi trap se pehle, har letter ko pin down karo, jo figure s01 ki picture se anchored hai.

$== s ==$ — wire se kisi point ki radial distance, wire ke perpendicular seedha bahar measured (sabse chhota gap). Units: metres.
$== R ==$ — ek mote wire (finite thickness wali real wire) ki poori outer radius, yaani uski dhatu axis se kitni door tak phaili hai. Units: metres. Dhyan raho $s$ andar ( $s < R$ ) ya bahar ( $s > R$ ) ho sakta hai.
$== n ==$ — solenoid ki turns per unit length: coil ke har metre mein kitne wire loops packed hain. Units: turns per metre ( $m^{- 1}$ ). Ghani coil ka $n$ bada hoga.
$== N ==$ — ek toroid (doughnut-shaped coil) par wound total number of turns. Seedha count, koi units nahi.
$r$ — toroid ke liye, circular Amperian loop ki radius jo doughnut ke andar chalti hai, doughnut ke central hole se measured. Units: metres.

Figure — Ampere's circuital law — magnetostatic form

Recall Pehle picture padho

Figure s01 mein, kaunsa arrow $s$ hai aur kaunsa shaded band $R$ se set hota hai? ::: $s$ wire ke centre se field point tak jaane wala horizontal arrow hai; $R$ shaded metal disc ki edge hai — yeh "wire ke andar" ( $s < R$ ) aur "bahar" ( $s > R$ ) ke beech ki boundary hai.

Sahi ya galat — justify karo

\oint B \cdot d l

ki circulation Amperian loop ki shape par depend karti hai.

Galat — yeh sirf enclosed current

μ_{0} I_{enc}

par depend karti hai; same current enclose karne wala koi bhi loop same value dega, aur yahi is law ki poori power hai.

Agar

\oint B \cdot d l = 0

ho toh loop ke har point par

B = 0

hoga.

Galat — zero sum bade fields ko hide kar sakta hai jo cancel ho jaate hain; jaise ek loop jo wire ke paas hai (lekin uske around nahi) uske upar real

B

hoga lekin zero circulation.

Ampère's law sirf approximately sahi hai; Biot–Savart exact wala hai.

Galat — magnetostatics mein dono exact aur equivalent hain; Ampère's law Biot–Savart se derived hai, isliye yeh kam exact nahi ho sakta.

Ampère's law hamesha

B

compute karne ke liye use ho sakta hai.

Galat — yeh hamesha sahi hai lekin sirf tab solvable hai jab symmetry

B

ko integral se bahar nikalne de (infinite wire, solenoid, toroid, sheets); warna yeh ek equation hai jisme bahut zyada unknowns hain.

Ek straight wire ke around circular loop ki radial distance

s

double karne se circulation aadhi ho jaati hai.

Galat —

B \propto 1/ s

girta hai lekin circumference

2 π s

same factor se badhta hai, isliye product

μ_{0} I

unchanged rehta hai; radius genuinely cancel ho jaata hai.

Ek current jo circular Amperian loop ke axis ke along flow karti hai lekin kabhi uski surface ko cross nahi karti woh phir bhi

I_{enc}

mein contribute karti hai.

Galat — sirf wahi current count hoti hai jo spanning surface ko pierce kare; loop ke apne plane mein chalti current kuch thread nahi karti.

Magnetostatic form

\oint B \cdot d l = μ_{0} I_{enc}

capacitor charging hone ke dauran bhi valid hai.

Galat — charging capacitor mein plates ke beech changing

E

hota hai; tumhe displacement-current term add karni hogi, warna same loop par do surfaces alag jawab denge.

Loop traverse karne ki direction ulti karne se computed

I_{enc}

ka sign flip ho jaata hai.

Sahi — loop direction aur surface normal right-hand rule se tied hain, isliye traversal ulta karne se "positive" current kis taraf point karta hai woh flip ho jaata hai.

Error dhundo

"Mere loop ke bahar

2 I

carry karne wala wire baitha hai, isliye

\oint B \cdot d l

jump karke

μ_{0} (I + 2 I)

ho jaata hai."

Error: bahar wala wire enclosed nahi hai. Uska field loop par local

B

mein add hota hai lekin circulation mein uske contributions exactly cancel ho jaate hain, aur

\oint B \cdot d l = μ_{0} I

rehta hai.

"Solenoid ke andar

B = μ_{0} n I

hai (

n

= turns per metre), isliye axis ke bilkul centre mein yeh double ho jaata hai."

Error:

B = μ_{0} n I

pehle se hi uniform interior value hai — yeh ek lambe solenoid ke andar position par depend nahi karta, isliye kahin bhi doubling nahi hoti.

"Outer radius

R

wali moti wire ke liye maine

s < R

par

I_{enc} = I

use kiya."

Error:

s < R

par sirf radius

s

ke andar wali current fraction enclosed hoti hai,

I_{enc} = I s^{2} / R^{2}

; poora

I

use karne se interior field overstate ho jaata hai.

"Ampère's law finite solenoid ke end ke paas bhi utni hi achhi tarah

B

deta hai jitna middle mein."

Error: recipe ideal infinite solenoid assume karti hai jahan outside

B \approx 0

; end ke paas field fringe aur leak karta hai, loop assumption tod deta hai — wahan Biot–Savart law use karo.

\oint B \cdot d l = μ_{0} I_{enc}

, isliye main kisi bhi loop ke liye

B (2 π s)

nikaal sakta hoon."

