## PHY450H1S. Relativistic Electrodynamics Lecture 27 (Taught by Prof. Erich Poppitz). Radiation reaction force continued, and limits of classical electrodynamics.

Posted by peeterjoot on May 3, 2011

# Reading.

Covering chapter 8 section 65 material from the text [1].

FIXME: Covering pp. 198.1-200: (last topic): attempt to go to the next order – radiation damping, the limitations of classical electrodynamics, and the relevant time/length/energy scales.

# Radiation reaction force.

We previously obtained the radiation reaction force by adding a “frictional” force to the harmonic oscillator system. Now its time to obtain this by continuing the expansion of the potentials to the next order in .

Recall that our potentials are

We can expand in Taylor series about . For the charge density this is

so that our scalar potential to third order is

Expanding the vector potential in Taylor series to second order we have

We’ve already considered the effects of the term, and now move on to . We will write as a total derivative

and gauge transform it away as we did with previously.

Looking first at the first integral we can employ the trick of writing , and then employ integration by parts

For the second integral, we have

so our gauge transformed vector potential term is reduced to

Now we wish to employ a discrete representation of the charge density

So that the second order vector potential becomes

We end up with a dipole moment

so we can write

Observe that there is no magnetic field due to this contribution since there is no explicit spatial dependence

we have also gauge transformed away the scalar potential contribution so have only the time derivative contribution to the electric field

To there is a homogeneous electric field felt by all particles, hence every particle feels a “friction” force

**Moral:** arises in third order term expansion and thus shouldn’t be given a weight as important as the two other terms. i.e. It’s consequences are less.

## Example: our dipole system

Here is the classical radius of the electron. For periodic motion

The ratio of the last term to the inertial term is

so

So long as , this approximation is valid.

# Limits of classical electrodynamics.

What sort of energy is this? At these frequencies QM effects come in

whereas the rest energy of the electron is

At these frequencies it is possible to create and pairs. A theory where the number of particles (electrons and positrons) is NOT fixed anymore is required. An estimate of this frequency, where these effects have to be considered is possible.

PICTURE: different length scales with frequency increasing to the left and length scales increasing to the right.

\begin{itemize}

\item , . LHC exploration.

\item , , the Compton wavelength of the electron. QED and quantum field theory.

\item , Bohr radius. QM, and classical electrodynamics.

\end{itemize}

here

is the fine structure constant.

Similar to the distance scale restrictions, we have field strength restrictions. A strong enough field (Electric) can start creating electron and positron pairs. This occurs at about

so the critical field strength is

**Is this real?**

Yes, with a very heavy nucleus with some electrons stripped off, the field can be so strong that positron and electron pairs will be created. This can be *observed* in heavy ion collisions!

# References

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

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