# Physics Using Geometric Algebra/Relativistic Classical Mechanics/The electromagnetic field

The electromagnetic field is defined in terms of the electric and magnetic fields as

Alternatively, the fields can be derived from a paravector potential as

where:

and

## Lorenz gauge[edit | edit source]

The Lorenz gauge (without t) is expressed as

The electromagnetic field is still invariant under a gauge transformation

where is a scalar function subject to the following condition

where

## Maxwell Equations[edit | edit source]

The Maxwell equations can be expressed in a single equation

where the current is

Decomposing in parts we have

- Real scalar: Gauss's Law
- Real vector: Ampere's Law
- Imaginary scalar: No magnetic monopoles
- Imaginary vector: Faraday's law of induction

### Electromagnetic Lagrangian[edit | edit source]

The electromagnetic Lagrangian that gives the Maxwell equations is

### Energy density and Poynting vector[edit | edit source]

The energy density and Poynting vector can be extracted from

where energy density is

and the Poynting vector is

## Lorentz Force[edit | edit source]

The electromagnetic field plays the role of a spacetime rotation with

The Lorentz force equation becomes

or equivalently

and the Lorentz force in spinor form is

### Lorentz Force Lagrangian[edit | edit source]

The Lagrangian that gives the Lorentz Force is

## Plane electromagnetic waves[edit | edit source]

The propagation paravector is defined as

which is a null paravector that can be written in terms of the unit vector as

A vector potential that gives origin to a polarization|circularly polarized plane wave of left *helicity* is

where the phase is

and is defined to be perpendicular to the propagation vector . This paravector potential obeys the Lorenz gauge condition. The right helicity is obtained with the opposite sign of the phase

The electromagnetic field of this paravector potential is calculated as

which is nilpotent