Topics in Classical Electrodynamics

1 Summary of Electrostatics
Classical electrodynamics is a theory of electric and magnetic fields caused by
macroscopic distributions of electric charges and currents. In these lectures,
we recapitulate the basic concepts of classical electrodynamics. Within the
field of electrodynamics, one can study electromagnetic fields under certain
static conditions leading to electrostatics (electric fields independent of time)
and magnetostatics (magnetic fields independent of time). First, we focus
on the laws of electrostatics. Then we derive Maxwell’s equations and study
some of their solutions. We end up with the discussion of two classical
radiation problems.
1.1 Laws of Electrostatics
Electrostatics is the study of electric fields produced by static charges. It
is based entirely on Coulomb’s law. This law defines the force that two
electrically charged bodies (point charges) exert on each other
⃗F (⃗x) = k q 1 q 2
⃗x 1 − ⃗x 2
|⃗x 1 − ⃗x 2 | 3 , (1)
where k is Coulomb’s constant (depends on the system of units used 1 ), q 1 and
q 2 are the magnitudes of the two charges, and ⃗x 1 and ⃗x 2 are their position
vectors (as presented in Figure 1).
One can introduce the concept of an electric field E ⃗ as the force per unit
charge
F ⃗ (⃗x)
⃗E (⃗x) = lim .
q→0 q
We have used the limiting procedure to introduce a test charge such that it
will only measure the electric field at a certain point and not create its own
field. Hence, using Coulomb’s law, we obtain an expression for the electric
1 In SI units, the Coulomb’s constant is k = 1
4πɛ 0
, while force is measured in newtons,
charge in coulombs, length in meters, and the vacuum permittivity ɛ 0 is given by
ɛ 0 = 107
4πc
= 8.8542 · 10 −12 F/m . Here, F indicates farad, a unit of capacitance being equal
2
to one coulomb per volt. One can also use the Gauss system of units (CGS). In CGS units,
force is expressed in dynes, charge in statcoulombs, length in centimeters, and the vacuum
permittivity then reduces to ɛ 0 = 1
4π . 2