Lorentz force

The Lorentz force is the force on a point charge due to electromagnetic fields.

If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force

( Lorentz force f on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time. )

Force on a current carrying wire

When a wire carrying an electrical current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force). 

By combining the Lorentz force law above with the definition of electrical current, the following equation results, in the case of a straight, stationary wire:

where ℓ is a vector whose magnitude is the length of wire, and whose direction is along the wire, aligned with the direction of conventional current flow I.

(Right-hand rule for a current-carrying wire in a magnetic field B)

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Ref: Wikipedia