General Relativity: An Introduction for Physicists · Problem 6.10

Electromagnetism — Hobson, Efstathiou & Lasenby

Physics · charged-particle-equation-of-motion

Problem

In Cartesian inertial coordinates, the equation of motion for a charged particle in an electromagnetic field is

m_0 \frac{du^\mu}{d\tau} = qF^\mu{}_{\nu}u^\nu.

Show that

\frac{d\vec{p}}{dt} = q(\vec{E} + \vec{u} \times \vec{B}) \qquad \text{and} \qquad \frac{d\mathcal{E}}{dt} = q\vec{E}\cdot\vec{u},

where

\vec{p}

and

\mathcal{E}

are the 3-momentum and the energy respectively of the particle in

S

. Interpret these results physically.