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I have trouble understanding Gaussian Units in relation to SI units: $1/c$ factors seem to appear almost at random, while other constant such as $\varepsilon _0$ or $4\pi$ disappear with seemingly comparable randomness.
The related Wikipedia page is less than helpful, stating things in the manner of: "one difference is that...", "another difference is..." and never giving a full picture on what is going on. How can I convert from SI to Gaussian and back? What is the complete set of rules?

But also.. Why do we bother? I have heard that theoretical physicist prefer to use Gaussian Units over SI units, why is it so? Is it simply a matter of ease of writing?

Noumeno
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  • Appendix D in Griffiths Electrodynamics book might help in understanding the rules of conversion. – TaeNyFan Dec 08 '20 at 17:37
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    Gaussian units eliminate pointless and confusing “concepts” like the permittivity and permeability of free space. – G. Smith Dec 08 '20 at 18:14

3 Answers3

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The SI system has 4 base units: length (m), time (s), mass (kg) and current (ampere). All mechanical measurements can be expressed as a combination of the 4 base units. For example, energy $\rightarrow{\rm kg\ m^{-2}\ s^{-2}}$ and electric charge $\rightarrow {\rm A\ s}$.

The Gaussian system has 3 base units: length (cm), time (s), mass (g). All mechanical measurements can be expressed a combination of these 3 base units. For example, energy $\rightarrow {\rm g\ cm^{-2}\ s^{-2}}$, electric charge $\rightarrow {\rm g^{1\over 2}\ cm^{3\over 2}\ s^{-1}}$ and current $\rightarrow {\rm g^{1\over 2}\ cm^{3\over 2}\ s^{-2}}$.

Note the awkwardness of the base unit ampere in the SI system. This is one of the reason why physicists prefer the Gaussian system while engineers prefer the SI system.

How the SI System Came About

When experimentalists observed that currents in two parallel wires can produce a force, they decided to define a new base unit, the ampere ($\rm A$), to quantify currents. In the SI system, 1 $\rm A$ is defined as the constant current between two parallel wires spaced 1 $\rm m$ apart that produces $2 \times 10^{-7 } {\rm N}$ of force per unit length.

Consequently, one can define the electric charge (Coulomb) as current $\times$ time. In the SI system, 1 Coulomb is defined as the quantity of electric charge carried in 1 $\rm s$ by a current of $1$ A.

The electric charge defined this way forces us to write Coulomb's law $\vec{F}=k\frac{q_1 q_2}{r^2}$ as $$\vec{F}=\frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2},$$ where the constant of proportionality $k$ is fixed to be equal to $\frac{1}{4\pi\epsilon_0}$.

So, in this system, we needed a new base unit to define currents. Also, we can't talk about electric charge without referring to currents. Is there a better way to do this?

Why the Gaussian System is Better

In the Gaussian system, the electric charge is defined directly using Coulomb's law $$\vec{F}=k\frac{q_1 q_2}{r^2},$$ where the constant $k$ is picked to be equal to 1. One statcoulomb is defined as the charge carried by two stationary objects 1 $\rm cm$ apart that produces a force of 1 dyne (i.e. 1 $\rm g\ cm\ s^{-2}$).

The current is then naturally defined as the quantity of charge that flows through a point per unit time. There is no need to define a fourth base unit just to measure current.

Compared to the SI system, the Gaussian system is defined in a simpler and more elegant way. No wonder theoreticians like it more!

PS: Another reason why the Gaussian system is better is that the electric and magnetic fields have the same units. This is nice since electric and magnetic fields transform into each other under Lorentz transformations.

TaeNyFan
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  • I’m afraid no one will ever convince me that Gaussian units are better. – Gilbert Dec 08 '20 at 19:07
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    @Gilbert I am not sure how the system of units with out-of-blue constants $\epsilon_0$, $\mu_0$ could come into existence at all... – Roger V. Dec 08 '20 at 19:20
  • Great answer. So at the end of the day the only difference between SI and Gaussian unit system is the definition of charge, right? – Noumeno Dec 09 '20 at 10:15
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To your question as to "why do we bother" here is a comment by Val Fitch [1] (https://en.wikipedia.org/wiki/Val_Logsdon_Fitch) on the same subject:

Many thanks for your editorial comments on units @Robert H. Romer, ‘‘Units: SI-Only, or Multicultural Diversity?’’ Am. J. Phys. 67(1), 13–16 (1999). As you imply, units are a cultural matter—not scientific. And, as you stopped just short of saying, any system that gives E and B different units, when they are related through a relativistic transformation, is on the far side of sanity.

