 Solution

The spatial part of the Lorentz force isPart (c), together with $\mathcal{E}=\gamma m{c}^2$, givesWriting $\mathbf{a}=d\mathbf{v}/dt$ and expanding the momentum derivative,Substitution of $d\gamma /dt$ gives the coordinate acceleration of a relativistic charged particle:

In the nonrelativistic limit $|\mathbf{v}|/c\to 0$, one has $\gamma \to 1$ and the final term is smaller by order $v^2/c^2$. Hencewhich is precisely Newton's second law with the ordinary Lorentz force.