Force on a Current-Carrying Conductor

Drape a loose loop of wire between the poles of a horseshoe magnet, close a switch, and the wire jumps — visibly, audibly, like something alive. Open the switch and it falls slack. Reverse the battery and it jumps the other way. This is the motor effect, the demonstration that has opened this topic in classrooms for two centuries, and it is the exact physics spinning every fan, every train motor, every phone's vibration unit and every loudspeaker cone on Earth. Something in the magnetic field grabs a current-carrying wire and pushes. But wait — a wire is electrically neutral, and a magnetic field only pushes on moving charges. The resolution is the whole story of this module: inside the wire, billions upon billions of electrons are drifting, each one earning a tiny magnetic kick of $qv_dB$ . The lattice of the wire collects those kicks — some $10^{22}$ of them per few centimetres — and delivers them as one honest, macroscopic force. When the bookkeeping is done, every microscopic detail cancels out and a three-letter formula survives: $F = BIL\sin\theta$ . This module derives that formula from the drift picture (a derivation boards explicitly ask for), arms you with Fleming's left-hand rule for directions, and drills the craft: which angle counts, which length counts, and which direction the wire jumps. It is the bridge between the particle physics of the last two modules and the machines of the rest of this chapter.

Key takeaways

A current-carrying conductor in a magnetic field feels a force because each drifting charge inside it feels $qv_dB$ — and the wire collects all the kicks.
Summing over the carriers turns microscopic drift into the macroscopic formula $F = BIL\sin\theta$ , where $\theta$ is the angle between the wire and the field.
Every microscopic quantity ( $n$ , $A$ , $v_d$ , $q$ ) cancels on the way — only the current $I$ , the length $L$ in the field, and the field $B$ survive.
Direction: Fleming's left-hand rule — First finger Field, seCond finger Current (conventional!), thuMb the Motion (force).
Maximum force when wire ⊥ field ( $\sin 90° = 1$ ); zero force when wire ∥ field; only the perpendicular geometry earns the full $BIL$ .
KEY IDEA: a magnetic field pushes on a current-carrying wire with $F = BIL\sin\theta$ — billions of per-charge kicks $qv_dB$ , banked by the lattice into one force you can see.

The core insight

A straight conductor of length $L$ carrying current $I$ at angle $\theta$ to a uniform magnetic field $B$ experiences a force $F = BIL\sin\theta$ , perpendicular to both the wire and the field (Fleming's left-hand rule) — the collective result of the magnetic force $qv_dB$ on every drifting charge carrier inside it.

Where this lives today

Roughly forty-five percent of the world's electricity ends its journey inside an electric motor — every one of them a machine for harvesting $F = BIL$ . Your phone's vibration alert is a tiny off-balance motor; a Mumbai local train leans on the same equation at four megawatts. The loudspeaker playing anything near you right now is the formula running forty times a millisecond: a coil in a magnet gap, pushed back and forth by an audio-frequency current, force faithfully proportional to the signal because $F$ is exactly linear in $I$ — that linearity is why recorded music sounds like music. The frontier version is the Kibble balance, the instrument that now defines the kilogram: it weighs masses against the magnetic force on a current-carrying coil, using this module's physics in metrological anger. And the ampere's own SI story runs through this force — the unit was defined for seventy years by the push between parallel currents, a tale told properly in Parallel Currents & the Definition of the Ampere, coming up after the field-making modules.

The physical picture

Think of the wire as a hose packed with charge. From Drift Velocity & the Microscopic Picture you know what the current really is: a vast crowd of electrons creeping along at fractions of a millimetre per second. Place the wire in a magnetic field and each drifting carrier becomes a moving charge in a field — exactly the situation of Force on a Moving Charge: qv×B — so each one feels a sideways kick of magnitude $qv_dB$ . One kick is pathetically small: about $10^{-23}$ newtons. But a mere 25 cm of ordinary 1 mm² copper contains roughly $2\times10^{22}$ drifting carriers, all creeping the same way, all kicked the same way. The carriers cannot leave the wire — the lattice holds them in. So each sideways kick is handed over to the wire itself, through collisions and internal electric forces, and the wire as a whole feels the sum: a real, macroscopic push, strong enough to jump a wire off a bench or spin a rotor. The invisible slowness of the drift is redeemed by the astronomical size of the crowd.

