January 13, 1999

Notes for Lecture #1

1 Introduction

1. Textbook and course structure

2. Motivation

3. Chapters I and 1 and Appendix of Jackson (a) Units - SI vs Gaussian

(b) Laplace and Poisson Equations

(c) Green’s Theorm

2 Units - SI vs Gaussian

Coulomb’s law has the form:

F = K_{C} \frac{q_{1}q_{2}}{r_{12}^{2}} .  (1)

Ampere’s law has the form:

F = K_{A} \frac{i_{1}i_{2}}{r_{12}^{2}} ds_{1} × ds_{2} × r̂_{12},  (2)

where the current and charge are related by i_{1} = dq_{1}/dt for all unit systems. The two constants K_{C} and K_{A} are related so that their ratio K_{C}/K_{A} has the units of (m/s)^{2} and it is experimentally known that in both the SI and CGS (Gaussian) unit systems, it the value K_{C}/K_{A} = c^{2}, where c is the speed of light. The choices for these constants in the SI and Gaussian units are given below:

CGS (Gaussian) SI
K_{C}  1  \frac{1}{4πϵ_{0}}
K_{A}  \frac{1}{c^{2}}  \frac{μ_{0}}{4π}

Here, \frac{μ_{0}}{4π} ≡ 10^{−7}N/A^{2} and \frac{1}{4πϵ_{0}} = c^{2} · 10^{−7}N/A^{2} = 8.98755 × 10^{9}N · m^{2}/C^{2}.

Below is a table comparing SI and Gaussian unit systems. The fundamental units for each system are so labeled and are used to define the derived units.

Variable  SI  Gaussian  SI/Gaussian
Unit Relation  Unit  Relation
length  m fundamental  cm  fundamental  100
mass  kg fundamental  gm  fundamental  1000
time  s fundamental  s  fundamental  1
force  N  kg · m^{2}/s  dyne  gm · cm^{2}/s  10^{5}
current A fundamental statampere statcoulomb/s  \frac{1}{10c}
charge  C  A · s  statcoulomb \sqrt{dyne · cm^{2}}  \frac{1}{10c}

One advantage of the Gaussian system is that all of the field vectors: E, D, B, H, P, M have the same dimensions, and in vacuum, B = H and E = D and the dielectric and permittivity constants ϵ and μ are unitless.

CGS (Gaussian)  SI
∇ · D = 4πρ  ∇ · D = ρ
∇ · B = 0  ∇ · B = 0
∇ × E = −\frac{1}{c} \frac{∂B}{∂t}  ∇ × E = −\frac{∂B}{∂t}
∇ × H = \frac{4π}{c}J + \frac{1}{c} \frac{∂D}{∂t}  ∇ × H = J + \frac{∂D}{∂t}
F = q(E + \frac{v}{c} × B  F = q(E + v × B
u = \frac{1}{8π}(E · D + B · H) u = \frac{1}{2}(E · D + B · H)
S = \frac{c}{4π}(E × H)  S = (E × H)