16.2: The CGS Electrostatic System
- Page ID
- 5516
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.