Figure

The motor effect — every drifting carrier kicked the same way, the wire banks the total

Two structural facts are visible already in the picture, and both are exam gold. First, the force on the wire is perpendicular to both the wire and the field — it inherits that from every carrier's sideways kick. Second, reversing the current reverses every kick, so the wire pushes the other way: that reversal is how a DC motor keeps turning, and The DC Motor & Its Cousins (coming up in this chapter) is engineered entirely around it.

From one charge to the whole wire

The derivation below is a favourite of CBSE, ISC and A-Level papers ("derive the expression for the force on a current-carrying conductor"), and it is worth owning cold, because it is the single cleanest example in this chapter of the microscopic world conspiring to produce a simple macroscopic law.

F = BIL\sin\theta \qquad (1)

where: $B$ — magnetic flux density (T); $I$ — conventional current (A); $L$ — the length of wire inside the field (m); $\theta$ — the angle between the wire's direction and the field. Read the structure: the force doubles with the current, doubles with the immersed length, and cares about geometry through $\sin\theta$ — full price when the wire runs square across the field, nothing at all when the wire runs along it. The tesla itself is calibrated by this law: $1\ \mathrm{T} = 1\ \mathrm{N}$ per ampere-metre, which is why a 1 m wire carrying 1 A across a 1 T field feels exactly 1 N.

Derivation AThe force on a wire from the drift of its carriers

We derive $F = BIL\sin\theta$ by adding up the magnetic force on every charge carrier drifting inside a straight segment of wire. The strategy: count the carriers in the segment, compute the kick on each from the qv×B law, multiply, and watch the microscopic quantities cancel against the definition of current from Drift Velocity & the Microscopic Picture. The cancellation is the point — the final law contains nothing you cannot read off an ammeter and a ruler.

Step 1. Count the carriers. A wire segment of length $L$ and cross-sectional area $A$ , with $n$ free carriers per unit volume, contains $N = nAL$ drifting charges.

Step 2. Kick each one. Each carrier of charge $q$ drifts at speed $v_d$ along the wire, at angle $\theta$ to the field, so each feels a magnetic force of magnitude $f = qv_dB\sin\theta$ (Force on a Moving Charge: qv×B). All the kicks point the same way, perpendicular to both wire and field, because all carriers drift the same way.

Step 3. Add them up. Forces in the same direction add as numbers:

F = N f = (nAL)(qv_dB\sin\theta) = (nAv_dq)\,LB\sin\theta.

Step 4. Recognise the current. The bracket is exactly the microscopic expression for current, $I = nAv_dq$ , owned by Drift Velocity & the Microscopic Picture. Substitute it:

F = BIL\sin\theta.

Why the cancellation matters: the carrier density n, the wire's area A, the drift speed and even the sign and size of the carrier charge have all vanished. A thick wire with many slow carriers and a thin wire with few fast ones feel the SAME force at the same current — which is why one formula serves copper, aluminium, salt water and semiconductor alike. *Verified numerically: copper wire, $A = 1.0\ \mathrm{mm^2}$ , $n = 8.5\times10^{28}\ \mathrm{m^{-3}}$ , $I = 4.0$ A gives $v_d = I/nAe = 2.9\times10^{-4}$ m/s; a 0.25 m segment holds $N = 2.1\times10^{22}$ carriers, each kicked with $ev_dB = 1.4\times10^{-23}$ N at $B = 0.30$ T ⊥ — total $N \times f = 0.30$ N, exactly $BIL = 0.30\times4.0\times0.25$ . The microscopic sum and the macroscopic formula agree to the last digit.*

Notice what the sine acts on. $\theta$ is the angle between the wire and the field — not between the force and anything. A wire at $30°$ to the field feels half the maximum force ( $\sin 30° = 0.5$ ); a wire along the field feels nothing, because its carriers drift parallel to $B$ and the qv×B force dies at $\theta = 0$ . The force itself, whenever it exists, is perpendicular to the plane containing the wire and the field — direction next.

