A current in a conductor consists of moving charge carriers. If the conductor lies in a magnetic field, the Lorentz forces on those moving charges add to produce a force on the conductor itself.

Consider a straight segment of wire with cross-sectional area $A$, carrier number density $n$, carrier charge $q$, and drift velocity $\mathbf v_d$. The current density is

$$\mathbf J=nq\mathbf v_d.$$

Each carrier experiences magnetic force

$$\mathbf F_q=q\mathbf v_d\times\mathbf B.$$

In a wire segment of length $L$, the number of mobile carriers is

$$N=nAL.$$

The total magnetic force on those carriers is therefore

$$\mathbf F=Nq\mathbf v_d\times\mathbf B =nALq\mathbf v_d\times\mathbf B.$$

Using

$$nqA\mathbf v_d=I\hat{\boldsymbol\ell},$$

where $\hat{\boldsymbol\ell}$ points in the conventional-current direction, we obtain

$$\boxed{\mathbf F=I\mathbf L\times\mathbf B},$$

where $\mathbf L$ is a vector of magnitude $L$ along the current direction.

The magnitude is

$$\boxed{F=ILB\sin\theta},$$

where $\theta$ is the angle between the current direction and $\mathbf B$.

General wire shape

For a small current element,

$$\boxed{d\mathbf F=I,d\boldsymbol\ell\times\mathbf B}.$$

The force on a curved conductor is found by adding these contributions along the current path:

$$\mathbf F=I\int d\boldsymbol\ell\times\mathbf B.$$

If the magnetic field varies in space, its local value must be used inside the integral.

Direction

For conventional current, use the right-hand rule for

$$\mathbf L\times\mathbf B.$$

The microscopic electron drift in a metal is opposite to conventional current, but the electron charge is negative, so the two sign reversals lead to the same macroscopic force direction.

Worked example

A straight wire segment of length

$$L=0.20,\mathrm m$$

carries current

$$I=5.0,\mathrm A$$

perpendicular to a uniform magnetic field

$$B=0.30,\mathrm T.$$

The force magnitude is

$$F=ILB=(5.0)(0.20)(0.30)=\boxed{0.30,\mathrm N}.$$

If the current direction is reversed, the magnetic force reverses. If the field direction is reversed, it also reverses. If both are reversed, the force direction is unchanged.

Force between parallel currents

Two long parallel wires exert magnetic forces on one another. The first wire creates magnetic field at the second, and the current in the second experiences force from that field.

For separation $r$, the field from wire 1 is

$$B_1=\frac{\mu_0I_1}{2\pi r}.$$

If a length $L$ of wire 2 is parallel to wire 1, the force magnitude is

$$F=I_2LB_1,$$

so

$$\boxed{\frac{F}{L}=\frac{\mu_0I_1I_2}{2\pi r}}.$$

Currents in the same direction attract; currents in opposite directions repel.

Magnetic force on current-carrying conductors is the macroscopic bridge from Lorentz force to motors, speakers, magnetic actuators, and forces between circuits.