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# 16.2: The CGS Electrostatic System

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

Definition. One CGS esu of charge (also known as the statcoulomb) is that charge which, if placed 1 cm from a similar charge in vacuo, will repel it with a force of 1 dyne.

The following exercises will be instructive.

Potential Difference

If the work required to move a charge of 1 esu from one point to another is 1 erg, the potential difference between the points is 1 esu of potential difference, or 1 statvolt.

It is often said that an esu of potential difference is 300 volts, but this is just an approximation. The exact conversion is

$1 \ \text{statvolt} = 10^{-8} c \ \text{V}.$

Capacitance

If the potential difference across the plate of a capacitor is one statvolt when the capacitor holds a charge of one statcoulomb, the capacitance of the capacitor is one centimetre. (No – that's not a misprint.)

$1 \ \text{cm} = 10^9 c^{-2} \text{F}.$

Here is a sample of some formulas for use with CGS esu.

Potential at a distance $$r$$ from a point charge $$Q$$ in vacuo = $$Q/r$$.

Field at a distance $$r$$ in vacuo from an infinite line charge of $$\lambda \ \text{esu/cm} = 2 \lambda /r$$.

Field in vacuo above an infinite charged plate bearing a surface charge density of $$\sigma \ \text{esu/cm}^2 = 2 \pi \sigma$$.

An electric dipole moment $$\textbf{p}$$ is, as in SI, the maximum torque experienced by the dipole in unit electric field. A debye is $$10^{-18}$$ esu of dipole moment. The field at a distance $$r$$ in vacuo along the axis of a dipole is $$2p/r$$.

Gauss's theorem: The total normal outward flux through a closed surface is 4$$\pi$$ time the enclosed charge.

Capacitance of a plane parallel capacitor = $$\frac{kA}{4 \pi d}$$.

Capacitance of an isolated sphere of radius $$a$$ in vacuo = $$a$$. Example: What is the capacitance of a sphere of radius 1 cm? Answer: 1 cm. Easy, eh?

Energy per unit volume or an electric field $$= E^2/(8 \pi)$$.

One more example before leaving esu. You will recall that, if a polarizable material is placed in an electrostatic field, the field $$\textbf{D}$$ in the material is greater than $$\epsilon_0 \textbf{E}$$ by the polarization $$\textbf{P}$$ of the material. That is, $$\textbf{D}= \boldsymbol{\epsilon} \textbf{E} + \textbf{P}$$. The equivalent formula for use with CGS esu is

$\textbf{D}=\textbf{E} + 4\pi \textbf{P}$

And since $$\textbf{P}= \chi_e \textbf{E}$$ and $$\textbf{D} = k\textbf{E}$$, it follows that

$k= 1 + 4 \pi \chi_e.$

At this stage you may want a conversion factor between esu and SI for all quantities. I'll supply one a little later, but I want to describe emu first, and then we can construct a table given conversions between all three systems.