# Does the New SI spoil the nice Coulomb law in Gaussian units?

Now, after the redefinition of SI, the elementary charge $$e$$ and the reduced Planck constant $$\hbar$$ (and also $$k_B$$) are exact quantities in the SI units, as is the velocity of light $$c$$. The magnetic and electric constants became non-exact quantities (having the uncertainty of order $$10^{-10}$$), as the fine-structure constant $$\alpha$$ always was.

The Gaussian units (and also units in other variants of CGS) are connected with the SI units via exact factors involving $$\pi$$ and $$c$$. Neither BIPM nor any other organization regulates the Gaussian system as BIPM regulates SI. So it seems that fixing $$e$$, $$\hbar$$, and $$c$$ in SI automatically fixes these constants in Gaussian unit system.

Generally, $$\alpha=k_Ee^2/\hbar c$$, and the Coulomb potential is $$\phi=k_Eq/r$$ (Coulomb law). In the Gaussian unit system, the units were chosen in such a way that $$k_E=1$$, nicely.

So apparently this means that the Coulomb constant $$k_E$$ in the Gaussian unit system now becomes an experimentally measured quantity approximately equal to $$1$$ with the same relative uncertainty. Is that true?

Another option is to make the conversion factor between the charge units in SI and the Gaussian system a non-exact quantity but keep $$k_E=1$$. But who will determine what quantity becomes non-exact ($$k_E$$ or the conversion factor)? Is it possible that different authors (in particular, textbook authors) will use different conventions?

• The dimensions of $q$ and $e$ are not the same in Gaussian and SI units. I think that might be the answer to your concern. Check out this: physics.stackexchange.com/questions/1673/… – KF Gauss May 21 '19 at 21:29
• @KFGauss, I've never said they are the same. They are connected via dimensional but exact conversion factors. – colt_browning May 21 '19 at 21:31
• But hmmm, the conversion factor itself is apparently connected with the SI k_E. So it seems that the correct answer is: the Gaussian k_E is still 1, but the conversion factors are non-exact now. – colt_browning May 21 '19 at 21:34
• Charge in the two unit systems isn't connected by exact conversion factors. They are connected by $\epsilon_0$ which has an uncertainty due to its relation to the fine structure constant. – KF Gauss May 22 '19 at 4:43
• $\epsilon_0$ was exact until the recent redefinition. Now it's not (in SI), and that's what I'm saying in the 1st paragraph of the question. "Charge in the two unit systems isn't connected by exact conversion factors" -- with the New SI, yes, and that's what I said in my previous comment. – colt_browning May 22 '19 at 7:08