Magnetic force on conductor

A current-carrying conductor in an external magnetic flux density $\mathbf{B}$ experiences a force per unit length

$\frac{d\mathbf{F}}{dl}=I\hat{\mathbf{l}}\times \mathbf{B}\text{(N/m)},$

where $I$ is the current and $\hat{\mathbf{l}}$ is the unit vector along the current direction. In differential form:

$d\mathbf{F}=Id\mathbf{l}\times \mathbf{B}.$

This is the basis of every electric motor. It also explains why current-carrying wires near each other attract or repel, why railguns and loudspeakers work, and why a wire in a magnet’s field jumps when you connect a battery.

Where it comes from

This is the same physics as the Lorentz force on a single moving charge, $\mathbf{F}=q\mathbf{u}\times \mathbf{B}$, summed over all the carriers in a wire segment.

Take a segment of wire with cross-section $S$, length $dl$, carrier density $n$, drift velocity $\mathbf{u}$. The total moving charge is $nqSdl$. The force on this segment:

$d\mathbf{F}=(nqSdl)\mathbf{u}\times \mathbf{B}=(nq\mathbf{u})Sdl\times \mathbf{B}=\mathbf{J}(Sdl)\times \mathbf{B},$

using $\mathbf{J}=nq\mathbf{u}$ for the Current density. Since $\mathbf{J}$ points along the wire, $\mathbf{J}S=I\hat{\mathbf{l}}$:

$d\mathbf{F}=Id\mathbf{l}\times \mathbf{B}.$

The lattice ions transmit this force to the bulk of the conductor. The carriers feel the $\mathbf{u}\times \mathbf{B}$ force first, get nudged sideways, and drag the entire wire with them.

Total force on a finite wire

For a finite wire in a non-uniform field, integrate:

$\mathbf{F}=I{\int }_{C}d\mathbf{l}\times \mathbf{B}.$

For a straight wire of length $L$ in a uniform field perpendicular to it:

$|\mathbf{F}|=BIL.$

This is the “BIL force” formula familiar from introductory physics.

Right-hand rule

The cross product encodes a sign rule: point fingers along the current, curl them toward $\mathbf{B}$, thumb points along the force. Equivalently, $\hat{\mathbf{l}}\times \hat{\mathbf{B}}=\hat{\mathbf{F}}$.

A current along $+\hat{\mathbf{x}}$ in a field along $+\hat{\mathbf{y}}$ produces a force along $+\hat{\mathbf{z}}$.

Reversing either the current direction or the field direction reverses the force. Reversing both leaves the force the same, which is why AC motors work: current and field reverse together.

Force between two parallel wires

Two long parallel wires separated by distance $d$, carrying currents $I_1$ and $I_2$ in the same direction.

Wire 1’s field at wire 2’s location (by Ampère’s law): ${\mathbf{B}}_1={\mu }_0{I}_1/(2\pi d)$, perpendicular to the connecting line.

Force per unit length on wire 2:

$\frac{F}{l}=I_2{B}_1=\frac{{\mu }_0{I}_1{I}_2}{2\pi d}.$

The force is attractive when the currents are parallel, repulsive when antiparallel. This was once the definition of the ampere: 1 A is the current that produces $2\times 10^{-7}$ N/m between two wires 1 m apart in vacuum.

Torque on a current loop

A rectangular current loop in a uniform field experiences zero net force (forces on opposite sides cancel) but nonzero torque:

$\mathbf{\tau }=\mathbf{m}\times \mathbf{B},$

where $\mathbf{m}=I\mathbf{A}$ is the magnetic moment ($\mathbf{A}$ being the loop’s vector area). The loop rotates to align $\mathbf{m}$ with $\mathbf{B}$, the same way a compass needle aligns with Earth’s field.

This is the working principle of DC motors: a current loop in a magnet’s field experiences torque, and a commutator flips the current direction each half-rotation to keep the torque pushing in the same rotational sense. Continuous rotation results.

Versus the moving-charge form

Two related but distinct statements:

Source	Force
Single moving charge	$\mathbf{F}=q\mathbf{u}\times \mathbf{B}$ (point force, depends on $\mathbf{u}$)
Current-carrying conductor	$d\mathbf{F}=Id\mathbf{l}\times \mathbf{B}$ (force per length)

The wire form is what you use in motors and force-on-current problems; the charge form is what you use for free-particle motion (cyclotron, cathode-ray tube, mass spectrometer). They’re the same physics packaged for different applications.

Like the moving-charge magnetic force, the wire form does no work on the carriers (perpendicular to their drift velocity). The work that turns a motor shaft comes from the source maintaining the current against the back-emf.