- #1
USeptim
- 98
- 5
Different four potentials can generate the same electromagnetic fields. Specifically the following degrees of freedom yield the same EM fields:
A'=A+(e/c)∇χ
[itex]\phi[/itex]'=[itex]\phi[/itex]+(∂χ/∂t)
Where A es the vector potential, [itex]\phi[/itex] is the scalar potential and ∇χ is any scalar field.
In the electrostatic, potential is well defined except for a constant that has to be the same on all the space. Besides, for a continuous charge distribution, the potential energy of the charges is the same than the energy stored in the field. So it looks like the potential can be uniquely defined in this special case.
In electrodynamics however, the gauge conditions are more complex and as far as I have seen, there is no document or textbook stating how to define exact potentials. Instead of that, “gauge-fixing” procedures are used, for example de Lorentz gauge: [itex]∂_{\alpha}[/itex][itex]A^{\alpha}[/itex]=0.
However Lienard-Wiechert potentials show the potential generated by a moving charge on every point of the space, I understand this is the “effect” of the moving charge on the point. So I think it should be possible to determine the exact potential by integrating the potential caused by all the charges at δ( -[itex]t_{0}[/itex]-t)|[itex]x_0{}[/itex] - x|) plus a term corresponding to the boundary radiated fields.
Is there anything that may be wrong in this assumption?
NOTE: [itex]t_{0}[/itex] and [itex]t_{0}[/itex] is the space-time point when we measure the potential and t and x are the position of the charges that cause effect on this point.
A'=A+(e/c)∇χ
[itex]\phi[/itex]'=[itex]\phi[/itex]+(∂χ/∂t)
Where A es the vector potential, [itex]\phi[/itex] is the scalar potential and ∇χ is any scalar field.
In the electrostatic, potential is well defined except for a constant that has to be the same on all the space. Besides, for a continuous charge distribution, the potential energy of the charges is the same than the energy stored in the field. So it looks like the potential can be uniquely defined in this special case.
In electrodynamics however, the gauge conditions are more complex and as far as I have seen, there is no document or textbook stating how to define exact potentials. Instead of that, “gauge-fixing” procedures are used, for example de Lorentz gauge: [itex]∂_{\alpha}[/itex][itex]A^{\alpha}[/itex]=0.
However Lienard-Wiechert potentials show the potential generated by a moving charge on every point of the space, I understand this is the “effect” of the moving charge on the point. So I think it should be possible to determine the exact potential by integrating the potential caused by all the charges at δ( -[itex]t_{0}[/itex]-t)|[itex]x_0{}[/itex] - x|) plus a term corresponding to the boundary radiated fields.
Is there anything that may be wrong in this assumption?
NOTE: [itex]t_{0}[/itex] and [itex]t_{0}[/itex] is the space-time point when we measure the potential and t and x are the position of the charges that cause effect on this point.