Foundations — Magnetic force on charge — F = qv × B

1.8.20 · D1 · Physics › Electromagnetism › Magnetic force on charge — F = qv × B

Isse pehle ki tum $\vec{F}=q\vec{v}\times \vec{B}$ par trust kar sako, uska har ek piece aisa hona chahiye jo tum dekh sako. Yeh page har ek symbol ko zero se build karta hai, ek aise order mein jahan har ek sirf apne pehle walon ko use karta hai.

1. Ek number vs. ek arrow — scalars aur vectors

Picture dekho. Arrow ki length uski size hai (the "how much"), aur jis taraf wo point karta hai wo uski direction hai (the "which way"). Ek scalar sirf length hai; ek vector pointing ko bhi saath rakhta hai.

Yeh topic ise kyun chahta hai: velocity $\vec{v}$, force $\vec{F}$, aur field $\vec{B}$ sab vectors hain. Agar tum sirf unki sizes yaad rakhte aur directions phek dete, tum kabhi nahi keh sakte ki force "sideways" hai — sideways ek direction statement hai, aur sirf arrows hi directions carry karte hain.

2. Ek arrow ki size — magnitude bars $|\cdot |$

Yeh topic ise kyun chahta hai: magnitude formula $|\vec{F}|=|q|vB\sin \theta$ dono meanings ko ek saath use karta hai — vectors ke aas paas bars ($|\vec{F}|$) length ka matlab hai, scalar charge ke aas paas bars ($|q|$) sign-stripped size ka matlab hai, aur plain $v$, $B$ lengths $|\vec{v}|$, $|\vec{B}|$ ke shorthand hain. Ek force magnitude hamesha $\ge 0$ hai, toh right side par har ingredient bhi $\ge 0$ hona chahiye — exactly yahi bars guarantee karte hain.

3. Ek arrow ko pieces mein todna — components

Figure mein, tilted arrow ek horizontal shadow ($v_x$) aur ek vertical shadow ($v_y$) mein split hota hai. Un do shadow-arrows ko tip-to-tail jodne se original rebuild ho jaata hai. Yahi "components" ka matlab hai: ek arrow ko reference axes ke along arrows ke sum ke roop mein likhna.

Yeh topic ise kyun chahta hai: parent note mein foolproof direction method ek component recipe hai, $(v_y{B}_z-v_z{B}_y,\dots )$. Tum ise use nahi kar sakte jab tak tum $\vec{v}$ aur $\vec{B}$ ko teen numbers ki list ke roop mein nahi padh sakte.

4. Do arrows ke beech ka angle — $\theta$

Yeh topic ise kyun chahta hai: magnitude $|\vec{F}|=|q|vB\sin \theta$ is angle par depend karta hai. Jab $\vec{v}$ $\vec{B}$ ke along point karta hai, $\theta =0^{\circ }$; jab yeh across point karta hai, $\theta =90^{\circ }$.

5. $\sin \theta$ kyun? — "kitna sideways hai" measure karna

Velocity arrow ko imagine karo jo field ke angle $\theta$ par jhuka hua hai. Use do pieces mein toddo: ek $\vec{B}$ ke parallel aur ek $\vec{B}$ ke perpendicular.

• Perpendicular (sideways) piece ki length $v\sin \theta$ hai.
• Parallel (along) piece ki length $v\cos \theta$ hai.

Jab $\theta =0^{\circ }$ (velocity field ke along), $\sin 0^{\circ }=0$ — koi sideways part nahi — toh koi force nahi. Jab $\theta =90^{\circ }$ (velocity fully across), $\sin 90^{\circ }=1$ — sab sideways — toh maximum force. Yahi exactly wo experimental pattern hai jo parent note mein pada jaata hai, aur $\sin \theta$ wo tool hai jo "kitna $\vec{v}$ $\vec{B}$ ke across hai" capture karta hai.

