Foundations — Ampere's circuital law — magnetostatic form

1.8.23 · D1 · Physics › Electromagnetism › Ampere's circuital law — magnetostatic form

Yeh page kuch bhi assume nahi karta. Parent note comfortably padhne se pehle, ${\oint }_C\vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$ mein har squiggle obvious lagna chahiye. Hum unhe ek-ek karke build karte hain, har ek pichle se.

0. Vector kya hota hai (arrow)

Ise ek plain number ("scalar") jaise temperature se compare karo — usme sirf size hoti hai, koi direction nahi. Magnetic field directional hai (har jagah kisi direction mein point karti hai), isliye hume arrows ki zaroorat hai ise describe karne ke liye.

Figure s01. Ek single vector: yeh ek point se start hota hai (black dot), iski length iski size hai, aur iski arrowhead (orange) iski direction hai. Dotted plum lines dikhate hain ki usi arrow ko ek horizontal part aur ek vertical part mein split kiya ja sakta hai — ek trick jo hum baad mein dot product ke liye reuse karte hain.

• Picture: ek arrow jo ek point se start hokar kahin point karta hai.
• Topic ko kyun chahiye: magnetic field $\vec{B}$ ek vector hai — space ke har point par iske paas ek strength aur ek direction hai. "Field ka path ke saath push" ke baare mein baat karna bina arrows ke possible nahi.

1. Magnetic field $\vec{B}$ (har point par ek arrow)

• Symbol: $\vec{B}$, measured in tesla (T).
• Picture: wire ko hug karte concentric circles of arrows.
• Topic ko kyun chahiye: $\vec{B}$ is show ka star hai — Ampère's law $\vec{B}$ ke baare mein ek statement hai.

2. Current $I$ aur current density $\vec{J}$

• Picture: $I$ = ek pipe mein total flow; $\vec{J}$ = pipe ke munh par har jagah flow kitna dense hai; $d\vec{A}$ = us munh se bahar nikalta arrow, aur hum sirf us flow ko count karte hain jo iske saath ja raha hai.
• Topic ko kyun chahiye: $\vec{B}$ ka source current hai. Parent ke Worked Example 2 (radius $R$ ki thick wire, dekho §8) ko $J=I/(\pi R^2)$ chahiye yeh jaanne ke liye ki ek chhote circle ke andar kitni current hai.

3. Angle idea aur dot product $\vec{B}\cdot d\vec{l}$

Bade integral se pehle hume samajhna hai ki field ka "tumhe path ke saath push karna" matlab kya hai. Woh word along hi dot product hai.

Figure s02. Step $d\vec{l}$ (orange) path ke saath lie karta hai. Field $\vec{B}$ (teal) angle $\theta$ par lean kar raha hai. $\vec{B}$ ko step direction par seedha giraao (dotted plum) aur thick plum segment along-path part hai, $|\vec{B}|\cos \theta$. Woh length, step length se multiply hokar, $\vec{B}\cdot d\vec{l}$ hai.

• Saare cases jo tumhe yaad rakhne chahiye:
  • $\theta =0^{\circ }$: $\cos \theta =1$ → poora push. (Wire loop mein use hota hai: $\vec{B}$ circle ke saath run karta hai.)
  • $\theta =90^{\circ }$: $\cos \theta =0$ → zero. (Solenoid mein use hota hai: rectangle ke chhote sides $\vec{B}$ ke perpendicular hote hain.)
  • $\theta =180^{\circ }$: $\cos \theta =-1$ → poora negative (opposing) contribution.
• Topic ko kyun chahiye: $\vec{B}\cdot d\vec{l}$ ki har appearance "field-along-path" hai. Clever loops choose karne ka poora method $\theta$ ko ya toh $0$ (constant push) ya $90^{\circ }$ (zero) banana hai.

4. $d\vec{l}$ — path ke saath ek tiny step

• Picture: ek curve hazaron tiny arrows mein toot jaata hai ek dusre ke saath nose-to-tail; har ek $d\vec{l}$ hai.
• Topic ko kyun chahiye: "path ke saath push add karne" ke liye hum path ko tiny steps $d\vec{l}$ mein chop karte hain, har ek ka along-part $\vec{B}\cdot d\vec{l}$ lete hain, aur unhe sum karte hain.

5. Closed loop integral ${\oint }_C$

Figure s03. Teal circle page se bahar point karne wale wire ke aas-paas Amperian loop $C$ hai. Orange arrows steps $d\vec{l}$ hain; yahan hum counter-clockwise chalte hain. $\vec{B}$ field same direction mein wrap karta hai, isliye har $\vec{B}\cdot d\vec{l}$ poora positive push hai, aur unka total circulation hai. Plum arrow perpendicular distance $s$ mark karta hai.

• Topic ko kyun chahiye: Ampère's law ka poora left-hand side yahi circulation hai. Magical claim yeh hai ki yeh total simple hai chahe har step complicated ho.

6. Orientation: walk direction ko $I_{\text{enc}}$ se link karna (right-hand rule)

Ampère's law do cheezein compare karta hai — woh direction jis taraf tum $C$ ke around chalte ho, aur woh direction jis taraf current surface se guzarne par positive count hoti hai. Inhe ek fixed convention se tie karna zaroori hai, warna signs bigad jaate hain.

• Picture: fingers counter-clockwise curl kar rahe hain (jaise Figure s03 mein) → thumb page se bahar point karta hai → page se bahar flow karne wali current positive hai.
• Topic ko kyun chahiye: iske bina usi wire ko encircle karne wala loop $+{\mu }_{0}I$ ya $-{\mu }_{0}I$ de sakta hai. Rule ambiguity hata deta hai, aur yahi right-hand rule hai jo batata hai $\vec{B}$ wire ke aas-paas kis taraf wrap karta hai (§9).

7. $I_{\text{enc}}$ — enclosed current

• Picture: ek ring jis par film hai; sirf woh current arrows count karo jo film ko stab karte hain, $+$ ya $-$ thumb direction se.
• Topic ko kyun chahiye: yeh poora right-hand side hai (${\mu }_0$ times). "Total push = ${\mu }_0$ times jo loop ko thread kare" hi law hai.

8. Constant ${\mu }_0$

• Topic ko kyun chahiye: yeh $\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$ ke dono sides ke units match karata hai, aur effect ki strength set karta hai.

9. $\pi$, circles, distance $s$, aur wire radius $R$

• Topic ko kyun chahiye: har geometric answer (wire, thick wire, solenoid, toroid) in circle facts ko loop length ya pierced area compute karne ke liye use karta hai. $s$ (distance out) aur $R$ (wire radius) ko alag names rakhna thick-wire case mein zaroori hai jahan $J=I/(\pi R^2)$ hai lekin tum chhote radius $s<R$ par evaluate karte ho.

10. Field ke twist ke liye right-hand rule

• Picture: thumb wire ke upar, fingers uske around counter-clockwise curl kar rahi hain (thumb side se dekha gaya).
• Topic ko kyun chahiye: yeh fix karta hai $\vec{B}$ tumhari loop par kis direction mein point karta hai, toh tum jaante ho $\vec{B}\cdot d\vec{l}$ add hoga ya subtract.

Foundations topic ko kaise feed karte hain

Equipment checklist

Khud test karo: right side cover karo aur reveal karne se pehle answer do.

Ready ho? Tab parent note par wapas jao — aur jab differential form $\nabla \times \vec{B}={\mu }_0\vec{J}$ ya Stokes' theorem aaye, ya Gauss's law for magnetism aur Biot–Savart law se compare karo, toh ab tumhare paas har symbol earned hai.