## PHY450H1S. Relativistic Electrodynamics Lecture 24 (Taught by Prof. Erich Poppitz). Non-relativistic electrostatic Lagrangian.

Posted by peeterjoot on March 30, 2011

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# Reading.

Covering chapter 5 \S 37, and chapter 8 \S 65 material from the text [1].

Covering pp. 181-195: the Lagrangian for a system of non relativistic charged particles to zeroth order in : electrostatic energy of a system of charges and .mass renormalization.

# A closed system of charged particles.

Consider a closed system of charged particles and imagine there is a frame where they are non-relativistic . In this case we can describe the dynamics using a Lagrangian only for particles. i.e.

If we work t order .

If we try to go to , it’s difficult to only use for particles.

This can be inferred from

because at this order, due to radiation effects, we need to include EM field as dynamical.

# Start simple

Start with a system of (non-relativistic) free particles

So in the non-relativistic limit, after dropping the constant term that doesn’t effect the dynamics, our Lagrangian is

The first term is where the second is .

Next include the fact that particles are charged.

Here, working to , where we consider the particles moving so slowly that we have only a Coulomb potential , not .

HERE: these are NOT ‘EXTERNAL’ potentials. They are caused by all the charged particles.

For we have have , but we won’t do this today (tomorrow).

To leading order in , particles only created Coulomb fields and they only “feel” Coulomb fields. Hence to , we have

What’s the , the Coulomb field created by all the particles.

\paragraph{How to find?}

or

where

This is a Poisson equation

(where the time dependence has been suppressed). This has solution

This is the sum of instantaneous Coulomb potentials of all particles at the point of interest. Hence, it appears that should be evaluated in 3.11 at ?

However 3.11 becomes infinite due to contributions of the a-th particle itself. Solution to this is to drop the term, but let’s discuss this first.

Let’s talk about the electrostatic energy of our system of particles.

The first term is zero since for a localized system of charges or higher as .

In the second term

So we have

for

Now substitute 3.11 into 3.14 for

or

The first term is the sum of the electrostatic self energies of all particles. The source of this infinite self energy is in assuming a \underline{point like nature} of the particle. i.e. We modeled the charge using a delta function instead of using a continuous charge distribution.

Recall that if you have a charged sphere of radius

PICTURE: total charge , radius , our electrostatic energy is

Stipulate that rest energy is all of electrostatic origin we get that

This is called the classical radius of the electron, and is of a very small scale .

As a matter of fact the applicability of classical electrodynamics breaks down much sooner than this scale since quantum effects start kicking in.

Our Lagrangian is now

where is the electrostatic potential due to all \underline{other} particles, so we have

and for the system

This is THE Lagrangian for electrodynamics in the non-relativistic case, starting with the relativistic action.

# What’s next?

We continue to the next order of tomorrow.

# References

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

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