Foundations — Magnetic field of straight wire, circular loop, solenoid, toroid

1.8.24 · D1 · Physics › Electromagnetism › Magnetic field of straight wire, circular loop, solenoid, to

parent note mein koi bhi formula padhne se pehle, uska har ek squiggle tumhare liye kuch matlab rakhna chahiye. Yeh page har ek cheez ko bilkul zero se build karta hai.

0. "Current" kya hai aur "field" kya hai?

Yeh topic ko kyun chahiye: magnetic fields currents se banti hain. $I$ nahi toh $\vec{B}$ nahi. Parent ke har formula mein $I$ numerator mein baitha hai — flow double karo, field double ho jaayegi.

Figure s01 mein blue circles wire ko wrap karne wala field hain. Dhyan do: koi shuruaat nahi, koi ant nahi — magnetic field lines apne aap mein close ho jaati hain. Yahi circling magnetism ki poori personality hai.

1. Vectors aur chota hat: $\vec{a}$, $\hat{r}$

Yeh topic ko kyun chahiye: Biot–Savart mein, $\hat{r}$ yeh jawab deta hai ki "wire ka yeh chota sa tukda apna influence kis direction mein aim karta hai?" Direction ($\hat{r}$) aur distance ($r$) ko alag karna hi allow karta hai ki $1/r^2$ ki weakening aur swirl-direction ko alag-alag handle kiya ja sake.

2. Distance $r$ aur perpendicular distance

Yeh topic ko kyun chahiye: magnetic influence distance ke saath fade hoti hai. Parent ka straight-wire law kehta hai ki field half ho jaata hai jab tum $r$ double karte ho — toh $r$ clearly define hona chahiye tab hi yeh baat sense banti hai.

3. Wire ke tiny pieces: $d\vec{l}$ aur uski length $dl$

Yeh topic ko kyun chahiye: Biot–Savart har $d\vec{l}$ ki contribution ko add karta hai. Jab bhi direction already handle ho chuki hoti hai, arithmetic mein sirf plain length $dl$ bachti hai — isliye tum jaanno chahiye ki $dl$ kuch bhi mysterious nahi, bas $|d\vec{l}|$ hai.

4. Cross product $\times$ — yeh swirl machine

Yeh woh symbol hai jo zyaadatar log yahan pehli baar dekhte hain, toh hum ise dheere-dheere build karte hain.

Size formula padhna: $\sin \theta$ yahan key hai. Jab dono arrows parallel hain ($\theta =0$, $\sin 0=0$) cross product zero hai — ek wire-piece apni khud ki direction mein koi field nahi bhejta. Jab woh perpendicular hain ($\theta =90^{\circ }$, $\sin 90^{\circ }=1$) field sabse strong hai. s03 mein pink dot dekho: result tab sabse bada hota hai jab dono black arrows L-shape banate hain.

5. Dot product $\cdot$ — yeh "kitna lines up hai" machine

Ampère's law ek alag product par bani hai, toh hume woh bhi chahiye.

s04 dekho: dot product sirf $\vec{a}$ par padi $\vec{b}$ ki shadow rakhta hai (blue projection). Jab $\vec{b}$ puri tarah $\vec{a}$ ke along jhukta hai shadow full-length hai; jab $\vec{b}$ seedha khada hota hai, shadow kuch nahi reh jaata.

Yeh topic ko kyun chahiye: $\oint \vec{B}\cdot d\vec{l}$ mein, ek achhe se chune gaye loop par field walk ke parallel hoti hai toh $\cos \theta =1$ aur $\vec{B}\cdot d\vec{l}=Bdl$. Yeh akela simplification hi Ampère's law ko ek easy $B(2\pi r)$ mein badal deta hai. Solenoid ke rectangle ki perpendicular sides kuch contribute nahi karti kyunki wahan $\cos 90^{\circ }=0$ hai.

6. Integral signs: $\int$ aur $\oint$

Yeh topic ko kyun chahiye: Biot–Savart defined hai ek integral se (har wire-piece ki contribution add karo). Ampère's law defined hai field ke ek closed-loop integral se. In dono laws mein se koi bhi nahi padh sakte agar yeh nahi jaante ki yeh do symbols ka matlab hai "tiny pieces add karo."

7. Angles $\theta$, $\alpha$, ${\theta }_1$, ${\theta }_2$ — inhe straight rakhna

Is topic mein ek angle sirf do kaam karta hai; ek baar naam de lo toh saari confusion khatam.

Yeh topic ko kyun chahiye: parent mein har trig factor ($\sin {\theta }_1$, $\cos \alpha$, har cross product mein chupta $\sin \theta$) inhi mein se ek hai. Subscript ya letter padho aur tum turant jaante ho woh angle kaunsa kaam kar raha hai.

8. Constants: ${\mu }_0$, $\pi$, aur turn counts $N$, $n$

Yeh topic ko kyun chahiye: yeh ek bare single-wire result ko ek real coil ke field mein turn karte hain. $N$ se multiply karna bhool jaana parent ka flagged classic mistake hai.

Prerequisite map

Ise ek chain ki tarah padho, upar se neeche — neeche wala har idea upar wale par tika hai:

1. Current $I$ flow hota hai → woh ek magnetic field $\vec{B}$ banata hai jo usse circles karta hai.
2. Vectors aur unit vector $\hat{r}$ har arrow ko ek direction dete hain.
3. Wire kaatna tiny pieces $d\vec{l}$ (length $dl$) deta hai.
4. Cross product ($\sin \theta$) naya perpendicular field direction banata hai → Biot-Savart Law ko power deta hai.
5. Dot product ($\cos \theta$) measure karta hai kitna lines up hai → Ampere's Law ko power deta hai.
6. Integrals $\int ,\oint$ saare pieces add karte hain; constants ${\mu }_0,\pi$ aur turn counts $N,n$ numbers supply karte hain.
7. Bahar aate hain parent topic ke straight wire, loop, solenoid, aur toroid ke fields.

Yeh tools kahan dobara aate hain

Ek baar yeh symbols aa gaye, toh yahi vocabulary neighbouring topics mein power karti hai: cross product ek current par force bhi deta hai, loop ka field actually ek magnetic dipole ki tarah hai, aur ek area par $\vec{B}$ add karna Magnetic flux deta hai, jo Faraday's law of induction ka ingredient hai.

Equipment checklist

Khud ko test karo — sirf answer ke baad reveal karo.