Coulomb calibration

The Coulomb calibration (according to its connection with the Coulomb potential (see below); also called radiation calibration or transverse calibration ) is a possible calibration of electrodynamics , i.e. it describes a restriction of the electrodynamic potentials .

No calibration of electrodynamics

In order to facilitate the solution of Maxwell's equations , the scalar potential and the vector potential are introduced for the electric field and the magnetic field , which describe the classic observable fields: ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle \Phi }$ ${\displaystyle {\vec {A}}}$

${\displaystyle {\vec {E}}({\vec {r}},t)=-\nabla \Phi -\partial _{t}{\vec {A}}({\vec {r}},t)}$

${\displaystyle {\vec {B}}({\vec {r}},t)=\nabla \times {\vec {A}}({\vec {r}},t)}$.

This definition allows calibration freedoms in the choice of scalar and vector potential, which have no effect on measurable quantities, in particular not on the electric field and magnetic flux density.

The Coulomb calibration

This calibration freedom is used in the Coulomb calibration to require the vector potential to be free from divergence :

${\displaystyle \nabla \cdot {\vec {A}}({\vec {r}},t)=0}$

Because of and follow from this the results noted in the next paragraph. ${\displaystyle \Delta =\nabla \cdot \nabla }$ ${\displaystyle {\frac {\partial }{\partial t}}\nabla =\nabla {\frac {\partial }{\partial t}}}$

The inhomogeneous Maxwell equations in the Coulomb calibration

If one uses this calibration to insert the potentials in the inhomogeneous Maxwell equations ( Gaussian law and the extended law of induction ), one obtains

${\displaystyle \Delta \Phi =-{\frac {\rho }{\varepsilon _{0}}}}$

and

${\displaystyle \Delta {\vec {A}}-{\frac {1}{c^{2}}}\partial _{t}^{2}{\vec {A}}=-\mu _{0}{\vec {j}}+{\frac {1}{c^{2}}}\nabla \partial _{t}\Phi \,\,(=:\,-\mu _{0}{\vec {j}}_{\mathrm {eff} })}$.

The first equation is solved by

${\displaystyle \Phi ({\vec {r}},t)={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho ({\vec {r}}^{\prime },t)}{\left|{\vec {r}}-{\vec {r}}^{\prime }\right|}}\mathrm {d} ^{3}r^{\prime }}$,

so in this calibration the scalar potential is identical to the Coulomb potential . ${\displaystyle \Phi }$

The second equation is an inhomogeneous wave equation with the solution obtained by the retarded potential method :

${\displaystyle {\vec {A}}({\vec {r}},t)={\frac {\mu _{0}}{4\pi }}\int {\frac {{\vec {j}}_{\mathrm {eff} }({\vec {r}}^{\prime },t')}{\left|{\vec {r}}-{\vec {r}}^{\prime }\right|}}\mathrm {d} ^{3}r^{\prime }}$.

The retarded time is given by  . Physically, the last specified difference corresponds to the time span that a light or radar signal needs to travel the distance from the starting point (the integration point) of the signals to the point of arrival (c is the speed of light ). ${\displaystyle t'}$ ${\displaystyle t':=t-{\frac {|{\vec {r}}-{\vec {r}}^{\prime }|}{c}}}$ ${\displaystyle {\vec {r}}'}$ ${\displaystyle {\vec {r}}}$

Using two different times in the integral -  t for the scalar potential, t '   for the vector potential - is the main advantage or disadvantage of the Coulomb calibration. The competing Lorenz calibration does not have this disadvantage, but is explicitly relativistically invariant in that it takes the retardation into account throughout.

If there are no sources ( charges and currents ), the equations simplify to

${\displaystyle \Delta \Phi =0}$

and

${\displaystyle \Delta {\vec {A}}-{\frac {1}{c^{2}}}\partial _{t}^{2}{\vec {A}}=0}$,

the vector potential thus satisfies the homogeneous wave equation.

literature

• John D. Jackson: Classical Electrodynamics . Walter de Gruyter Berlin New York, 2006, ISBN 978-3-11-018970-4 .