Magnetic force on charge — F = qv × B — Hinglish notes

1.8.20 · Physics › Electromagnetism

LAW kya hai?

Magnitude: $∣ F ∣ = ∣ q ∣ v B sin θ$ jahan $θ$ , $v$ aur $B$ ke beech ka angle hai.

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

Cross product kyun? (First principles se Derivation)

Hum formula ko postulate nahi karte — hum isse experiments se padhte hain, phir pattern ko encode karte hain.

Observation 1 — Motion zaroori hai. Ek magnet ke field mein stationary charge koi magnetic force feel nahi karta. Toh $F \propto v$ (yeh zero ho jaata hai jab $v = 0$ ).

Observation 2 — Sideways component chahiye. Ek charge ko field lines ke saath fire karo: koi force nahi. Ise cross karo: maximum force. Toh effective speed woh part hai jo $v$ ka $B$ ke perpendicular hai, yani $v sin θ$ . Isliye $F \propto v sin θ$ .

Observation 3 — Force dono ke perpendicular hai. Deflection measure karo: yeh hamesha $⊥ v$ aur $⊥ B$ hota hai. Woh unique vector operation jo (a) do inputs ke perpendicular result deta hai aur (b) magnitude $\propto sin θ$ hoti hai, woh hai cross product.

Observation 4 — Charge ke proportional, sign matter karta hai. $q$ double karo → force double; $q$ ka sign flip karo → direction flip.

Charon ko ek vector statement mein bundle karo: $F = q v \times B$ Proportionality constant SI mein exactly $1$ hai — kyunki tesla defined hi aise kiya gaya hai taki aisa ho. Toh formula aadha experiment hai, aadha units ka ek clean choice.

Direction kaise nikaalay — right-hand rule

Component form (foolproof method):

\begin{vmatrix} \hat i & \hat j & \hat k\\ v_x & v_y & v_z\\ B_x & B_y & B_z\end{vmatrix} =(v_yB_z-v_zB_y)\,\hat i+(v_zB_x-v_xB_z)\,\hat j+(v_xB_y-v_yB_x)\,\hat k$$ --- ## Circular motion: sabse important consequence Agar $\vec v\perp\vec B$ (toh $\sin\theta=1$), toh force ki magnitude constant $qvB$ hai aur hamesha motion ke perpendicular hai → uniform **circular motion**. **Radius derive karo.** Centripetal requirement: $\dfrac{mv^2}{r}=qvB$. Ek $v$ cancel karo: $$\boxed{r=\frac{mv}{qB}}$$ **Period** (ek loop ke liye time): distance $2\pi r$ speed $v$ par: $$T=\frac{2\pi r}{v}=\frac{2\pi m}{qB},\qquad \omega=\frac{qB}{m}$$ > [!intuition] Cyclotron motion ka surprise > $T$ mein **koi $v$ nahi hai!** Ek fast particle bada circle trace karta hai, slow wala chhota, lekin dono ek lap **same time** mein finish karte hain. Isliye cyclotrons kaam karte hain. Frequency $\omega=qB/m$ ko ==cyclotron frequency== kehte hain. Agar $\vec v$ ka $\vec B$ ke saath bhi koi component hai, toh woh component **koi force feel nahi karta**, toh particle circle karte hue aage drift karta hai → ek **helix**. --- ## Worked Examples > [!example] 1 — Magnitude formula mein plug karo > Ek proton ($q=1.6\times10^{-19}\,$C) $v=2\times10^6$ m/s se $30^\circ$ par field $B=0.5$ T mein move karta hai. $|\vec F|$ nikalo. > > $|\vec F|=qvB\sin\theta=(1.6\times10^{-19})(2\times10^6)(0.5)\sin30^\circ$ > *Yeh step kyun?* Magnitude ke liye sirf $\sin\theta$ chahiye, koi vector bookkeeping nahi. > $=(1.6\times10^{-19})(2\times10^6)(0.5)(0.5)=8\times10^{-14}\,$N. > [!example] 2 — Full vector cross product > $q=+2$ C, $\vec v=(3,0,0)$ m/s, $\vec B=(0,4,0)$ T. $\vec F$ nikalo. > > $\vec v\times\vec B=(v_yB_z-v_zB_y,\ v_zB_x-v_xB_z,\ v_xB_y-v_yB_x)=(0-0,\ 0-0,\ 3\cdot4-0)=(0,0,12)$. > *Yeh step kyun?* Sirf $v_xB_y$ bachta hai, ek $+\hat k$ result deta hai — force page se bahar point karta hai. > $\vec F=q(\vec v\times\vec B)=2(0,0,12)=(0,0,24)\,$N. > [!example] 3 — Electron, sign dhyaan rakho > Ek electron ($q=-1.6\times10^{-19}$ C) $+\hat x$ mein $\vec B$ ke through $+\hat z$ mein move karta hai. Force ki direction? > > $\hat x\times\hat z=-\hat y$. Toh $\vec v\times\vec B$, $-\hat y$ mein point karta hai. > *Yeh step kyun?* $\hat x\times\hat z=-\hat y$ (cyclic order hai $x\to y\to z$; $x\to z$ jaana backwards hai → minus). > Negative charge se multiply karo: $\vec F\propto q(-\hat y)=(-)(-\hat y)=+\hat y$. Electron $+\hat y$ mein deflect hota hai. > [!example] 4 — Electron ke circle ka radius > Electron, $v=1\times10^7$ m/s, $B=0.01$ T, $m=9.1\times10^{-31}$ kg. > $r=\dfrac{mv}{qB}=\dfrac{(9.1\times10^{-31})(10^7)}{(1.6\times10^{-19})(0.01)}\approx 5.7\times10^{-3}\,$m $\approx 5.7$ mm. > *Yeh step kyun?* $r=mv/qB$ use kiya $|q|$ ke saath (radius ek magnitude hai). --- ## Common Mistakes (Steel-manned) > [!mistake] "Magnetic force kaam karta hai aur particle ki speed badhata hai." > **Kyun sahi lagta hai:** yeh ek force hai, aur forces aam taur par speed change karte hain (Newton ki training). > **Fix:** $P=\vec F\cdot\vec v=0$ kyunki $\vec F\perp\vec v$ *hamesha*. Yeh **direction** change karta hai, **speed** nahi. Sirf electric field kisi charge par kaam karta hai. > [!mistake] "$\sin\theta$ bhool jaana / poora $v$ use karna." > **Kyun sahi lagta hai:** simplest problems mein $\vec v\perp\vec B$, toh $\sin\theta=1$ aur tumhein yeh kabhi dikhta nahi. > **Fix:** force sirf perpendicular part $v\sin\theta$ par depend karta hai. Agar $\vec v\parallel\vec B$, force $=0$. > [!mistake] "Electron proton ki tarah same direction mein curve karta hai." > **Kyun sahi lagta hai:** same right-hand rule, same $\vec v$ aur $\vec B$. > **Fix:** right-hand rule $\vec v\times\vec B$ deta hai; **negative** charge ke liye result flip ho jaata hai. Same velocity wale electron aur proton **opposite** directions mein curve karte hain. > [!mistake] "Period speed par depend karta hai." > **Kyun sahi lagta hai:** fast cheezein aam taur par laps jaldi khatam karti hain. > **Fix:** $T=2\pi m/qB$ — speed cancel ho jaati hai. Faster ⇒ bada circle, same lap time. --- > [!recall]- Feynman: ek 12-saal ke bachche ko explain karo > Socho tum ice par slide kar rahe ho aur ek invisible haath tumhe hamesha **sideways** dhakelta hai, kabhi aage ya peeche nahi. Tum na tez ho sakte ho na dheema, lekin tum lagaataar mud'te rehte ho — toh tum **circle** mein jaate ho. Ek magnet ek chalti hui charged ball ke saath exactly yahi karta hai. Agar ball chal nahi rahi, toh haath uski parwah nahi karta. Aur jitna tez ball phenko, utna bada circle banata hai — lekin funny baat yeh hai ki ek chakkar mein hamesha **same amount of time** lagta hai. > [!mnemonic] Yaad rakho > **"Sideways Steering, No Speeding."** Magnetic force = steering wheel (tumhein mod'ta hai), kabhi gas pedal nahi (koi kaam nahi). Aur $\vec F=q\vec v\times\vec B$ mein **FvB** order — *"Force from velocity-cross-field."* --- ## Active Recall #flashcards/physics Charge par magnetic force ka full vector formula kya hai? ::: $\vec F = q\,\vec v\times\vec B$ Magnetic force ki magnitude kya hai? ::: $|F| = |q|\,v\,B\sin\theta$ Magnetic force particle ki speed kyun kabhi nahi badal sakti? ::: Kyunki $\vec F\perp\vec v$, toh power $\vec F\cdot\vec v=0$; kinetic energy constant rehti hai. Chalti hui charge par magnetic force kab zero hoti hai? ::: Jab $v=0$, ya jab $\vec v\parallel\vec B$ ($\sin\theta=0$). $\vec v\perp\vec B$ ke liye circular motion ka radius? ::: $r=\dfrac{mv}{qB}$ Cyclotron motion ka period, aur iske baare mein surprising kya hai? ::: $T=\dfrac{2\pi m}{qB}$; yeh speed $v$ se independent hai. Cyclotron (angular) frequency? ::: $\omega=\dfrac{qB}{m}$ Electron ki curving direction proton se kaise compare hoti hai (same $\vec v,\vec B$)? ::: Opposite, kyunki negative charge $\vec v\times\vec B$ ko flip kar deta hai. Agar $\vec v$ ke components $\vec B$ ke saath aur cross dono mein hain toh motion ka shape kya hoga? ::: Ek helix (circle + $\vec B$ ke saath drift). B ki SI unit aur uski definition? ::: Tesla; $1\,\text{T}=1\,\text{N}/(\text{A·m})$. --- ## Connections - [[Lorentz force law]] — full $\vec F=q(\vec E+\vec v\times\vec B)$, electric part bhi add karta hai. - [[Cross product]] — direction ke peeche ka math engine. - [[Centripetal force and circular motion]] — $mv^2/r$ provide karta hai jo $r$ nikaalane mein use hota hai. - [[Cyclotron]] aur [[Mass spectrometer]] — $r=mv/qB$ par built devices. - [[Magnetic force on a current-carrying wire]] — $\vec F=I\vec L\times\vec B$, same law jo chalti charges par summed hai. - [[Velocity selector]] — $qE$ aur $qvB$ ko balance karna. ## 🖼️ Concept Map ```mermaid flowchart TD LAW["F = q v x B"] MOTION["Charge must move"] CROSS["Cross product v x B"] MAG["Magnitude q v B sin theta"] PERP["Force perp to v and B"] RHR["Right-hand rule"] SIGN["Sign of q flips direction"] WORK["No work done"] SPEED["Speed constant, path bends"] TESLA["Tesla defined so constant = 1"] MOTION -->|F vanishes if v=0| LAW LAW -->|direction from| CROSS CROSS -->|gives| MAG CROSS -->|result| PERP CROSS -->|direction via| RHR LAW -->|proportional to q| SIGN PERP -->|F dot v = 0| WORK WORK -->|KE constant| SPEED LAW -->|SI units| TESLA ```