Re: VRD - coulombs, physical constants

The root of your problem is that the mysterious 10^-7 number is not dimensionless! I had to work a bit to figure out how you came up with that particular formula, because none of my physics references use it in the form you showed. I think that's because your form, while easier to calculate, is misleading in that the units are hidden.

Let me try to make this clear. The 10^-7 comes from a fundamental physical constant called the permeability of free space, generally symbolized as the greek letter mu with a zero subscript, "mu-naught." I'll use mu0. It has the units of henrys per meter (H/m), where the henry is the standard unit of inductance. The value of mu0 is exactly 4*pi*10^-7 H/m.

The other constant of interest is the permittivity of free space, symbolized by epsilon-naught (ep0). Its units are farads per meter (F/m), with the farad the unit of capacitance.

Now, it turns out that these two constants are related, because together they determine the speed of light in a vacuum. This makes sense, because the capacitance and inductance of free space will determine how the electric and magnetic fields in a light wave interact. That, in turn, determines how fast the wave propagates.

The relationship formula is: c^2=1/(ep0 * mu0). You can see that this relates two electronic units (farads and henrys) to a velocity. The farad unit is proportional to coulombs-squared. Henrys are proportional to 1/ coulomb^2. When you multiply them together, the charge cancels out and the other units end up simplifying to the inverse-square of a velocity.

Now, the standard formula for the classical electron radius, which I will refer to as "r" like you do, is:

r= e^2/(4*pi*ep0*m*c^2)

You'll notice that in this formulation, the coulomb-squared part of ep0 cancels very neatly the e^2, eliminating the problem you had with the units. To get the formula you used, substitute the value of mu0 to get an expression for ep0=1/4*pi*c^2*10^-7. When you plug this in to the above formula for r, the 4*pi*c^2 cancels out nicely, leaving only the 10^-7. If you work through the original expressions, that 10^-7 has to have units of kg*m/C^2, which makes your formula come out sensibly.

I hope this helps. It was good of you to check the units as you did. Often that provides more insight into the physics behind the formula.