Magnetic Force

$\vec{F}=\int \vec{v}\times \vec{B}\mathrm{d}q=\int \vec{I}\times \vec{B}\mathrm{d}{\ell }^{\prime }=\int \vec{K}\times \vec{B}\mathrm{d}{a}^{\prime }=\int \vec{J}\times \vec{B}\mathrm{d}{\tau }^{\prime }$

• $\vec{v}$ is the velocity of the charge
• $\vec{I}=\lambda \vec{v}$ and is the current
• $\vec{K}=\sigma \vec{v}$ and is the surface current density
• $\vec{J}=\rho \vec{v}$ and is the current density