# Derivation of Lienard-Wiechart

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
Mentor
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
Homework Helper
Gold Member
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 Fαβ = ∂αAβ - ∂βAα. The thing on the right is the four dimensional curl. The thing on the left is the Farady tensor.

Meir Achuz
Homework Helper
Gold Member
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.

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.