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snoopies622
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Can anyone point me to a derivation of the Lienard-Wiechart potential formulas? I assume that they can be derived from Maxwell's equations alone.
Thanks.
Thanks.
snoopies622 said:Is
[tex]
\vec B = \nabla \times \vec A
[/tex]
still valid if A is four dimensional? Is the curl of a four-dimensional vector field even defined?
[tex](\phi,{\vec A})[/tex] is chosen to be a 4-vector in a LT so that the continuity equationsnoopies622 said:That is very helpful, thank you.
What makes
[tex]
( \frac {\phi }{c} , A^x , A^y , A^z )
[/tex]
a four-vector? That is, why is applying a Lorentz transformation to it valid? Is the electric potential somehow the time-component of the magnetic potential?
dx said:The thing on the right is the four dimensional curl.
clem said:[tex](\phi,{\vec A})[/tex] is chosen to be a 4-vector in a LT so that the continuity equation [tex]\partial_\mu A^\mu=0[/tex] will hold in any LT so charge conservation will hold in any Loentz system.
The Lienard-Wiechert potential is a mathematical expression used in classical electromagnetism to describe the electromagnetic field generated by a moving electric charge. It takes into account both the velocity and acceleration of the charge, and can be used to calculate the electric and magnetic fields at any point in space and time.
The Lienard-Wiechert potential was independently derived by French physicist Emile Lienard and Dutch physicist Hendrik Lorentz in the late 19th century. However, it was not until the early 20th century that it was fully recognized and applied in the field of classical electromagnetism.
The Lienard-Wiechert potential is significant because it provides a complete and accurate description of the electromagnetic field generated by a moving electric charge. It allows for the prediction and understanding of various electromagnetic phenomena, such as the Doppler effect, radiation from accelerating charges, and the behavior of charged particles in electric and magnetic fields.
The derivation of the Lienard-Wiechert potential is based on the assumptions that the charge is point-like, has a constant velocity, and is non-relativistic. It also assumes that the electromagnetic field is described by Maxwell's equations, and that the medium in which the charge is moving is homogeneous and isotropic.
The Lienard-Wiechert potential has numerous practical applications in various fields, including telecommunications, radar technology, and particle accelerators. It is also used in the development of new technologies, such as wireless power transfer and electromagnetic propulsion. Additionally, it is an essential tool in the study of astrophysical phenomena, such as the emissions from pulsars and the behavior of charged particles in the Earth's magnetosphere.