[NOTES/EM-01014] Gaussian Units

Gaussian Units

We briefly explain a system of units, called the Gaussian system of units that was commonly used before the SI units were introduced. The connection with SI system of units is briefly explained.

It must be remembered that the Gaussian system is no longer in use. Our motivation for a brief discussion of Gaussian comes from a desire to trace appearance of \(c\) in Maxwell's equations in the Gaussian system and its relationship to constants \(\epsilon_0\) and \(\mu_0\) that appear in SI system of units. We also give reference to an article about the original experimental paper, where the numerical value of \(c\) was first measured, and the facts related to the measurement of \(c\).
 
 Apart from mechanical cm-gram-sec (cgs) units, two types of electrical units were used in early days. The electrostatic units were based on the Coulomb's law of force between charges
\begin{equation}\label{EQ01}
  \vec F  = \frac{q_1 q_2 (\vec r_1-\vec 2)}{|\vec r_1-\vec r_2|^3}.
\end{equation}
This corresponds to replacing \(1/(4\pi\epsilon_0)\) with unity in the SI expression for the Coulomb's law. The esu units define a unit of charge with dimensions of \(M^{1/2}L^{1/2} T^{-1}\)

The magnetic units, called the electromagnetic unit (emu) were defined by means of the force between  two  elements carrying currents \(I_1\) and \(I_2\).\footnote{\eqref{EQ02} is needed only for dimensional reasoning, a detailed understanding of this expression is not required at this stage.
\begin{equation}\label{EQ02}
  \vec F = I_1I_2 \oint_1\oint_2 \frac{ d{\pmb{\ell}}_2\times \big(d{\pmb{\ell}}_1\times(\vec r_1-\vec r_2)\Big)}{|\vec r_1-\vec r_2|^3}.
\end{equation}
If mechanical cgs units are used the above expression defines 1 emu  of  current with dimensions \(M^{1/2}L^{1/2} T^{-1}\).  The current density and the charge density, in SI units, are related by  \begin{equation}\label{EQ13}
 \vec j =\rho \vec v \end{equation}  It is to be noted that  dimension of \(\rho\) in esu and the dimension of \(\vec j\) in emu are the same.
 
 Hence we need to write the relation  \eqref{EQ13} in the form \begin{equation}
   J \text{(emu)} = \rho \frac{\vec v }{c} \text{ esu}
\end{equation}  In the above equation, the constant \(c\) has dimensions of velocity.

The equations of the theory, in a consistent set of Gaussian units using esu and emu both, carry \(c\) as the only constant, whose value is to be determined by experiments. Such an experiment to determine the value was first carried out by Weber and Kohrausch in 1855.

Thus in cgs system of units we have \begin{equation}\label{EQ03}
\text{1 dyne} =  \text{(1 emu of   current)}^2 = \text{(10 ampere )}^2 = 10^2 \frac{\text{(coulomb)^2}}{\text{(second)}^2}
\end{equation} where we have used the definition  1 ampere = (1/10) emu of current.

Relationship between Gaussian and SI systems of units}
In SI system  \eqref{EQ02} is replaced by \begin{equation}
 \label{EQ04}   \vec F = \frac{\mu_0}{4\pi}J_1J_2 \oint_1\oint_2 \frac{ d{\pmb{\ell}}_2\times d{\pmb{\ell}}_1\times(\vec r_1-\vec r_2))}{|\vec r_1-\vec r_2|^3}. \end{equation}
where \(\mu_0\) is to be fixed by writing all the quantities in emu system. Therefore, we get \begin{eqnarray} \label{EQ05}
 1 \text{dyne} &=& \frac{\mu_0}{4\pi} \text{ (10 ampere)}^2\\
10^{-5}\text{newton} &=& 10^2\Big(\frac{\mu_0}{4\pi}\Big) \frac{\text{(coulomb)}^2}{\text{(second)}^2}.  \end{eqnarray}
This gives \begin{equation}
 \frac{\mu_0}{4\pi} =10^{-7} \text{henry/meter}
\end{equation} in SI units and an   appropriate SI unit has been indicated for  the right hand side.
 
 Further the constant \(\epsilon_0\) is obtained from the relation \(\epsilon_0 \mu_0 =1/c^2\) which is a result of demanding that electromagnetic theory equations in SI units be consistent with the equations in cgs system and we get  \begin{eqnarray}
\epsilon_0 &=&\frac{10^7}{4\pi c^2} \, \frac{\text{coulomb}^2 \text{second}^2}{\text{kilogram (meter)}^3}\\
&\approx& \frac{1}{36 \pi\times 10^9} \frac{1 \text{farad}}{1 \text{meter}}.
\end{eqnarray} here, in the last step use has been made of the value  \(c=3\times 10^8 m/s\) and of the relation
\[1\, \frac{\text{coulomb}^2 \text{second}^2}{\text{kilogram (meter)}^3} = 1 \frac{\text{farad}}{\text{meter}}.\]
A few derived units were commonly used in the cgs system. For example,  Maxwell(Mx) for the flux,  Gauss for the magnetic field intensity \(B\) and Oersted (Oe) for \(H\) field.   So we have
1 maxwell = 1 gauss × (centimetre)\(^{-2}\)
1 oersted = 1dyn/ (1 maxwell)
The units maxwell, gauss, Oersted can be related to corresponding SI units. So for example, we have \begin{equation}
  1\text{ Mx }= 10^{-8}.\text{ Wb}\qquad  1\text{ gauss} = 10^{-4}. T
\end{equation}  It must be remembered that the definitions and expressions of different quantities and physical equations appear differently in different sets of units. The constants \(c, \mu_0, \epsilon_0\)  in general appear in such equations.
For details of translation of expressions and equations and of conversions of units from one system to another system  we refer to the appendix of Jackson

 As already remarked the cgs system is not longer used in physics books. They may appear in older books and engineering books. In this book only SI units will be used. Often derived units with special names can be written in several different ways depending on the relation that is taken as the defining relation.