Electrodynamics
A quick review of basic ED needed in plasma physics
• Maxwell’s equations and Lorentz force
• Ohm’s law
• Scalar and vector potentials
• Lorenz and Coulomb gauges
• Poynting’s theorem
• Electromagnetic waves
• Units in plasma physics
Maxwell’s equations
We call B magnetic field. It actually gives the density of magnetic flux
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It is not quite clear how to define D or P for plasma, i.e.
for a set of free charges!
However the polarization current:
and the magnetization current:
are useful plasma concepts.
Electric and magnetic fields are empirically determined through
the Lorentz force
or
Because
only E performs work
It is often stated that a changing magnetic field can accelerate particles,
but actually particles are then accelerated by the induced electric field!
Ohm’s law
Ohm’s law
relating the electric current and electric field
is similar to the other constitutive equations
and
The conductivity , permittivity , and permeability depend on the
electric and magnetic properties of the media considered. They may be
scalars or tensors, and there does not need to be a local constitutive
relation at all, not even Ohm’s law!
A medium is called linear if
of time and space.
are scalars and they are not functions
Note that also in linear media = ( ,k), which is a very important
relationship in plasma physics (whether linear or non-linear)!
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Scalar and vector potentials
How to solve Maxwell’s equations?
A is called vector potential
thus
scalar potential
Inserting these into remaining Maxwell’s equations we get
By solving A and from
these we get E and B
as their derivatives
Because E and B are derivatives of
A and , we have certain freedom to
manipulate them and yet get the same
physical fields
Gauge transformations:
The Lorenz gauge is defined
by condition
Four inhomogeneous wave equations, for which there are well-known
solution methods leading to retarded potentials (c.f. any good ED text-book!)
The potentials take into account the finite speed of information (c) in a
relativistically correct way. In 4-vector notation the wave equation is
where
&
The Coulomb gauge condition is
The Coulomb gauge is very useful in radiation problems, because the
radiation fields can then be calculated from the vector potential alone.
The Coulomb gauge separates the static and inductive electric fields
but this separation is not Lorentz covariant
(be careful with moving frames of reference!)
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If there are no local currents, i.e., the magnetic field is determined by
external sources only, the field can be expressed as a gradient of a
magnetic scalar potential
This greatly simplifies the calculations because the methods of potential
theory, similar to electrostatics become available. Even the rather
complicated magnetic field of the Earth can be expressed using
spherical harmonic expansions
Another presentation of B is to use the Euler potentials
defined by
Thus B is perpendicular to both and , i.e.,
they are constant along the magnetic field
lines. This is useful when one needs to trace
a field line from one location to another.
Conservation of EM energy
Poynting’s theorem
The energy of electromagnetic field is given by
Strating from Maxwell’s equations it is a straightforward exercise to get
Poynting’s theorem
where
is the Poynting vector
Integrating over volume V (and using Gauss’s law for the divergence)
work performed
by the EM field
energy flux through
the surface of V
change of
energy in V
Conservation law of electromagnetic energy
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Electromagnetic waves
in plasma physics
•
Plasmas can host a large variety
of waves
•
– plasma sources of radiation
(bremsstrahlung, cyclotron and
synchrotron emissions)
– plasma effects between the
source and observer
– emissions from shocks
– plasma oscillations, oscillations
related to gyromotion, hybrids of
these
•
EM-waves propagating through
the plasma are
•
Energy and momentum transfer
between fields and particles
Observations of remote objects
(astronomy)
– reflected, refracted, absorbed
•
Plasma diagnostics
– laboratories
– space (in situ and remote)
– acceleration, heating, resonances
•
Plasma effects on
– satellite positioning
– telecommunications
– spacecraft charging
EM waves in vacuum
In absence of charges and currents, Maxwell’s equations reduce to
where we simply denote B =
0H
homogeneous wave equations
propagation speed:
Consider propagation to ±z – direction and plane wave solutions.
For a plane wave there is a plane propagating with the wave where
the electric field is constant. Such a solution can be written as
or in vector form
amplitude
angular frequency
wave number
phase speed
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A wave can be considered as a plane wave only far from the source.
Sometimes it is necessary to consider spherical waves, i.e., waves,
for which there exists a spherical surface where E is constant.
An example is the field of a radiating dipole:
(this is an approximation up to terms of the order of 1/r3; the exact
solution can be expressed in terms of Bessel functions)
Plane waves are most convenient
to treat using the complex notation
Now the differential operations
reduce to multiplications
Important!
and Maxwell’s equations
become algebraic equations
From vacuum to dielectric media
If there are no free charges or currents, but
0
and
0
(but constants)
Now
The ratio
is the index of refraction
dispersion equation
phase velocity (propagation of the constant phase)
group velocity (propagation of energy and infromation!)
If in addition J = E ( constant), the situation is more complicated
one form of so-called telegrapher’s equations;
a standard class-room example of suing Fourier
transformations to solve partial differential equations
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Instead of making the Fourier transformations, let’s start from Maxwell’s
equations and make the plane wave assumption
Clearly k, E and H are all perpendicular
to each other: transversal wave
Choose: k || ez, E || ex, H || ey
dispersion equation
now
!
write
The solution is
select the phase of so
that the wave is damped!
impedance of the medium:
skin depth
Wave polarization
Consider an EM wave propagating in +z direction
Focus on the plane z = 0, denote = Ey/Ex = – Hx/Hy;
1) If is real, Ey and Ex are in the same phase
Direction of E is (1, ,0) (if = , E is in the y-direction)
This is a linearly polarized wave
2) If = +i, there is a phase shift ( = /2) between Ey and Ex
This is a right-hand circularly polarized wave (positive helicity)
3) If = –i , there is a phase shift ( = – /2) between Ey and Ex
This is a left-hand circularly polarized wave (negative helicity)
4) If
is a general complex number, the wave is elliptically polarized
All polarizations can be obtained from left- and right-hand polarized
waves by superposition.
WARNING: In optics the left- and right-hand polarizations are defined
in the opposite way!!
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Comments on unit systems
We use the SI unit system, but many plasma physics books and almost
all astrophysics books use the cgs Gaussian unit system
Maxwell’s equations transform from SI to cgs in the following way:
In plasma physics we
often give temperature
in eV: 1 eV
11600 K
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