Download - Potentials - Liénard-Wiechart Potentials

- Potentials
- Liénard-Wiechart Potentials
- Larmor’s Formula
- Dipole Approximation
- Beginning of Cyclotron & Synchrotron
Maxwell’s equations in a vacuum become
A basic feature of these eqns is the existence of
traveling wave solutions that carry energy.
Taking the curl of 3rd eqn & using 4th, :
Using the vector identity
Vector wave equation
â1 and â2 are perpendicular to each other and
transverse to the direction of propagation, k
- Easy to show that E0 = B0 and phase velocity = c
Where â1 and â2 are unit
vectors; E0 & B0 are
complex constants and
k=kn is the wave vector
and ω is the frequency.
R&L 2.5: We can describe the electromagnetic
field in terms of a vector potential, A and a scalar
potential, φ
Can alter A and φ by an arbitrary scalar function to simplify the
associated equations
Most important Gauge transformation for EM waves
is the Lorentz gauge
Evaluated at the
retarded time…
Solutions:
The retarded time is interpreted as the time t at which the field is
observed minus the light propagation time from the point r’ where
the source was at time t. This is due to the fact that the
electromagnetic field propagates from the source to the point of
observation at the speed of light. In other words, the field
measured by an observer at r is determined by the configuration of
charges and currents not as they currently are, but as they were a
time t ago equal to the light travel time between the source and
observer.
Goal: Develop a formula which describes the power radiated by a
non-relativistic particle
1)  Consider the potential due to a charge with arbitrary trajectory
2)  Derive the Poynting vector associated with the electromagnetic
field of this particle
3)  Write down an expression for the power radiated away in the
electromagnetic field
define
These potentials differ from those of static em-theory in 2 ways:
1.  The factor κ, which becomes very important at velocities close to c where it tends to
concentrate the potentials into a narrow cone about the particle velocity (beaming)
2.  The quantities are all evaluated at retarded time tret. The major consequence of this
is that it makes it possible for the particle to radiate, as we will now show.
-  From ϕ and A we can derive E and B
Velocity field
acceleration field
Acceleration field = radiation field
- The E at time t=1 from a charge
moving at uniform speed at t< 0 which
comes to an abrupt stop at t=0 and x=0.
If the charge would have continued at
its original speed, at t=1 it would have
been at x=1. The wave front travels
outward with a velocity c and drops off
as 1/x.
-  When β <<1 (non-relativistic) Poynting vector is in the direction of n
Hence the emitted power is Integrating over all
solid angles gives
Larmor’s formula for
emission from a
single accelerated
charge q: 3
- Consider a system of many accelerating charges in a system of extent L
-  Have to calculate E for each charge independently – very complicated
-  Can simplify if the light travel time across the system is << than the timescale for
changes in the system, i.e., when β <<1 (non-relativistic) Recall
-  Define the dipole moment
1933: Karl Jansky (Bell Labs) builds an antenna designed to receive radio
waves at 20.5 MHz
- Discovers a steady source of radio emission from the sky that varied on a cycle
of 23 hours 56 minutes (sidereal day)
-  Comparing his observations to optical astronomical maps, he realized that this
radiation was coming from the Milky Way and was strongest towards
Sagittarius, the center of our Galaxy
-  What is this mysterious radiation?
- Synchrotrons are large particle accelerators that accelerate
particles in a magnetic field
- 1947: General Electric discover “an arc in the tube” of their
synchrotron accelerator
-  coined the term ‘synchrotron radiation’
-  1950: Hannes Alfven and Bernt
Herlofsen suggest that the mysterious
‘radio stars’ generate their radio
emission by synchrotron radiation
-  1950: Kiepenheuer suggests that the
galactic radio emission was due to
cosmic-ray electrons gyrating in the
galactic magnetic field producing
synchrotron radiation
-  Synchrotron radiation now known to be
one of dominant sources of radiation in
our universe
General Electric synchrotron accelerator
built in 1946, the origin of the discovery
of synchrotron radiation. The arrow
indicates the evidence of radiation
(source: Wikipedia)
- Radio emission (<30 GHz) from ‘normal’
galaxies
- Optical and X-ray continuum emission from
AGN
Radio image of the whole sky at 408 MHz
Credit: Haslam et al. 1982
- Non-thermal optical emission from the Crab
Nebula (and pulsar)
HST Image of the synchrotron jet of M87
Credit: NASA and The Hubble Heritage Team (STScI/AURA)
Crab pulsar
CHIMERA optical image of the Crab Nebula and pulsar
Synchrotron radiation of relativistic and
ultra-relativistic electrons is a dominant
process in high energy astrophysics
Interstellar medium:
10-6 Gauss
Crab Nebula:
10-3 Gauss
This room:
0.3 Gauss
Earth’s magnetic pole:
0.5 Gauss
Jupiter’s magnetic pole:
14 Gauss
Refrigerator magnet:
50 Gauss
Sunspot:
1500 Gauss
Brown Dwarf:
5000 Gauss
Strongest lab magnetic field
(without destroying the lab):
106 Gauss
White Dwarf:
up to 109 Gauss
Neutron Star:
up to 1015
Gauss-4
1 Gauss = 10 Tesla
-  Consider an electron moving with velocity
- The magnetic force is given by the Lorentz force
-  The magnetic force is perpendicular to the particle velocity
-  No power is transferred to the electron and its kinetic energy
(mev2/2) remains constant
-  Therefore
therefore
is constant.
remains constant. However,
is also constant,
- In a constant magnetic field, the electron moves along the magnetic
field line on a uniform helical path with constant linear and angular
speeds
-  In the inertial frame of the electron, the electron orbits in a circle
perpendicular to the magnetic field with angular velocity ω needed to
balance centripetal and magnetic forces
Electron cyclotron
Let
frequency
-  Example 1: Hercules X-1 (X-ray
binary)
-  Absorption feature at 34 KeV in X-ray
spectrum
-  Cyclotron absorption by hot gas near
poles of magnetized neutron star
-  Can be used to measure magnetic
field strength of the neutron star!