Why does magnetic field do no work on an electric charge?

There is a qualitative and a quantitative way to show that the magnetic field does no work on electric charges. The qualitative description requires a picture; essentially, one looks at the circular orbits of moving particles in a magnetic field and notices that the force on the charge is always central (perpendicular to the direction of movement). If the force is central then F.dx is always zero and there can be no work done by the field. Interestingly, this is the case for any central force (e.g. gravity).
Quantitatively, one can prove this statement by looking at the equation describing the force on a charge due to a magnetic field, the lorentz force formula F = q(E + v x B), we can assume no electric field to be present so the equation becomes F = q(v x B). Similarly, the rate of work done can be found to be P = F.v thus for forces due to magnetic fields we find P = q(v x B).v. But this is equal to zero because v x B is perpendicular to both v and B, and hence the dot product is zero.