[1] Fitch: "THE FAR SIDE OF SANITY", Am. J. Phys., Vol. 67, No. 6, June 1999, p467

hyportnex
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  • Doesn't the argument from relativistic transformation between E and B apply equally to the units of time and distance? – Adam Zalcman Dec 08 '20 at 17:28
  • @Adam_Zalcman set $c=1$ – hyportnex Dec 08 '20 at 17:41
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    The point I was trying to make is that if systems of units such as SI are "on the far side of sanity" for giving E and B different units, then presumably systems of units such as those derived from CGS including Gaussian Units are also "on the far side of sanity" for giving distance and time different units. In other words, in Gaussian Units $c \neq 1$, so the quote above doesn't answer the question "why do we bother with Gaussian Units?". – Adam Zalcman Dec 08 '20 at 19:59
  • @AdamZalcman The $\vec{E}$ and $\vec{B}$ fields are actually defined by the Lorentz force law $\vec{F}=q(\vec{E}+\vec{v}\times{B})$. The quantity we directly measure is the $\vec{F}$. So we can choose whatever units for $\vec{E}$ and $\vec{B}$ as long as $\vec{F}$ has the correct units of mass $\times$ acceleration. But for time and distance, it is more natural to stick to $\rm s$ and $\rm m$ because these are how they are directly measured. In the CGS system, the force still has the familiar units of $\rm g\ cm\ s^{-2} $. – TaeNyFan Dec 09 '20 at 00:23
  • @AdamZalcman So the Gaussian system isn't actually a system that is completely convenient for the theory of relativity. After all, it is a system that meant to be used to make measurements in the lab, instead of being completely theoretical. However, since E and B fields are just imaginary concepts used to describe the E and B force, why not make them the same units? It certainly would be weird to want the fields to have different units when the E and B force have the same units. – TaeNyFan Dec 09 '20 at 00:30
  • @AdamZalcman yes, the sane thing to do is to set $c=1$ and measure time and space with the same units. CGS is a bit closer to sanity than SI. – fqq Dec 11 '20 at 01:18
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You may want to look at the answer that I posted here. In a nutshell: choosing a system of units is a matter of convenience, and different fields of science and engineering use different units (including different systems used in different domains of theoretical physics).

Roger V.
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    Yes, it is obvious that is a matter of convenience. But in this case what is the convenience? It is convenient because equations get shorter? Or because in some context the equations are easier to understand using Gaussian Units? Or something else? – Noumeno Dec 08 '20 at 12:43
  • Also in my question I asked for a complete set of rules to change unit system, to properly understand how Gaussian Units work. In your liked answer you do not provide any of this. Your exposition seems in line with what I disliked about the Wikipedia explanation. – Noumeno Dec 08 '20 at 12:47
  • E.g., I worked in nanostructure physics - most people in this field use Gaussian units, so using anything else would be an extra complication. Although in practice I found it convenient to set to $1$ the planck constant, the electron charge and the speed of light - all of them are easily reintroduced in the final expression from general considerations and dimensional analysis. – Roger V. Dec 08 '20 at 12:50
  • Correctly constructing a system of units is a rather complex business. You certainly can dig into books and find how to convert properly from one system to the other. However, I suspect that most active physicists never bothered to do this - I certainly never went beyond one lecture that we had during my undergraduate studies. I could name the book, but it is in Russian. – Roger V. Dec 08 '20 at 12:52
  • This answer and the Wikipedia article explain rather well that the issue is not a homework problem with a defined answer. The problem is that unit systems have changed over time and as a function of the discipline. The quest for a complete set of rules may as well be the quest to write an historical tome across a few different scientific languages. – Jeffrey J Weimer Dec 08 '20 at 14:32
  • @JeffreyJWeimer I am not sure what your comment refers to. – Roger V. Dec 08 '20 at 14:47
  • Your answer was downvoted. I note that I think it cannot be improved, other than that it could be as expansive as the reference that you give in it. The OP seeks a complete set of rules. I don't know that such can be obtained writing a book on the history of the evolution of the unit system across different disciplines. – Jeffrey J Weimer Dec 08 '20 at 15:07
  • @JeffreyJWeimer I think that creating systems of units is a complex subject - I have written that much in the comments above. On the other hand, conversion between the existing systems is completely unambiguous - although I am not familiar with all the rules, it is certainly possible to formulate them. – Roger V. Dec 08 '20 at 15:16
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    I see and agree with what you are saying. – Jeffrey J Weimer Dec 08 '20 at 15:41