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

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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

Contents

Lorenz gaugeEdit

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 EquationsEdit

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 LagrangianEdit

The electromagnetic Lagrangian that gives the Maxwell equations is

 

Energy density and Poynting vectorEdit

The energy density and Poynting vector can be extracted from

 

where energy density is

 

and the Poynting vector is

 

Lorentz ForceEdit

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 LagrangianEdit

The Lagrangian that gives the Lorentz Force is

 

Plane electromagnetic wavesEdit

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