6. Cross product — "do arrows ke perpendicular ek arrow"

Experiments kehte hain ki force $\vec{v}$ aur $\vec{B}$ dono ke perpendicular hai ek saath, size $\sin \theta$ ke proportional hai. Exactly ek arrow-operation hai jo dono karta hai. Woh Cross product hai.

Figure mein do arrows ek flat plane mein hain aur cross-product arrow seedha us plane se bahar upar khada hai — ek saath dono ke right angle par hone ka yahi ek tarika hai.

Component recipe kahan se aati hai? Teen unit vectors se shuru karo. Do rules upar waale (dono ke perpendicular, length $\sin \theta$) ko axis arrows par khud apply karo. Kyunki $\hat{i},\hat{j},\hat{k}$ mein har ek length $1$ hai aur ek doosre ke $90^{\circ }$ par hain, do alag waalon ka har cross product length $1\cdot 1\cdot \sin 90^{\circ }=1$ rakhta hai aur teesre axis ke along point karta hai; right-hand rule (hamara right-handed set) sign fix karta hai: $\hat{i}\times \hat{j}=\hat{k},\hat{j}\times \hat{k}=\hat{i},\hat{k}\times \hat{i}=\hat{j}$ (cycle $x\to y\to z\to x$ mein aage jaana $+$ deta hai; peeche jaana $-$ deta hai, jaise $\hat{j}\times \hat{i}=-\hat{k}$). Saath hi koi bhi arrow khud se cross karne par $\vec{0}$ deta hai (angle $0$, $\sin 0=0$): $\hat{i}\times \hat{i}=\vec{0}$.

Ab sirf $\vec{v}\times \vec{B}$ ko term by term multiply karo, $\times$ ko distribution ki tarah treat karo aur un nine chhoti identities use karo: $\vec{v}\times \vec{B}=(v_x\hat{i}+v_y\hat{j}+v_z\hat{k})\times (B_x\hat{i}+B_y\hat{j}+B_z\hat{k})$ Har "same-axis" term mar jaata hai ($\hat{i}\times \hat{i}=\vec{0}$), aur surviving cross-terms cycle rules ke through collapse ho ke yeh ban jaata hai:

Yeh bas upar wali unit-vector algebra hai, likhi hui — determinant/box pattern sirf usi same result ke liye ek memory aid hai.

Yeh topic ise kyun chahta hai: yeh single operation parent note ke Observations 2, 3 aur 4 ko package karta hai — perpendicularity aur $\sin \theta$ magnitude — ek clean symbol mein.

7. Charge $q$ aur uska sign

Yeh topic ise kyun chahta hai: $q$ force ko scale karta hai (double $q$, double force) aur uska sign direction ko flip karta hai. Ek negative $q$ force ko right-hand-rule thumb ki opposite direction mein bhejta hai — isi liye ek electron ek proton ke opposite curve karta hai. Magnitude formula mein hum $|q|$ (absolute value) use karte hain kyunki ek length kabhi negative nahi hoti.

8. Magnetic field $\vec{B}$

Yeh topic ise kyun chahta hai: $\vec{B}$ cross product mein feed hone wale do arrows mein se ek hai. Field ke liye ek direction ke bina, "field ke sideways" ka koi matlab nahi hoga.

9. Force $\vec{F}$, aur kyun perpendicular force cheezein ghuma deta hai

Yeh topic ise kyun chahta hai: dot product woh tool hai jo "no work, no speed change" ke peeche hai — magnetic force ki sabse important personality trait.

10. Ye sab topic ko kaise feed karte hain

Ise upar se neeche padho: arrows aur unki sizes raw material hain; angle tumhe $\sin \theta$ deta hai; ye cross product mein combine hote hain; charge aur field ise feed karte hain; puri cheez law ban jaati hai, jo phir "no work" aur circular motion dono explain karta hai.

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Connections

• ↑ Parent topic
• Cross product
• Lorentz force law
• Centripetal force and circular motion