Coulomb's constant kC is simply the fine structure constant alpha, dressed in the appropriate natural units. Alpha is often written as 1/137 for conciseness, although it's known to considerably better accuracy.
kC = alpha Planck force (Planck length)2/e2 = alpha h-bar c/e2.
Coulomb's constant (or constants artificially made out of it) enters Maxwell's equations when the effects of current and static charge are included. For historical reasons the constants made from kC are:
epsilono = (4 pi kC)-1
muo = 4 pi kC/c2
To see how Coulomb's constant and its surrogates appear in Maxwell's equations, let's have a look at the equations. In the symmetric version the fields E and B are commensurable in the sense that they are expressed in the same unit. Here are the equations in free space, without current and static charge:
div B
div E
curl B = 1/c Et
curl E = -1/c Bt
When introducing the charge density and current density into the equations, so as not to have to define notation I've simply put the word itself in square brackets.
div B = 0
div E = 4 pi kC [charge density]
curl B = 1/c (Et + 4 pi kC
[current density])
curl E = -1/c Bt
When Maxwell's equations are used with Planck units, E and B expressed in the Planck field unit: (Planck force)/e and defined by their appearance in the Lorentz force law with dimensionless speed beta= v/c:
F = q(E + beta×B)
There is not very much to remember, kC is just 1/137 followed by the appropriate string of units: Planck force area/charge2 = FP lP2/e2. Before continuing let me express my conviction that a self-respecting person never conceals a factor of 4 pi or Coulomb's constant. To do so is a low vice on par with cheating at cards.
A few simple substitutions will convert the above equations into their standard SI textbook versions. The main thing is to use the phoney "B" which is a code-name for B/c. This hides symmetry and contorts the equations by making the two fields have different field units. Then, instead of letting Coulomb's constant be visible, one introduces a pair of surrogate constants: epsilono and muo devised to conceal the Coulomb constant and the number 4 pi. As was said earlier, the phoney constants are defined as follows:
epsilono = (4 pi kC)-1
muo = 4 pi kC/c2
Because of the way the two are defined (both are screwed up versions of Coulomb's constant) multiplying the two together gives 1/c2. Using the surrogate constants enables us to arrive at the clumsier and less symmetric version of Maxwell's equations in conformity with SI (International System) preference:
div "B" = 0
div E = [charge density]/epsilono
curl "B" =
epsilonomuoEt + muo
[current density]
curl E = -"B"t
Compare this, for instance, with the equations in a widely-used textbook, Giancoli's Physics for Scientists and Engineers, with Modern Physics. In a professor teaching the second year Electricity and Magnetism course, it is a sign of discernment and moral fiber to disregard SI preference and teach Maxwell's equations, as indeed some do, in their symmetric form.
In the Planck units the value of Coulomb's constant, namely the fine structure constant, is experimentally determined. The NIST gives its inverse as 137.035 999 76 with standard uncertainty 0.000 000 50, which is relative standard uncertainty 3.7 x 10-9.
http://physics.nist.gov/cgi-bin/cuu/Value?alphinv|search_for=abbr_in!
In SI units it is not possible to measure the values for kC or the two surrogates because they are established by convention. This could change if the 1990 CPGM electrical standards (which are already used in the national laboratories for careful measurement), or something like them, were adopted as official. But at least for the time being we have the arbitrary assigned value:
kC = muoc2/ 4 pi = 10-7newton ampere-2×2997924582 meter2 second-2 = garbage × newton meter2/coulomb2. The numerical garbage term (which plays the role of the fine structure constant here) is 10-7×2997924582
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Copyright © 2002 by Leonard Cottrell. All rights reserved.