How to derive Lienard-Wiechert potential from Maxwell's equation?

In summary, the conversation discusses the lack of a rigorous derivation of a topic in Feynman Lectures on Physics. Jackson's derivation in chapter 14 is mentioned, but it requires familiarity with relativistic formalism and Green's functions. Another possible derivation is found in Griffiths' "Electrodynamics" text, which is considered self-contained and intuitive. The conversation ends with the person expressing their intention to look into Griffiths' derivation.
  • #1
kof9595995
679
2
I've seen one derivation on Feynman Lectures on Physics, but the derivation is not really rigorous(he took a very special case for the derivation),I googled about the topic and couldn't find a satisfactory one. So can anybody give me a rigorous one?
Thanks in advance
 
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  • #2
Jackson derives it in 6 equations at the start of chapter 14, but you need to be familiar with the relativistic formalism (current and potential as four-vectors) and retarded Green's functions. If you are familiar with these, than the LW potentials follow from:

[tex]\mathbf{A} = \frac{4\pi}{c} \int d^4x' G(x-x') \mathbf{J}(x') [/tex]

[tex]\mathbf{J}(x') = \int d\tau \mathbf{v}(\tau) \delta^4(x' - r(\tau))[/tex]

where r is the trajectory (four-vector), and v is the four-velocity. All you do is sub the second eq into the first and crank it out to derive the LW potentials.
[tex]
 
  • #3
Thanks.But I'm not quite familiar with the manipulation of those, I'll give a shot.
And are there any other derivations avaliable?
 
  • #4
I'm not a big fan of Griffiths, but his "Electrodynamics" text has a pretty good derivation of it in chapter 10 - self contained and quite intuitive.


-----
Assaf
http://www.physicallyincorrect.com" [Broken]
 
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  • #5
ozymandias said:
I'm not a big fan of Griffiths, but his "Electrodynamics" text has a pretty good derivation of it in chapter 10 - self contained and quite intuitive.


-----
Assaf
http://www.physicallyincorrect.com" [Broken]
Thanks, I will have a look at it.
 
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1. What is the Lienard-Wiechert potential and how is it related to Maxwell's equations?

The Lienard-Wiechert potential is a mathematical expression that describes the electromagnetic field of a moving charged particle. It is derived from Maxwell's equations, which are a set of fundamental equations that describe the behavior of electric and magnetic fields.

2. What are the steps involved in deriving the Lienard-Wiechert potential from Maxwell's equations?

The derivation of the Lienard-Wiechert potential involves several steps, including solving for the electric and magnetic fields using Maxwell's equations, incorporating the Lorentz transformation to account for the motion of the charged particle, and applying the boundary conditions to determine the final expression for the potential.

3. Why is the Lienard-Wiechert potential important in the field of electromagnetism?

The Lienard-Wiechert potential is important because it allows us to understand and predict the behavior of electromagnetic fields generated by moving charged particles. This has numerous applications in physics, engineering, and technology, including the development of wireless communication and particle accelerators.

4. Are there any limitations to the Lienard-Wiechert potential?

Yes, there are some limitations to the Lienard-Wiechert potential. It is a classical theory and does not take into account quantum effects. It also assumes that the charged particle is point-like and does not have a finite size, which may not be accurate for certain scenarios.

5. How is the Lienard-Wiechert potential used in practical applications?

The Lienard-Wiechert potential has numerous practical applications, including in the design and optimization of radio antennas, radar systems, and particle accelerators. It is also used in the development of wireless communication technologies and in understanding the behavior of electromagnetic radiation in various environments.

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