Retarded potential

The retarded potential (German: delayed potential ) is the designation for the mathematical form of the potential in electromagnetic field theory or other field theories , in which changes in the field spread at finite speed ( speed of light ) and not instantaneously . It occurs when investigating time-dependent problems such as: B. in the emission of electromagnetic waves .

In contrast, time dependencies are neglected in electrostatics , magnetostatics and Newton's theory of gravity .

Mathematical formulation

Mathematically, the potential is the solution of the inhomogeneous wave equation ( following from Maxwell's equations ) in three spatial dimensions ${\displaystyle \,u(x,t)}$

${\displaystyle {\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\sum _{i=1}^{3}\left({\frac {\partial ^{2}u}{\partial x_{i}^{2}}}\right)={\frac {1}{c^{2}}}\,{\frac {\partial ^{2}u}{\partial t^{2}}}-\Delta u=v(t,x_{1},x_{2},x_{3})\,.}$

with a source term on the right . The solution ${\displaystyle \,v(t,\mathbf {x} )}$

${\displaystyle u_{\text{retardiert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int \!\mathrm {d} ^{3}y\,{\frac {v\left(t-{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}}$

is called retarded potential . At the moment, it only depends on the inhomogeneity on the backward light cone of . The inhomogeneity affects the solution with a delay (hence the name) at the speed of light . ${\displaystyle \mathbf {x} }$ ${\displaystyle t}$ ${\displaystyle v}$ ${\displaystyle \mathbf {x} }$

The solution

${\displaystyle u_{\text{avanciert}}(t,\mathbf {x} )={\frac {1}{4\pi }}\int \!\mathrm {d} ^{3}y\,{\frac {v\left(t+{\frac {|\mathbf {x} -\mathbf {y} |}{c}},\mathbf {y} \right)}{|\mathbf {x} -\mathbf {y} |}}}$

means correspondingly advanced potential. This describes e.g. B. a sink that absorbs an existing field.

The emission and absorption of fields can thus be described with retarded and advanced potential.

literature

• Richard Courant and David Hilbert : Methods of Mathematical Physics Volume 2. Second edition, Springer Verlag, 1968

Web links

• Norbert Dragon, keywords and additions to calculation methods in physics (PDF; 1.9 MB), chapter 18