# Electron charge^2 as a product of radius, mass, and c^2?

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1. May 13, 2015

### Shawnyboy

Hello Physics Peeps,

It just came up in the notes for my electrodynamics class that an electrons charge squared can be expressed as the radius times the mass times the speed of light squared.

$e^2 = m_er_ec^2$

I don't understand the motivation for doing this. I've tried to search for other people doing this online to no avail. Is this just some kind of weird approximation or is it actually valid? If so why?

As far as I can see the units don't even match up although I may be missing something we have energy times distance on the right, charge on the left.

For a little more context this came up in a problem about synchroton radiation where we were using the formula for power emitted by an electron in a synchrotron which is:

$P = \frac{2e^2}{3c}\omega^2\beta^2\gamma^4$

And by using the relation in question we simplified the question by saying:

$\frac{2e^2}{3c} = \frac{2(.5MeV)(2.8*10^{-13}cm)}{3c} \approx 3 * 10^{-24 }MeV s$

2. May 13, 2015

### rude man

What unit system are you using? cgs perhaps? and mind the equivalence of e-v and Joules.

3. May 13, 2015

### Shawnyboy

Ah-hah! cgs! That at least explains the units, because according to wikipedia a Statcoulomb = erg^1/2 * cm^1/2. However it is still not entirely clear the motivation. I mean you can't just do something because the units work.

4. May 13, 2015

### nasu

It's maybe a definition of some electron "radius".
e^2/r is an electrostatic energy. mc^2 is the rest energy.

5. May 13, 2015

### Staff: Mentor

I think that nasu is right. The electron is a quantum "point particle" meaning that it has no internal structure. The electron doesn't have a radius, per se, not in quantum mechanics and not classically either.

6. May 13, 2015

### BvU

Easy Googling, folks! Compton radius is the name for what you get when equating rest mass with electrostatic energy.

But beware, you end up with superstition and fantasy in no time...
For all we know, electrons are pointlike. And yet they aren't obeying all the expected accompanying behaviour rules.
Can't find a reasonable quotation for the upper limit; thought it was around 10-30 m. Anyone know where to look ?

7. May 14, 2015

### Shawnyboy

Thank You! That's exactly what it is. That explains it perfectly, although a strange concept this "Compton Radius". I wonder how useful it actually could be. It seems very incorrect to think of an electron in this way.