Fleming's left-hand rule

You could find the force direction by applying the right-hand rule of Force on a Moving Charge: qv×B to the conventional current. But a century of exam candidates has voted for the dedicated tool. Hold your left hand with thumb, first finger and second finger mutually at right angles:

First finger — Field. Point it along $B$ , north pole to south pole.

seCond finger — Current. Point it along the conventional current $I$ (positive-terminal-to-negative-terminal direction).

thuMb — Motion. It now points along the force on the wire — the direction the wire jumps.

Figure

Fleming's left-hand rule — three fingers, three directions, one verdict

Two disciplines keep the rule honest. Conventional current only: the second finger follows positive flow, always — if a question hands you electron flow, reverse it first, once, before your hand comes out. Left hand for motors: when you reach Electromagnetic Induction next chapter, a right-hand rule appears for generators; mixing the two hands is a classic mark-loser, so anchor the pairing now — motors are in your left hand. If you ever doubt your fingers, fall back on the one memorized anchor from Force on a Moving Charge: qv×B and check consistency: current east, field into the page, force upward.

Which length? Which angle? The craft

Exam setters have three standard ways to dress up equation (1), and each has a clean counter.

Only the length in the field counts. A 2 m wire passing through a 5 cm magnet gap has $L = 0.05$ m in the formula. The rest of the wire sits in zero field and feels nothing.

Only the perpendicular arrangement pays full price. If the wire makes angle $\theta$ with the field, multiply by $\sin\theta$ . The two limiting geometries — wire ⊥ field ( $F = BIL$ ) and wire ∥ field ( $F = 0$ ) — bracket every possible question, and quoting them is often worth a reasoning mark on its own.

A bent wire acts like its straight shortcut. In a uniform field, a wire that wanders from point P to point Q feels the same total force as a straight wire running directly from P to Q carrying the same current. Verify it on the simplest bend: a right angle of arms 0.30 m and 0.40 m, carrying $I = 2.0$ A in a field $B = 0.20$ T into the page. The arms feel $F_1 = 0.20\times2.0\times0.30 = 0.12$ N and $F_2 = 0.20\times2.0\times0.40 = 0.16$ N, at right angles to each other; the resultant is $\sqrt{0.12^2 + 0.16^2} = 0.20$ N. The straight shortcut has length $\sqrt{0.30^2 + 0.40^2} = 0.50$ m and predicts $BIL = 0.20\times2.0\times0.50 = 0.20$ N — identical. One immediate corollary: a closed loop in a uniform field feels zero net force (its shortcut has zero length), which is why loops in fields rotate rather than fly away — the torque story that Current Loops as Magnetic Dipoles (coming up) makes precise.

Limiting cases

case	result	meaning
wire ⊥ field ( $\theta = 90°$ )	$F = BIL$	maximum force — the geometry every device engineers for
wire ∥ field ( $\theta = 0$ )	$F = 0$	carriers drift along $B$ ; the qv×B force vanishes
$\theta = 30°$	$F = BIL/2$	the sine, not the cosine — check which angle the question gives
current reversed	force reverses	the DC motor's commutator trick, previewed
bent wire, uniform field	force of the straight P-to-Q shortcut	wandering path, same total push
closed loop, uniform field	zero net force	loops twist instead — torque, next modules
doubled current AND doubled length	force ×4	$F$ scales linearly in each factor separately

Worked examples

Board level — magnitude and direction (CBSE/IGCSE pattern, 3 marks)

A straight wire of length $0.25\ \mathrm{m}$ carries a current of $4.0\ \mathrm{A}$ , directed horizontally toward the east. It sits in a uniform magnetic field $B = 0.30\ \mathrm{T}$ pointing vertically downward. (a) Find the magnitude of the force on the wire. (b) Find its direction. (c) What does the force become if the wire is tilted so it makes $30°$ with the field?

