Foundations — Magnetic force on current-carrying conductor

1.8.21 · D1 · Physics › Electromagnetism › Magnetic force on current-carrying conductor

Is page pe koi assumption nahi hai. Jab tak tum $\vec{F}=I\vec{L}\times \vec{B}$ ko parent note pe nahi milte, us formula ka har letter, arrow, aur cross pehle se obvious lagna chahiye. Neeche hum har piece ko ground up se build karte hain, usi order mein jisme woh stack hote hain.

1. Arrows: "vector" kya hota hai

Hume arrows ki zaroorat kyun hai? Kyunki is topic mein direction hi sab kuch hai. Current ek direction mein jaati hai, field doosri direction mein point karti hai, aur force teesri direction mein end hoti hai jo dono mein se koi nahi hai. Ek plain number (jaise "5") teen alag-alag directions kabhi store nahi kar sakta — ek arrow kar sakta hai.

2. Direction labels: $\hat{x}$, $\hat{y}$, $\hat{z}$

Inhe ek kamre ke kone ki teen edges ki tarah socho:

• $\hat{x}$ → east (right),
• $\hat{y}$ → north (tumse door),
• $\hat{z}$ → upar.

Koi bhi real arrow tab "itna east, itna north, itna upar" hota hai. Is tarah hum current ko ek direction mein aur field ko doosri mein point kar sakte hain aur phir bhi bookkeeping seedhi rakh sakte hain.

3. Charge $q$ aur electron

Parent note $q$ ki kyun parwah karta hai? Kyunki ek single carrier pe force uske charge ke proportional hoti hai. Charge nahi, toh magnetic push nahi. Yeh $q$ woh seed hai jisse poori derivation ugti hai.

4. Velocity $\vec{v}$ aur drift velocity ${\vec{v}}_d$

Wire ke andar, electrons zyada speed pe randomly jhangate hain, lekin jab battery connected hoti hai toh woh ek overall direction mein bhi dheere creep karte hain. Woh dheema overall creep hi drift velocity ${\vec{v}}_d$ hai.

5. Marching charges se current $I$ tak

Ab wire ke andar ki cheezein assemble karo. Wire ko do:

• cross-sectional area $A$ ("pipe" ki size, ${\mathrm{m}}^2$ mein),
• $n$ = har cubic metre mein packed charge carriers ki sankhya ("crowd density", ${\mathrm{m}}^{-3}$ mein),
• har carrier charge $q$ ka, $v_d$ speed se drift karta hua.

Ek second mein har carrier $v_d$ distance aage move karta hai. Toh $v_d$ length aur $A$ area ke ek box ke andar ke saare carriers finish line cross kar lete hain. Us box mein $nA{v}_d$ carriers hain, har ek charge $q$ carry karta hai, jo deta hai:

6. Length vector $\vec{L}$

Yeh "wire kitni lambi hai" aur "current kaunsi direction mein jaati hai" ko ek object mein package karta hai — exactly wahi jo ek cross product khana chahta hai.

7. Magnetic field $\vec{B}$

8. Cross product $\times$ — sab kuch ka dil

Yeh woh akela symbol hai jo logon ko dara deta hai, isliye hum isse dheere build karte hain.

Cross product kyun, aur ordinary multiplication kyun nahi? Kyunki magnetic force sideways hai. Ordinary multiplication ($5\times 3$) ek number banata hai aur koi direction nahi rakhta; dot product bhi ek number deta. Sirf cross product ek teesra, perpendicular arrow manufacture karta hai — jo precisely woh sideways shove hai jo hum observe karte hain. Nature ne ek perpendicular result demand kiya, isliye hum ek aise tool ke paas jaate hain jo ek produce karta hai. (Zyada geometry Cross product (vectors) mein.)

9. Sab kuch jodna

Parent ke boxed formula mein har symbol ka ab ek chehra hai:

$\vec{F}=\underset{\S 5}{\underbrace{I}}\underset{\S 6}{\underbrace{\vec{L}}}\underset{\S 8}{\underbrace{\times }}\underset{\S 7}{\underbrace{\vec{B}}}$

• $I$ — har second mein kitna charge march karta hai,
• $\vec{L}$ — wire (aur current) kitni dur aur kaunsi direction mein chalti hai,
• $\times$ — woh machine jo "wire direction" aur "field direction" ko ek perpendicular push mein badalta hai,
• $\vec{B}$ — woh field jo push kar rahi hai.

Aur magnitude $F=BIL\sin \theta$ sirf §8 ka cross-product length formula hai jisme letters fill in kiye gaye hain. Kuch naya nahi — tum already har piece ke malik ho.

Prerequisite map

Equipment checklist

Cover the right side and answer out loud before revealing.

Connections

• Magnetic force on current-carrying conductor — woh topic jise yeh foundations feed karte hain.
• Lorentz force on a moving charge — ek carrier pe force, $\vec{F}=q\vec{v}\times \vec{B}$.
• Drift velocity and current — jahan $I=nAq{v}_d$ janam leta hai.
• Cross product (vectors) — §8 ki geometry poori tarah se.
• Torque on a current loop · Electric motor · Moving-coil galvanometer — jahan yeh sab le jaata hai.
• Force between two parallel currents — yeh do forces ek dusre se milti hain.