Derivation of Lienard-Wiechart

[snoopies622]

Main Question or Discussion Point

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.

 

[jtbell]

http://farside.ph.utexas.edu/teaching/em/lectures/node124.html

 

[snoopies622]

That is very helpful, thank you.

A follow-up question, if I may:

What makes

$$( \frac {\phi }{c} , A^x , A^y , A^z )$$

a four-vector? That is, why is applying a Lorentz transformation to it vaild? Is the electric potential somehow the time-component of the magnetic potential? Is

$$\vec B = \nabla \times \vec A$$

still valid if A is four dimensional? Is the curl of a four-dimensional vector field even defined?

 

[dx]

Is

$$\vec B = \nabla \times \vec A$$

still valid if A is four dimensional? Is the curl of a four-dimensional vector field even defined?

The analog of that equation is F_αβ = ∂_αA_β - ∂_βA_α. The thing on the right is the four dimensional curl. The thing on the left is the Farady tensor.

 

[clem]

That is very helpful, thank you.
What makes
$$( \frac {\phi }{c} , A^x , A^y , A^z )$$
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?

$$(\phi,{\vec A})$$ is chosen to be a 4-vector in a LT so that the continuity equation
$$\partial_\mu A^\mu=0$$ will hold in any LT so charge conservation will hold in any Loentz system.
They are no longer called the electric and magnetic potential, but just the 4-vector potential.

 

[snoopies622]

The thing on the right is the four dimensional curl.

OK, good. Thanks.

$$(\phi,{\vec A})$$ is chosen to be a 4-vector in a LT so that the continuity equation $$\partial_\mu A^\mu=0$$ will hold in any LT so charge conservation will hold in any Loentz system.

I'm sorry - could you re-word this? I'm not sure I get your meaning.