Step 1. Part (a): wire (east, horizontal) and field (vertically down) are perpendicular, so $\sin\theta = 1$ : $F = BIL = 0.30 \times 4.0 \times 0.25 = \boxed{0.30\ \mathrm{N}}$ .

Step 2. Part (b): Fleming's left hand. First finger points down (field), second finger points east (current); the thumb lands pointing horizontally toward the south. State it as a compass direction — "perpendicular to the wire" alone does not earn the mark, because two horizontal directions are perpendicular to an east-west wire.

Step 3. Part (c): now $\theta = 30°$ between wire and field: $F = BIL\sin 30° = 0.30 \times 0.5 = \boxed{0.15\ \mathrm{N}}$ . The direction is still perpendicular to both the wire and the field — tilting the wire changes how hard it is pushed, not the rule for which way.

Step 4. The sentence that earns the method mark: "Each charge drifting in the wire experiences a magnetic force; the resultant on the wire is $F = BIL\sin\theta$ , directed by Fleming's left-hand rule using conventional current."

Competitive level — the floating wire (JEE-Main/NEET pattern, about 2.5 minutes)

A straight horizontal wire of mass per unit length $\lambda = 0.020\ \mathrm{kg/m}$ is placed in a uniform horizontal magnetic field $B = 0.50\ \mathrm{T}$ directed perpendicular to the wire. (a) Find the current required for the magnetic force to hold the wire in mid-air, and state its direction requirement. (b) If the current is switched to twice this value, find the wire's initial acceleration. (Take $g = 9.8\ \mathrm{m/s^2}$ ; CBSE and NEET papers state $g$ — use whatever the paper gives.)

Step 1. Part (a): levitation demands the magnetic force per metre balance the weight per metre. For a length $L$ : $BIL = (\lambda L)g$ — the length cancels, a sign this is really a per-metre balance:

I = \frac{\lambda g}{B} = \frac{0.020 \times 9.8}{0.50} = \boxed{0.39\ \mathrm{A}}.

Step 2. Direction: the force $BIL$ must point vertically up. With $B$ horizontal and perpendicular to the wire, Fleming's left hand fixes the current direction uniquely — reverse the current and the wire is pressed down instead. (A statement, not a calculation: "current directed so that the left-hand rule gives an upward force.")

Step 3. Part (b): at $I' = 2I$ , the upward force is $2\lambda L g$ against weight $\lambda L g$ ; the net upward force is $\lambda L g$ on mass $\lambda L$ . Newton's second law: $a = \dfrac{\lambda L g}{\lambda L} = g = \boxed{9.8\ \mathrm{m/s^2}}$ , upward. Doubling the current doesn't double the acceleration from rest to $2g$ — the weight still subtracts; the elegant answer $a = g$ is the setter's fingerprint. Timing note: two and a half minutes. The length-cancellation in step 1 is the speed move — work per metre and $\lambda$ drops straight in; if you introduce an unknown $L$ and carry it, you lose thirty seconds and gain nothing. Part (b) is pure bookkeeping: net force $= (2-1)\times$ weight, so $a = g$ without touching a calculator.

Exam traps

Electron flow versus conventional current. Fleming's second finger follows conventional current. If the question specifies electron flow, reverse the direction once, before applying the rule — students who "remember to flip" while their hand is mid-air routinely flip twice.

The wrong hand. Left hand for the motor effect (this module); the right hand serves the generator rule in Electromagnetic Induction later. Papers with both effects in one question are exactly where the hands get swapped.

Length in the field. $L$ is the immersed length only. Magnet gaps are centimetres; wires are metres. Read the geometry.

The sine's angle. Between wire and field. If the problem quotes the angle to the normal of the field, convert. And $\sin\theta$ , never $\cos\theta$ — check the two limits (parallel: zero; perpendicular: max) if unsure.

Direction answers need a direction. "Perpendicular to the wire" is not an answer; "horizontally, toward the south" is. Boards award the mark for a named direction (compass point, up/down, into/out of page).

Units. $B$ in tesla, but data often arrive in millitesla or gauss-flavoured trivia; convert to SI before $BIL$ . Force comes out in newtons — a jump-rope wire feeling kilonewtons means a slipped exponent.

Scope notes. IGCSE 0625: the motor effect is tested qualitatively — demonstration, direction reversals (reverse current or reverse field), and the left-hand rule at Supplement level; the $F = BIL\sin\theta$ calculation is not required. IB (SL and HL): the full formula is standard-level content. CBSE/ISC/JEE/NEET/AP-2/A-Level: quantitative throughout, and CBSE/ISC love the drift-picture derivation as a 3-mark item.

Origins · 1 minJohn Ambrose Fleming and how this was discovered

John Ambrose Fleming

1849–1945 · University College London

Fleming was the great translator of electromagnetism into engineering — consultant to Edison's London company, then to Marconi, and the first professor of electrical engineering at UCL. The left-hand rule was born there, out of classroom desperation: his students kept mangling the three mutually perpendicular directions of field, current and force in exams, so Fleming packaged the geometry into fingers, where it has lived ever since. (He supplied the right-hand twin for generators at the same stroke.) In 1904 he capped a career of practical inventions with the thermionic valve — the first electronic device that could rectify a radio signal — built, fittingly, from a leftover effect he had studied in Edison's lamps. He lectured with legendary clarity into his nineties, deaf, precise, and still demonstrating with both hands.

History

1820

— Ørsted's compass twitches beside a current; within months the reverse question is asked — does a magnet push on a current?

1821

— Faraday makes a current-carrying wire circle a magnet: the first electric motor, and the motor effect made visible.

1832

— Pixii builds the first commutated machine; reversing the current each half-turn keeps the force turning one way.

1885

— Fleming, teaching at UCL, condenses the force geometry into the left-hand rule his students still use.

1948

— The ampere itself is defined via the force between current-carrying conductors — the motor effect promoted to metrology.

Going further

Texts. NCERT Physics Class 12, Ch. 4 (Moving Charges and Magnetism, §4.4) · HC Verma, Concepts of Physics Vol. 2, Ch. 34 · CIE 9702 Topic 20.1-20.2 · IGCSE 0625 §4.5.4 (motor effect, qualitative + left-hand rule) · IB Physics Guide, Theme D.3.

Provenance. All results are standard. A verifier should recompute: the drift-sum closure ( $v_d = 2.9\times10^{-4}$ m/s, $N = 2.1\times10^{22}$ , per-carrier $1.4\times10^{-23}$ N, total 0.30 N $= BIL$ ); board example $F = 0.30$ N and $0.15$ N at $30°$ ; bent-wire check $\sqrt{0.12^2+0.16^2} = 0.20$ N $= BIL_{\rm chord}$ with chord $0.50$ m; floating wire $I = 0.020\times9.8/0.50 = 0.39$ A and $a = g$ at doubled current; demonstration force $0.05$ N.

Prerequisites. Force on a Moving Charge: qv×B · Drift Velocity & the Microscopic Picture · Electric Current & Charge Flow. Leads to. Parallel Currents & the Definition of the Ampere (coming up) · Current Loops as Magnetic Dipoles: Torque & Moment (coming up) · The Moving-Coil Galvanometer (coming up) · The DC Motor & Its Cousins (coming up) · Magnetic Force on Current-Carrying Wires (UG deep dive).

If the wire is neutral, why does the field push it? I thought B only acts on charges.

