# Liénard-Wiechert potential derivation

• Snaab
In summary, the question pertains to the missing factor of γ in the current density when it is boosted from the stationary frame to the observing frame. This is due to the γ being swallowed by the delta function in the transformation from r to r'. This can be understood by using the identity δ(x) = c δ(cx).
Snaab
Hi there,

I'm studying some classical field theory, and I have a question about the derivation of the Liénard-Wiechert potential for a moving point charge, see http://en.wikipedia.org/wiki/Liénard–Wiechert_potential#Derivation" for the derivation and the meaning of the symbols I use.

The current density in the observing frame is said to be given by:
$j^\mu (\mathbf{r},t) = e(1,\mathbf{v_0}) \delta(\mathbf{r'} - \mathbf{r_0}(t))$.

Now comes my question.
In the stationary frame, the 4-current density should be given by:
$j^\mu(\mathbf{r}) = (c e \delta(\mathbf{r}),0)$

So, when this gets boosted, shouldn't one have the following current density:
$j^\mu = (\gamma c e \delta(\mathbf{r'} - \mathbf{r_0}(t)), \gamma e \mathbf{v_0}\delta(\mathbf{r'} - \mathbf{r_0}(t)))$

In other words, why isn't there a factor $\gamma$ in the density like posed on wikipedia? Also in my own literature there is no such factor.

Cheers,

Snaab

Last edited by a moderator:
The missing γ has been swallowed by the delta function. There's a γ in the transformation from r to r', and one must use the identity δ(x) = c δ(cx). In other words, δ(r) = (1/γ)δ(r').

## 1. What is the Liénard-Wiechert potential derivation?

The Liénard-Wiechert potential derivation is a mathematical process used to calculate the electric and magnetic potentials created by a moving charged particle. This derivation is based on Maxwell's equations and takes into account the relativistic effects of the particle's motion.

## 2. Why is the Liénard-Wiechert potential derivation important?

The Liénard-Wiechert potential derivation is important because it allows us to understand and predict the behavior of electromagnetic waves and particles. It is also essential in the study of electrodynamics and has applications in various fields, such as astrophysics and particle physics.

## 3. How is the Liénard-Wiechert potential derivation derived?

The Liénard-Wiechert potential derivation is derived by solving Maxwell's equations for a point charge moving at a constant velocity. This involves using the Lorentz transformation to convert the electric and magnetic fields in the particle's rest frame to the fields in the observer's frame.

## 4. What is the significance of the Liénard-Wiechert potential derivation in relativity?

The Liénard-Wiechert potential derivation is significant in relativity because it takes into account the effects of special relativity on the electric and magnetic fields of a moving charged particle. This allows us to accurately describe the behavior of electromagnetic radiation, which is important in understanding the behavior of particles traveling at high speeds.

## 5. Are there any limitations to the Liénard-Wiechert potential derivation?

While the Liénard-Wiechert potential derivation is a very useful tool, it does have some limitations. It only applies to point charges moving at constant velocities, and it does not take into account the quantum nature of particles. Additionally, it does not account for higher-order relativistic effects, such as acceleration or gravitational fields.

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