Magnetic forces do no work ($(q\mathbf{v}\times \mathbf{B})\cdot \mathbf{v}=0$)

The magnetic part of the Lorentz force is perpendicular to the particle velocity. Its power is therefore zero, so with no other forces the kinetic energy remains constant. This remains true relativistically because $dE/dt=\mathbf{F}\cdot \mathbf{v}$; a magnetic force can change momentum direction without changing total energy.

A tangential force integral around an instantaneous moving loop is not generally the force work along the actual particle trajectory. For an expanding circular guide, the particle has both radial and tangential velocity; the magnetic radial and tangential powers cancel exactly even when the tangential circulation is nonzero.