- 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 Where â1 and â2 are unit vectors; E0 & B0 are complex constants and k=kn is the wave vector and ω is the frequency.

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

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

Solutions:

Evaluated at the retarded time…

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

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.

define

-  From ϕ and A we can derive E and B

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.

Velocity field acceleration field

-  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’

General Electric synchrotron accelerator built in 1946, the origin of the discovery of synchrotron radiation. The arrow indicates the evidence of radiation (source: Wikipedia)

-  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

- Non-thermal optical emission from the Crab Nebula (and pulsar)

Synchrotron radiation of relativistic and ultra-relativistic electrons is a dominant process in high energy astrophysics

- Radio emission (<30 GHz) from ‘normal’ galaxies - Optical and X-ray continuum emission from

AGN

HST Image of the synchrotron jet of M87 Credit: NASA and The Hubble Heritage Team (STScI/AURA)

Radio image of the whole sky at 408 MHz Credit: Haslam et al. 1982

CHIMERA optical image of the Crab Nebula and pulsar

Crab pulsar

Interstellar medium:

Crab Nebula:

This room:

Earth’s magnetic pole:

Jupiter’s magnetic pole:

Refrigerator magnet:

Sunspot:

Brown Dwarf:

Strongest lab magnetic field (without destroying the lab):

White Dwarf:

Neutron Star:

10-6 Gauss

10-3 Gauss

0.3 Gauss

0.5 Gauss

14 Gauss

50 Gauss

1500 Gauss

5000 Gauss

106 Gauss

up to 109 Gauss

up to 1015 Gauss

1 Gauss = 10-4 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 remains constant. However, is also constant, therefore is 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 frequency

Let

-  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!