# 16.2: The CGS Electrostatic System

- Page ID
- 5516

**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.