**Required math: algebra**

**Required physics: special relativity**

Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 4; Problems 4.7.

In the last post, we introduced the electromagnetic field tensor* *:

In terms of , the electric and magnetic (Lorentz) force laws for a charge can be combined into a single equation:

where is the four-velocity. The three spatial components give the force law, but what about the time component? The time component of the four-momentum is the relativistic energy which, for a particle of rest mass is . If we expand this in a Taylor series for small , we get

Thus the relativistic energy is the rest mass plus the Newtonian kinetic energy (plus higher order terms). In the small- limit, and the component of 2 therefore is

Since the rest mass doesn’t change, this equation is saying that the rate of change of kinetic energy is . Does this make sense?

The force on a charge in an electric field is , so the work done by this field in moving the charge through a distance is . This work accelerates the charge, thus increasing its kinetic energy. The rate at which the kinetic energy increases is therefore .

Since the magnetic force on a charge is always perpendicular to the direction of motion, magnetic forces do no work, so there is no contribution to the kinetic energy from the magnetic field.

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