# Magnetic force in relation to velocity

I'm misunderstanding something very important regarding magnetic force and its relation to velocity.

According to the Lorentz force, the magnetic force $${\mathbf {F_B}}=q{\mathbf {v}}\times {\mathbf {B}}$$. Assuming the charge is not moving, then $${\mathbf {v} = \begin{pmatrix}0&0&0\end{pmatrix}}$$. Therefore, $${\mathbf {F_B}=0}$$.

So why when I hold a magnet close to a piece of metal, I can feel the magnet applying a force on the piece of metal? Since the piece of metal and the magnet are not moving, shouldn't the net magnetic force be 0? I assume the force I am feeling is from the magnetic field from the magnet and the charges in the piece of metal.

The magnet and the metal are both made of magnetic dipoles. These can be thought of as tiny loops of electric current that generate the kind of magnetic field that you would expect out of the ends of a macroscopic bar magnet. Unlike electric charges, magnetic dipoles can feel a force when not moving, as long as they are exposed to a field that changes across space. Namely, the force on a dipole with moment $$\vec{m}$$ by an external field $$\vec{B}$$ is:
$$\vec{F}=\vec{\nabla}(\vec{m}\cdot\vec{B})$$