Magnetic force on current-carrying conductor

Deriving the force from first principles

Setup. Take a straight wire of cross-sectional area $A$ , length $L$ , carrying current $I$ . Let $n$ = number of free charge carriers per unit volume, each of charge $q$ , drifting with velocity $\vec v_d$ .

Step 1 — Force on one carrier. $\vec F_1 = q\,\vec v_d \times \vec B$ Why this step? This is the Lorentz force, the only magnetic force law we ever need.

Step 2 — Count the carriers. Number of carriers in the segment: $N = n \times (\text{volume}) = n\,A\,L$ Why this step? Total force = (force per carrier) × (how many carriers there are).

Step 3 — Total force. $\vec F = N\,\vec F_1 = nAL\,(q\,\vec v_d \times \vec B)$

Step 4 — Recognise the current. Recall the drift-current relation $I = nAq\,v_d \quad\Rightarrow\quad nAq\,\vec v_d = I\,\vec L$ where $\vec L$ is a vector of magnitude $L$ pointing along the conventional current direction. Why this step? It lets us replace the microscopic stuff $(n,A,q,v_d)$ with the macroscopic, measurable current $I$ .

For a bent/curved wire we sum infinitesimal pieces: $d\vec F = I\,d\vec\ell \times \vec B \quad\Rightarrow\quad \vec F = I\!\int d\vec\ell \times \vec B$

Figure — Magnetic force on current-carrying conductor

Direction: the cross product / Fleming's Left-Hand Rule

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Forecast-then-verify: before reading on, predict the force on a wire

parallel to $\vec B$ . Prediction: zero. Verify: $\sin 0^\circ = 0 \Rightarrow F = 0$ . ✓ The carriers' drift velocity is along $\vec B$ , and $\vec v \parallel \vec B \Rightarrow \vec v\times\vec B = 0$ .

Recall Feynman — explain to a 12-year-old

Imagine a crowd of kids running through a windy field. The wind only shoves a kid sideways when the kid runs across the wind — if you run straight into the wind, this special "magnetic wind" doesn't push you. A wire with current is a crowd of running electric charges. Line up the charges across the magnetic field and the whole wire gets shoved sideways. Turn the wire to run along the field and nobody gets pushed. That sideways shove is what makes motors spin.

Flashcards

Force on a straight current wire (vector form)?

\vec F = I\,\vec L \times \vec B

Magnitude of force on a wire at angle

\theta

B

F = BIL\sin\theta

Why does a wire parallel to

\vec B

feel no force?

\vec L\parallel\vec B\Rightarrow\vec L\times\vec B=0

(

\sin0^\circ=0

Which rule gives the force direction on a wire?

Fleming's Left-Hand Rule (F=Field=first finger, C=Current=second, M=Motion=thumb).

Net force on a closed current loop in a uniform field?

Zero (

\oint d\vec\ell=0

); but there can be a net torque.

How do we go from

q\vec v\times\vec B

I\vec L\times\vec B

Use

nAq\vec v_d = I\vec L

and multiply force per carrier by

N=nAL

carriers.

Force on a curved wire between two ends in uniform

B

Same as a straight wire joining the ends:

\vec F = I\vec L_{\text{eff}}\times\vec B

Force on

0.5

m wire,

4

0.3

T, perpendicular?

F=BIL=0.60

Connections

Lorentz force on a moving charge — the parent law we derived this from.
Drift velocity and current — supplies $I = nAqv_d$ .
Torque on a current loop — the loop's net force is zero but torque drives motors.
Electric motor / Moving-coil galvanometer — direct applications.
Force between two parallel currents — each wire sits in the other's field.
Cross product (vectors) — the geometry behind the perpendicularity.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek current-carrying wire basically charges ki ek bheed hai jo move kar rahi hai. Hum jaante hain ki ek single moving charge ko magnetic field mein force lagta hai: $\vec F = q\vec v\times\vec B$ . Ab wire mein toh crore-crore aise charges ek saath chal rahe hain, toh poori wire ko force lagega. Saare charges ka force jod do aur use $I = nAqv_d$ se simplify karo, toh seedha formula nikalta hai: $\vec F = I\vec L\times\vec B$ , magnitude $F = BIL\sin\theta$ .

Sabse important baat — yeh ek cross product hai. Iska matlab force na current ke direction mein hai, na field ke direction mein, balki dono ke perpendicular. Agar wire field ke parallel ho, toh $\theta=0$ , $\sin0=0$ , force zero! Direction nikalne ke liye Fleming's Left-Hand Rule: pehli ungli Field, doosri ungli Current, aur angootha Force/Motion batata hai.

Yeh concept kyun zaroori hai? Kyunki yahi har electric motor, loudspeaker aur galvanometer ka dil hai — bijli ko motion mein badalna. Ek aur trick yaad rakho: uniform field mein ek curved wire ka net force utna hi hota hai jitna seedhi wire ka jo uske dono ends ko jodti hai. Aur ek closed loop ka net force zero hota hai (lekin torque ho sakta hai — wahi motor ghumata hai). Exam mein $\sin\theta$ kabhi mat bhoolna, aur left hand use karna, right hand nahi!

Magnetic force on current-carrying conductor

Deriving the force from first principles

Direction: the cross product / Fleming's Left-Hand Rule

Worked examples

Common mistakes (steel-manned)

Active recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections