Magnetic force on current-carrying conductor: Visual walkthrough

Step 1 — One charge, one shove

Before any symbol: a magnetic field is an invisible influence filling a region of space. We draw it as a bundle of parallel arrows. We write it $B$ — the little arrow on top means it has a direction, not just a size. The size $B$ is measured in tesla (T); bigger $B$ = stronger push.

A charge is a particle carrying electricity; its amount is $q$ (unit: coulomb, C). It drifts — crawls slowly — with velocity $v_{d}$ (the "d" is for drift). Velocity is also an arrow: which way, how fast.

Figure — Magnetic force on current-carrying conductor

The magnitude (the plain size, no direction) of a cross product is $∣ v_{d} \times B ∣ = v_{d} B sin θ,$ where $θ$ is the angle between the two arrows. Hold that $sin θ$ — it survives all the way to the final formula.

Step 2 — What the cross product looks like (the sideways rule)

Put $v_{d}$ and $B$ tail-to-tail. They span a flat sheet (a plane). The cross product $v_{d} \times B$ points straight out of that sheet, perpendicular to both arrows. Its length is the area of the parallelogram the two arrows make — and that area is $v_{d} B sin θ$ .

Recall Two arrows point the same way. What is their cross product?

Zero. ::: Same direction ⇒ $θ = 0 \Rightarrow sin 0^{\circ} = 0$ , so the parallelogram has no area.

Step 3 — A wire is a crowd of these charges

New pictures, new symbols, all earned now:

$A$ — the cross-sectional area of the wire (the size of the circular face you'd see if you cut it, unit $m^{2}$ ).
$L$ — the length of the segment we study (unit m).
$n$ — the number density: how many free charge carriers sit in each cubic metre (unit $m^{- 3}$ ).

Step 4 — Add up every push

This is correct but ugly: it is full of microscopic things we can't easily measure — how many carriers, how fast they crawl, how big each charge is. Step 5 hides all of that inside one number you can read off an ammeter: the current $I$ .

Step 5 — Trade the crowd for the current $I$

Current $I$ (unit: ampere, A) is how much charge flows past a point each second. The drift-current relation connects it to the crowd: $I = n A q v_{d} .$

$n$ — carriers per volume, $A$ — area, $q$ — charge each, $v_{d}$ — drift speed.
Multiply them and you get charge-per-second: exactly the current.

Now attach a direction. Define $L$ = a vector of length $L$ pointing along the conventional current direction (the way positive charge appears to flow). Then the messy clump lines up neatly: $n A q v_{d} = I L .$

Substitute this straight into the raw force from Step 4: $F = (n A q v_{d}) \times B = I L \times B .$

Taking magnitudes (Step 2's rule): $F = B I L sin θ,$ with $θ$ the angle between the wire and the field. That surviving $sin θ$ is the same one we met on the single electron.

Step 6 — All the angle cases (edge behaviour)

$θ = 9 0^{\circ}$ (wire $⊥$ field). $sin 9 0^{\circ} = 1$ , so $F = B I L$ — maximum. The drift velocity cuts straight across the field: biggest parallelogram, biggest shove.
$θ = 0^{\circ}$ (wire $∥$ field). $sin 0^{\circ} = 0$ , so $F = 0$ — no force. The charges drift along the field; there is no "across" for the field to grab. (Compare the windy-field kid: run straight into the wind, no sideways push.)
Between, e.g. $θ = 3 0^{\circ}$ . Only the part of the wire across the field counts; $sin 3 0^{\circ} = 0.5$ gives half the maximum.

Step 7 — Curved and closed wires (the shortcut)

For an infinitesimal piece: $d F = I d ℓ \times B \Rightarrow F = I \int d ℓ \times B .$

$d ℓ$ — a tiny straight length arrow along the current at that spot.
$\int$ — "add up all the tiny pieces" (the integral sign, just a fancy summation).

The one-picture summary

Read it left to right: one charge feels $q v_{d} \times B$ → count $n A L$ of them → total $n A L (q v_{d} \times B)$ → rename $n A q v_{d} = I L$ → out pops $F = I L \times B$ , whose size is $B I L sin θ$ . The Lorentz force on a moving charge went in; the wire law came out. See Force between two parallel currents for one wire sitting in another's field.

Recall Feynman retelling — say the whole walkthrough to a 12-year-old

Imagine one kid running across a windy field; a strange magnetic wind shoves them sideways — and only when they run across the wind, never straight into it. Now fill a hose-shaped path with a whole crowd of these running kids: that's a wire carrying current. Count how many kids fit in the hose (that's $n A L$ ), give each the same sideways shove, and add. The bookkeeping is messy — full of kid-counting — so we swap the whole crowd for one thing we can actually measure: the current $I$ . Out comes a clean rule: the wire's shove is $I$ times its length arrow "crossed" with the field, and it's biggest when the wire runs across the field, zero when it runs along it. That sideways shove is what makes motors turn.

Recall Rebuild the formula from memory in five words each step.

Q: The five moves? ::: 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's seed law.
Drift velocity and current — supplies $I = n A q v_{d}$ used in Step 5.
Cross product (vectors) — the geometry of Steps 1–2.
Torque on a current loop — what survives when net force vanishes (Step 7).
Electric motor / Moving-coil galvanometer — the payoff.
Force between two parallel currents — each wire in the other's field.