Magnetic force on current-carrying conductor: Visual walkthrough

1.8.21 · D2 · Physics › Electromagnetism › Magnetic force on current-carrying conductor

Step 1 — Ek charge, ek dhakka

Kisi bhi symbol se pehle: ek magnetic field ek aisi invisible influence hai jo space ke kisi region ko fill karti hai. Hum use parallel arrows ke ek bundle ki tarah draw karte hain. Hum ise $B$ likhte hain — upar chhota arrow matlab iska ek direction hai, sirf size nahi. Size $B$ ko tesla (T) mein measure karte hain; bada $B$ = zyada push.

Ek charge ek aisa particle hai jo electricity carry karta hai; uski quantity $q$ hai (unit: coulomb, C). Yeh drift karta hai — bahut dheere crawl karta hai — velocity $v_{d}$ ke saath ("d" drift ke liye hai). Velocity bhi ek arrow hai: kaunsi taraf, kitna fast.

Figure — Magnetic force on current-carrying conductor

Cross product ki magnitude (plain size, koi direction nahi) yeh hai: $∣ v_{d} \times B ∣ = v_{d} B sin θ,$ jahan $θ$ do arrows ke beech ka angle hai. Is $sin θ$ ko yaad rakho — yeh final formula tak survive karta hai.

Step 2 — Cross product kaisa dikhta hai (sideways rule)

$v_{d}$ aur $B$ ko tail-to-tail rakho. Yeh ek flat sheet (ek plane) banate hain. Cross product $v_{d} \times B$ us sheet se seedha bahar point karta hai, dono arrows ke perpendicular. Iski length woh area hai jo do arrows milkar parallelogram banate hain — aur woh area $v_{d} B sin θ$ hai.

Recall Do arrows ek hi taraf point kar rahe hain. Unka cross product kya hai?

Zero. ::: Same direction ⇒ $θ = 0 \Rightarrow sin 0^{\circ} = 0$ , toh parallelogram ka koi area nahi.

Step 3 — Wire in charges ki ek bheed hai

Nayi pictures, naye symbols, sab ab earned hain:

$A$ — wire ka cross-sectional area (agar aap use kaat dein toh aapko circular face ka size dikhega, unit $m^{2}$ ).
$L$ — us segment ki length jo hum study kar rahe hain (unit m).
$n$ — number density: har cubic metre mein kitne free charge carriers hain (unit $m^{- 3}$ ).

Step 4 — Har push ko add karo

Yeh sahi hai lekin ugly hai: yeh microscopic cheezaon se bhara hai jo hum easily measure nahi kar sakte — kitne carriers hain, kitna dheere crawl karte hain, har charge kitna bada hai. Step 5 in sab ko ek number ke andar chhupa deta hai jo aap actually ammeter par padh sakte ho: current $I$ .

Step 5 — Bheed ko current $I$ se trade karo

Current $I$ (unit: ampere, A) woh hai ki har second mein ek point se kitna charge flow karta hai. Drift-current relation ise bheed se connect karta hai: $I = n A q v_{d} .$

$n$ — carriers per volume, $A$ — area, $q$ — har ek ka charge, $v_{d}$ — drift speed.
Inhe multiply karo aur aapko charge-per-second milega: exactly current.

Ab ek direction attach karo. $L$ define karo = length $L$ ka ek vector jo conventional current direction mein point karta hai (woh taraf jidhar positive charge flow hota dikhta hai). Toh messy clump neatly line up ho jata hai: $n A q v_{d} = I L .$

Ise seedha Step 4 ki raw force mein substitute karo: $F = (n A q v_{d}) \times B = I L \times B .$

Magnitudes lete hain (Step 2 ka rule): $F = B I L sin θ,$ jahan $θ$ wire aur field ke beech ka angle hai. Yeh bachne wala $sin θ$ wahi hai jo hum single electron par mila tha.

Step 6 — Saare angle cases (edge behaviour)

$θ = 9 0^{\circ}$ (wire $⊥$ field). $sin 9 0^{\circ} = 1$ , toh $F = B I L$ — maximum. Drift velocity field ko seedhi taraf kaatti hai: sabse bada parallelogram, sabse bada dhakka.
$θ = 0^{\circ}$ (wire $∥$ field). $sin 0^{\circ} = 0$ , toh $F = 0$ — koi force nahi. Charges field ke saath saath drift karte hain; field ke pakadne ke liye koi "across" nahi hai. (Windy-field wale bache se compare karo: hawa ke seedhe andar bhago, koi sideways push nahi.)
Beech mein, jaise $θ = 3 0^{\circ}$ . Sirf wire ka woh hissa count hoga jo field ke across hai; $sin 3 0^{\circ} = 0.5$ se maximum ka aadha milta hai.

Step 7 — Curved aur closed wires (shortcut)

Ek infinitesimal piece ke liye: $d F = I d ℓ \times B \Rightarrow F = I \int d ℓ \times B .$

$d ℓ$ — us jagah current ke saath ek tiny straight length arrow.
$\int$ — "saare tiny pieces ko add karo" (integral sign, sirf ek fancy summation).

Ek-picture summary

Left se right padho: ek charge $q v_{d} \times B$ feel karta hai → unhe count karo $n A L$ → total $n A L (q v_{d} \times B)$ → rename $n A q v_{d} = I L$ → bahar aata hai $F = I L \times B$ , jiska size $B I L sin θ$ hai. Lorentz force on a moving charge andar gaya; wire law bahar aaya. Ek wire ke doosre wire ke field mein hone ke liye Force between two parallel currents dekho.

Recall Feynman retelling — poora walkthrough ek 12-saal ke bache ko batao

Sochlo ek bachcha ek windy field mein cross karke daud raha hai; ek ajeeb magnetic hawa use sideways dhakelta hai — aur sirf tab jab woh hawa ke across daude, kabhi seedhe iske andar nahi. Ab is daudne wale raste ko puri bheed se bharo: yahi current carry karne wali wire hai. Count karo ki hose mein kitne bache fit hote hain (yahi $n A L$ hai), har ek ko same sideways dhakka do, aur add karo. Bookkeeping messy hai — bache-counting se bhari — toh hum poori bheed ko ek aisi cheez se swap karte hain jo hum actually measure kar sakte hain: current $I$ . Bahar aata hai ek clean rule: wire ka dhakka $I$ times uska length arrow hai jo field ke saath "crossed" ho, aur yeh sabse bada hota hai jab wire field ke across chale, zero jab woh field ke saath chale. Yahi sideways dhakka motors ko ghoomata hai.

Recall Formula ko memory se paanch words har step mein rebuild karo.

Q: Paanch moves kya hain? ::: One-charge force → count carriers → total force → rename as $I L$ → wire law $F = I L \times B$ .

Quick self-test

Force on a

0.5

m wire,

4

0.3

T, perpendicular?

F = B I L = 0.3 \times 4 \times 0.5 = 0.60

Same wire at

3 0^{\circ}

F = B I L sin 3 0^{\circ} = 0.60 \times 0.5 = 0.30

Net force on a closed loop in a uniform field?

Zero, since

\oint d ℓ = 0

Force on a semicircle of radius

R

(uniform

B

out of page)?

F = I (2 R) B

, along the diameter's perpendicular.

Connections

Lorentz force on a moving charge — Step 1 ka seed law.
Drift velocity and current — Step 5 mein use hone wala $I = n A q v_{d}$ supply karta hai.
Cross product (vectors) — Steps 1–2 ki geometry.
Torque on a current loop — jab net force vanish ho jaaye toh kya bachta hai (Step 7).
Electric motor / Moving-coil galvanometer — payoff.
Force between two parallel currents — har wire doosri wire ke field mein.