It does only act on charges — on the moving ones inside. The wire's electrons drift and collect kicks of $qv_dB$ each; the stationary positive lattice feels essentially nothing magnetic. The drifting electrons cannot escape sideways, so they press against the inside of the wire (through collisions and tiny internal electric fields), and that pressure is the force on the wire. Neutrality kills the electric story — no net $qE$ force — but current is organised motion of charge, and organised motion is exactly what a magnetic field grips. A neutral wire with no current is genuinely invisible to the field.

The drift velocity is a fraction of a millimetre per second. How can such a crawl produce a force that lifts wires?

By headcount. In the verified numbers of the derivation, each carrier's kick is $1.4\times10^{-23}$ N — hopeless alone. But a 25 cm length of 1 mm² copper at 4 A contains $2.1\times10^{22}$ drifting electrons, and the kicks all point the same way. The product is 0.30 N — visible, bench-jumping force. This is the same resolution as in Drift Velocity & the Microscopic Picture, where a crawling drift still delivers amperes because the crowd is astronomical. Slow times astronomical is the recurring arithmetic of this chapter.

Does the force depend on what the wire is made of?

No — and the derivation shows why, which makes this a favourite conceptual question. Material properties enter through $n$ (carrier density) and $v_d$ (drift speed), but their product is locked by the current: $I = nAv_dq$ . A material with fewer carriers must drift them faster to carry the same $I$ , and the force bookkeeping $N \times qv_dB$ comes out identical. At the same current, in the same field, copper, aluminium and salty water feel the same force per metre. What the material does change is the resistance — how hard it is to drive that current in the first place (Resistivity & Geometry).

Which direction does the wire jump if I use electron flow in Fleming's rule?

Point the second finger along the conventional current — the direction positive charge flows, opposite to the electrons — and the rule works as designed. If you insist on thinking in electron flow: reverse the electron direction to get conventional current first, then apply the left hand normally. Never "correct" mid-gesture. The reason the rule survives with either carrier sign is buried in the derivation: negative charge drifting backward is the same current as positive charge drifting forward, and the force $(nAv_dq)LB$ is identical for both descriptions. Nature doesn't know which convention you chose; your fingers must pick one and stay loyal.

Why does a closed loop feel zero net force but still rotate in a field?

The bent-wire shortcut rule answers both halves. Net force: a closed loop starts and ends at the same point, so its straight shortcut has zero length — zero net force in a uniform field. But zero net force is not zero force everywhere: opposite sides of the loop are pushed in opposite directions, and if those pushes act along different lines they form a couple that twists the loop. That twist is the torque every motor rotor and galvanometer coil lives on — quantified in Current Loops as Magnetic Dipoles: Torque & Moment, coming up. The loop cannot be dragged across a uniform field, but it can absolutely be spun in place.

Is the jumping-wire demonstration dangerous to reproduce? The currents seem huge.

The classroom version is tame: 10 cm of wire in the 0.1 T gap of a strong horseshoe magnet at 5 A feels $F = 0.1\times5\times0.1 = 0.05$ N — enough to visibly kick a light, loosely hung wire, harmless to fingers. What needs respect is the circuit: 5 A through a thin wire warms it quickly (Joule heating, from Electrical Energy, Power & Joule Heating), so demonstrations use short pulses. The industrial cousins are genuinely violent — fault currents of kiloamperes in switchyard busbars generate forces that bend steel, which is why parallel busbars are braced. Same formula, five orders of magnitude apart.

Does the magnetic force on the wire do work? The last module said magnetic forces never do work.

Sharp catch — and the resolution is subtle enough that boards stay away from it, so file this as insight rather than exam ammunition. On each charge, the magnetic force is always perpendicular to that charge's velocity and does no work — that theorem survives intact. When the wire moves, the carriers' velocities acquire the wire's motion too, the force directions tilt, and the energy delivered to the moving wire is exactly siphoned from the electrical circuit driving the current (the battery works harder). The magnetic field acts as a broker, converting electrical energy to mechanical, paying out nothing of its own. The full audit belongs to the UG tier; at HS level: motors convert electrical to mechanical energy, and the field is the intermediary.