# Is the magnetic force on a current carrying conductor dependent on velocity?

A current carrying conductor is kept stationary in a region of uniform magnetic field, and another conductor, carrying the same amount of current, is moved with a certain velocity in a region of uniform magnetic field, of same magnitude (as the previously mentioned field).

Will the forces on the two be different?

I know that the mathematical formula of magnetic force on a current carrying wire does not account for such inequality, but still are we missing something in this formula ?

The external force on the whole current-carrying wire in region $V$ can be expressed as
$$\mathbf F = \int_V \rho\mathbf E_{ext} + \mathbf j \times \mathbf B_{ext}\,d^3\mathbf x.$$ where $\mathbf E_{ext}, \mathbf B_{ext}$ is the external electric and magnetic field and $\rho,\mathbf j$ is the charge and current density.
Stationary and moving wire will have different $\rho,\mathbf j$, so the external force will also be different.