## PHY450H1S. Relativistic Electrodynamics Lecture 19 (Taught by Prof. Erich Poppitz). Lienard-Wiechert potentials.

Posted by peeterjoot on April 26, 2011

# Reading.

Covering chapter 8 material from the text [1].

Covering lecture notes pp. 136-146: the Lienard-Wiechert potentials (143-146) [Wednesday, Mar. 9…]

# Fields from the Lienard-Wiechert potentials

(We finished off with the scalar and vector potentials in class, but I’ve put those notes with the previous lecture).

To find and need

, and

where

implicit definition of

In HW5 you’ll show

and then use this to show that the electric and magnetic fields due to a moving charge are

where everything is evaluated at the retarded time .

This looks quite a bit different than what we find in section 63 (63.8) in the text, but a little bit of expansion shows they are the same.

# Check. Particle at rest.

With

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{particleAtRestTrCalc}

\caption{Retarded time for particle at rest.}

\end{figure}

As illustrated in figure (\ref{fig:particleAtRestTrCalc}) the retarded position is

for

and

which is Coulomb’s law

# Check. Particle moving with constant velocity.

This was also computed in full in homework 5. The end result was

Writing

We can introduce an angular dependence between the charge’s translated position and its velocity

and write the field as

Observe that .

The electric field has a time dependence, strongest when perpendicular to the instantaneous position when , since the denominator is smallest ( largest) when is not small. This is strongly dependent.

Compare

Here we used

and

FIXME: he writes:

I don’t see where that comes from.

FIXME: PICTURE: Various ‘s up, and perpendicular to that, strongest when charge is moving fast.

# Back to extracting physics from the Lienard-Wiechert field equations

Imagine that we have a localized particle motion with

The velocity vector

doesn’t grow as distance from the source, so from 2.4, we have for

The acceleration term will dominate at large distances from the source. Our Poynting magnitude is

We can ask about

In the limit, for the radiation of EM waves

The energy flux through a sphere of radius is called the radiated power.

# References

[1] L.D. Landau and E.M. Lifshitz. *The classical theory of fields*. Butterworth-Heinemann, 1980.

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