Error:

B

tab hi factor out ho sakta hai jab woh loop ke along constant aur

d l

ke parallel ho; ek random loop mein dono nahi hote, isliye

\oint B \cdot d l \neq = B \times (length)

"Toroid ke bahar field chhota hai lekin nonzero hai kyunki bahut saare turns hain."

Error: toroid ke bahar Amperian loop equal aur opposite current enclose karta hai (windings baahir jaati hain aur wapas aati hain), isliye

I_{enc} = 0

aur ideally

B = 0

exactly outside hota hai.

Why questions

Wire ke Amperian circle ki radial distance

s

answer se kyun gayab ho jaati hai?

Kyunki field se

B \propto 1/ s

aur circumference se length

\propto s

multiply karke kuch

s

-independent banta hai — wahi cancellation precisely is law ko geometry-free banati hai.

Amperian loop closed kyun hona chahiye?

Result is baat par based hai ki angle

ϕ

poora

2 π

sweep kare (enclosing) ya start par wapas aaye (

0

); ek open path ka koi well-defined "enclosed" nahi hota aur Stokes' theorem — Stokes' theorem aur figure s02 dekho — ko surface ki boundary chahiye.

Loop ke bahar wali current har jagah field banane ke bawajood zero circulation kyun contribute karti hai?

Uske field ka loop ke through koi net "twist" nahi hai: jab tum around chalte ho, toh woh along-path pushes zero sum karte hain kyunki woh jo angle subtend karta hai woh apne start par wapas aa jaata hai,

\oint d ϕ = 0

Ampère's law ko "Gauss's law ka magnetic cousin" kyun kaha jaata hai?

Gauss's law flux ko enclosed charge (divergence ka source) se jodata hai; Ampère circulation ko enclosed current (curl ka source) se jodata hai — dono ek hard integral ko symmetry shortcut se replace karte hain.

Gauss's law for magnetism ke right side par zero kyun hai jabki Ampère ke paas

μ_{0} I_{enc}

hai?

Koi magnetic charges (monopoles) nahi hain jo flux sources ki tarah kaam karein, isliye magnetic flux hamesha zero hota hai; lekin currents haan circulation sources ki tarah kaam karte hain, isliye Ampère's law nonzero hai.

Solenoid ke rectangular loop ki short sides kuch contribute kyun nahi karti?

Andar,

B

axial hai jabki short sides axis ke perpendicular chalti hain, isliye wahan

B \cdot d l = 0

hota hai; sirf lambi andar wali side, jo

B

ke parallel hai, contribute karti hai.

Differential form

\nabla \times B = μ_{0} J

mein same physics kyun hai jitni integral form mein?

Stokes' theorem (figure s02) curl ke surface integral ko

B

ke loop integral mein convert karta hai, isliye dono forms ek hi statement ke small-loop aur finite-loop faces hain — Maxwell's equations dekho.

Edge cases

Ek loop jo do antiparallel wires mein se har ek

I

carry karne wale ko enclose karta hai, uska

I_{enc}

kya hai?

Zero — currents right-hand rule se opposite signs ke saath count hote hain, isliye

\oint B \cdot d l = 0

hoga chahein loop par

B

kitna bhi bada aur messy kyun na ho.

Exactly

s = R

par (uniform moti wire ki surface par, jahan

R

uski outer radius hai) kya andar aur bahar ke formulas agree karte hain?

Haan — andar

B = μ_{0} I s / (2 π R^{2})

μ_{0} I / (2 π R)

aata hai aur bahar

B = μ_{0} I / (2 π s)

s = R

par same value milti hai; field boundary par continuous hai.

Straight wire ke axis par (

s \to 0

) exterior formula se

B

kya hai?

Formula

B = μ_{0} I / (2 π s)

diverge karta hai, lekin yeh idealized zero-thickness limit hai; real wire mein

B \to 0

centre par hota hai kyunki

I_{enc} \to 0

zyada tezi se hota hai (

\propto s^{2}

Ek loop same wire ko do baar link karta hai (uske around do poore turns wrap karta hai).

I_{enc}

kya hai?

2 I

— angle

4 π

sweep karta hai, isliye

\oint d ϕ = 4 π

aur circulation

2 μ_{0} I

hai; enclosure multiplicity ke saath count hoti hai.

Ek region mein kahin bhi steady current zero hai lekin

B \neq = 0

wahan (door ki currents ka field). Us region mein loop ke liye Ampère kya deta hai?

\oint B \cdot d l = 0

kyunki koi current enclosed nahi hai, phir bhi

B

genuinely nonzero hai — nonzero field ke saath zero circulation ka ek saaf example.

Toroid (

N

total turns) ke liye

B

andar

r

par depend kyun karta hai jabki solenoid ka nahi karta?

Toroid ki loop length

2 π r

r

ke saath badhti hai jabki

I_{enc} = N I

fixed rehta hai, isliye

B \propto 1/ r

; solenoid ki enclosed current

n L I

(

n

= turns per metre) loop length

L

ke saath badhti hai, isliye ratio constant rehta hai — Solenoid and toroid fields dekho.

Recall One-line self-test

Agar tum sirf ek trap-buster yaad rakh sako, toh kaunsa? ::: "Enclosed, not present" — Ampère's law ke right side par sirf wahi current count hoti hai jo surface ko pierce kare, aur $B$ factor out karne ke liye symmetry chahiye, sirf truth nahi.