- #1

lailola

- 46

- 0

## Homework Statement

I have to find the Lienard-Wiechert potentials

[itex]\vec{A}=\frac{\vec{qv}}{R-\vec{R}\vec{v}/c}[/itex]

[itex]\phi=\frac{q}{R-\vec{R}\vec{v}/c}[/itex]

(both evaluated in t_r)

with [itex]\vec{R}=\vec{r}-\vec{x}(t_r)[/itex]. [itex]\vec{r}=x\hat{x}+y\hat{y}+z\hat{z}[/itex] is the vector asociated to the observation point, [itex]\vec{x}(t)=\vec{a}+ct \hat{z}[/itex] is the trajectory of the particle.

Then I have to find E, B and verify the conservation theorem for the energy density.

Finally, I have to compare my results with the results obtained in a frame that is moving with velocity [itex]\vec{v}=v\hat{z}[/itex]

## Homework Equations

[itex]R=c(t-t_r)[/itex]

[itex]\vec{v}=\frac{dx(t_r)}{dt_r}=c \hat{z}[/itex]

## The Attempt at a Solution

If I calculate the denominator:

[itex]R-\vec{R}\vec{v}/c = c(t-t_r)-(zc-ac-c^2t_r)/c= ct-z-a[/itex]

Substituting in the expressions I get A and [itex]\phi[/itex].

For calculating E and B I use:

[itex]\vec{E}=-grad(\phi)-\frac{1}{c}\frac{\partial\vec{A}}{\partial t}[/itex]

[iex]\vec{B}=rot(\vec{A})[/itex]

I find B=0, so [itex]\vec{S}=0[/itex]

The energy density is [itex]\varepsilon=\frac{c}{4\pi}(E^2+B^2)=\frac{c}{4\pi}(E^2)[/itex]

Finally, the conservation theorem says:

[itex]grad(\vec{S})+\frac{\partial \varepsilon}{\partial t}= 0[/itex]

One of my problems is that this last part is not satisfied, and I can't find the error.

The other problem is to solve the problem in the new inertial frame. I don't know how to do it.

Tthanks for any help.

Last edited: