Magnetic Force on a Current-Carrying Wire

Equivalent forms

F = B I L \sin\theta

F = B I L

The cross product encodes the right-hand rule geometry: thumb (I), fingers (B), palm push (F) — three orthogonal directions interlocked.

Symbol	Name	SI unit	Dimension	Range
$F$	Magnetic forceoutput Force on the current-carrying wire	N	M·L·T⁻²	0 – 50
$I$	Current Current flowing through the wire	A	I	0 – 20
$L$	Wire length Length of the wire segment inside the field	m	L	0.01 – 5
$B$	Magnetic flux density Strength of the external magnetic field	T	M·T⁻²·I⁻¹	0 – 2
$θ$	Angle Angle between current direction and B field	rad	dimensionless	0 – 3.14159

Unit systems

Symbol	SI	CGS	Imperial
$F$	N	dyn	lbf
$I$	A	statA	A
$L$	m	cm	ft
$B$	T	G	T

Where it holds

Valid for straight wire segments in a uniform B field, or as an integrand

dF = I dL \times B

along curved wires in nonuniform fields. Assumes steady current and non-relativistic drift speeds.

Dimensional analysis

$[F] = [I][L][B]$ = (A $)(m)(T) =$ (A $)(m)(kg\cdot s^{-2}\cdot A^{-1}) = kg\cdot m\cdot s^{-2} = N \checkmark$

Discovery

André-Marie Ampère / Pierre-Simon Laplace · 1820

Weeks after Ørsted's discovery that current deflects a compass, Ampère and Laplace formalized the force a magnetic field exerts on a current — the foundation of every electric motor that followed.

Try this

How does a 5 W earbud speaker move air fast enough to make sound — purely from electricity?

A 0.5 m wire carrying 4 A is placed in a 0.3 T magnetic field perpendicular to the current. Find the force on the wire.

Research status: stable

Real-world applications

Every DC and brushless motor on Earth
Loudspeaker voice coils — current in a field pushes the diaphragm
Railguns and electromagnetic launchers
MHD pumps for circulating liquid metals in nuclear reactors

Common misconceptions

Force is perpendicular to the wire, not along it
Doubling the current doubles the force only if B and L are unchanged
The wire itself isn't magnetic — the force comes from the external field acting on the moving charges

Experimental verification

Place a wire on a sensitive balance between the poles of an electromagnet; the reading change when current flows equals BIL to better than 1%. Re-derived in current-balance experiments that originally defined the ampere.

Derivation

A charge q moving with drift velocity v_d in field B feels

f = qv_d \times B

.

For n charges per volume in a wire of cross-section A and length L, total force

F =

(nqAL

) v_d \times B = I L \times B

, since

I =

nqAv_d.

Limiting cases

\theta = 0

⟶

F = 0

Wire parallel to the field feels no magnetic force — the cross product vanishes.

\theta = \pi /2

⟶

F = BIL

Maximum force when current is perpendicular to the field.

I \to 0

⟶

F \to 0

No current means no moving charges, so no magnetic force on the wire.

What if…

What if the current direction reverses?

The force reverses direction — the magnitude is unchanged. This is how AC motors produce torque each half-cycle.

What if the wire is bent into a square loop in a uniform B?

Net force is zero (opposite sides cancel), but a torque remains — the basis of every electric motor.

What if you double both I and L?

Force quadruples — F scales linearly with each factor.

1

Force on a current loop edge

Given ·

I:: 4
L:: 0.5
B:: 0.3
θ:: 1.5708

Find · F

Steps

Identify: $I = 4$ A, $L = 0.5\,\mathrm{m}$ , $B = 0.3\,\mathrm{T}$ , $\theta = 90^{\circ}$
Apply $F = BIL \sin \theta$
$\sin 90^{\circ} = 1$ , so $F = 0.3 \times 4 \times 0.5 \times 1$
$F = 0.6\,\mathrm{N}$ — direction given by right-hand rule

Result ·

F = 0.6\,\mathrm{N}

2

Wire tilted 30° to the field

Given ·

I:: 10
L:: 0.2
B:: 0.5
θ:: 0.5236

Find · F

Steps

$\sin 30^{\circ} = 0.5$
$F = 0.5 \times 10 \times 0.2 \times 0.5 = 0.5\,\mathrm{N}$
Force is perpendicular to the plane containing I and B

Result ·

F = 0.5\,\mathrm{N}

Lorentz Force Law→biot savartAmpère's Law (with Maxwell's correction)→