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How does a magnetic field exert force on a current-carrying wire?

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Diagram showing a circular loop of wire with an electric current, surrounded by curved blue lines representing magnetic field lines that loop around the wire and pass through its center. — magnetism Some lines of the magnetic field B (blue) for an electric current I (gold) in a loop.

When a current-carrying wire is placed in an external magnetic field B, the field exerts a force on the wire because the electric current represents the movement of charged particles. The Lorentz force acts on these individual moving charges, and because the charges are bound to the conductor, these forces are transferred to the wire itself.12

The magnitude of the force on a small length of the wire, dl, is calculated as:
$d F = i d l B \sin ϕ$
where i is the current and $ϕ$ is the angle between the magnetic field and the wire segment. The force is at its maximum when the current and the field are perpendicular to each other, and it is zero when the current flows parallel or anti-parallel to the field lines ( $ϕ = 0^{\circ}$ or $180^{\circ}$ ).12

The direction of this force is determined by the right-hand rule and is always perpendicular to both the wire segment and the magnetic field.12