1.8.23
Intuition The big pictureEk steady current apne aas-paas ke magnetic field ko "stir" karta hai. Agar tum ek closed loop mein chalo aur add karo ki field B along kitna push karti hai, toh total sirf uss current par depend karta hai jo tumhare loop se guzar rahi hai — loop ki shape par nahi . Yeh Gauss's law ka magnetic cousin hai: ek symmetry shortcut jo ek mushkil integral ko algebra ki ek line mein badal deta hai.
Definition Ampère's circuital law (magnetostatic form)Steady (time-independent) currents ke liye, kisi bhi closed loop C B μ 0 net current ke barabar hota hai jo C ∮ C B ⋅ d l = μ 0 I enc == I enc == enclosed current hai (current jo loop ko pierce karti hai), μ 0 = 4 π × 1 0 − 7 T⋅m/A C Amperian loop kehte hain.
WHAT it gives you: B agar geometry mein kaafi symmetry ho.WHAT it is NOT: jaise likha hai waise sirf magnetostatics ke liye valid hai (koi changing electric fields nahi). Full Maxwell version mein displacement-current term μ 0 ϵ 0 d Φ E / d t
Hum isse postulate nahi karte — hum isse simplest case ke liye derive karte hain aur generality ka argument dete hain.
Intuition Deeper "why" (field ka twist)Differential form: ∇ × B = μ 0 J curl (circulation) ka source hai, jaise charge E ∮ B ⋅ d l = ∫ ( ∇ × B ) ⋅ d A = μ 0 ∫ J ⋅ d A = μ 0 I enc
Worked example Strategy — the 80/20 core
Symmetry pehchano (kya B Ek Amperian loop choose karo jiske along B Likho ∮ B ⋅ d l = B ( length ) Nikalo I enc Solve karo B
Worked example Worked example 2 — Inside a thick wire (radiusR J s < R B
J = π R 2 I Kyun? current uniformly cross-section par spread hai.Loop: circle radius s s I enc = J ⋅ π s 2 = I R 2 s 2 Yeh step kyun? Sirf woh current count hoti hai jo loop ko thread kare; fraction = area R area s B ( 2 π s ) = μ 0 I R 2 s 2 ⇒ B = 2 π R 2 μ 0 I s ∝ 1/ s
Worked example Worked example 3 — Long solenoid (n I
Loop: rectangle, ek lamba side (L B ≈ 0 B ∮ B ⋅ d l = B L Kyun? sirf andar wala side contribute karta hai.I enc = n L I Kyun? loop n L I B L = μ 0 n L I ⇒ B = μ 0 n I
Worked example Worked example 4 — Toroid (N
Loop: circle radius r B ( 2 π r ) = μ 0 N I ⇒ B = 2 π r μ 0 N I I enc = 0 ⇒ B = 0
Recall Predict before checkingEk doosri wire 2 I bahar hai. Kya ∮ B ⋅ d l Answer: Nahi — ∮ B ⋅ d l enclosed current par depend karta hai, isliye yeh unchanged rahega (= μ 0 I B zaroor change hoga (bahari wire field add karti hai); integral ke contributions bas cancel out ho jaate hain. Yahi subtlety hai jo log miss karte hain.
∮ B ⋅ d l = 0 B = 0 Kyun sahi lagta hai: zero total matlab zero har jagah lagta hai.
Fix: Zero sum ≠ zero har jagah — positive aur negative contributions cancel ho sakte hain. B circulation zero hai (jaise loop wire ke paas ho lekin enclose na kare).
I enc Kyun sahi lagta hai: har current ek field banata hai.
Fix: Sirf woh current count hoti hai jo chosen surface ko pierce kare . External currents local B
Common mistake "Ampère's law se main kisi bhi geometry ke liyeB Kyun sahi lagta hai: law hamesha sahi hai (magnetostatics mein).
Fix: Yeh hamesha sahi hai lekin B tabhi useful hai jab symmetry se B
Common mistake Isse changing currents/fields ke saath use karna.Kyun sahi lagta hai: yeh general lagta hai.
Fix: Magnetostatic form tab fail karta hai jab ∂ E / ∂ t = 0
Ampère's law (integral magnetostatic form) I enc Loop C
Ampère's law se infinite straight wire ka field B = 2 π s μ 0 I
Wire ke liye radius cancel kyun hota hai? B ∝ 1/ s ∝ s s
Uniform thick wire (radius R s < R B = 2 π R 2 μ 0 I s s
Long solenoid ke andar field B = μ 0 n I
Radius r N B = 2 π r μ 0 N I
Ampère's law ka differential form Magnetostatic form GALAT kab hota hai? Jab electric fields time ke saath change hon; displacement current μ 0 ϵ 0 ∂ t Φ E
Kya ∮ B ⋅ d l = 0 B = 0 Haan — contributions cancel hote hain; circulation ≠ pointwise value.
Ampère's law USEFUL kab hai (sirf sahi nahi)? Jab symmetry se
B chosen loop ke along constant aur parallel (ya zero) ho.
Recall Feynman: explain to a 12-year-oldSocho ek pipe mein river current beh rahi hai (yahi electric current hai). Yeh apne aas-paas ke paani ko circles mein swirl karti hai (yahi magnetic field hai). Ampère ka rule kehta hai: agar tum pipe ke around ek full lap chalo aur track karo ki swirling tumhe tumhare steps ke along kitna push kar rahi hai, toh total push sirf iss par depend karta hai ki tumhare lap ke andar river kitni strong hai — koi farq nahi padta ki tum bada circle chalo ya wobbly weird loop, jab tak tum pipe ke around ek baar ghum lo. Agar tum ek khaali jagah ke around ghum lo (andar koi current nahi), toh pushes cancel ho jaate hain aur tumhe zero total milta hai.
"CLIME" — C irculation of B μ 0 L oop-I ntegral's M agnetic source = μ 0 I enc "Sirf woh matter karta hai jo threaded ho."
generalize by polar split
B of infinite wire B=u0 I / 2 pi s
2 pi s cancels 1 / 2 pi s
Displacement current term
