Electromagnetism

1
Electromagnetism

• Christopher R Prior

ASTeC Intense Beams Group Rutherford Appleton
Laboratory
Fellow and Tutor in Mathematics Trinity College,
Oxford
2
Contents

• Review of Maxwells equations and Lorentz Force
  Law
• Motion of a charged particle under constant
  Electromagnetic fields
• Relativistic transformations of fields
• Electromagnetic energy conservation
• Electromagnetic waves
• Waves in vacuo
• Waves in conducting medium
• Waves in a uniform conducting guide
• Simple example TE01 mode
• Propagation constant, cut-off frequency
• Group velocity, phase velocity
• Illustrations

3
Reading

• J.D. Jackson Classical Electrodynamics
• H.D. Young and R.A. Freedman University Physics
  (with Modern Physics)
• P.C. Clemmow Electromagnetic Theory
• Feynmann Lectures on Physics
• W.K.H. Panofsky and M.N. Phillips Classical
  Electricity and Magnetism
• G.L. Pollack and D.R. Stump Electromagnetism

4
Basic Equations from Vector Calculus
Gradient is normal to surfaces ?constant
5
Basic Vector Calculus
6
What is Electromagnetism?

• The study of Maxwells equations, devised in 1863
  to represent the relationships between electric
  and magnetic fields in the presence of electric
  charges and currents, whether steady or rapidly
  fluctuating, in a vacuum or in matter.
• The equations represent one of the most elegant
  and concise way to describe the fundamentals of
  electricity and magnetism. They pull together in
  a consistent way earlier results known from the
  work of Gauss, Faraday, Ampère, Biot, Savart and
  others.
• Remarkably, Maxwells equations are perfectly
  consistent with the transformations of special
  relativity.

7
Maxwells Equations

• Relate Electric and Magnetic fields generated by
  charge and current distributions.

E electric field D electric displacement H
magnetic field B magnetic flux density ?
charge density j current density ?0
(permeability of free space) 4? 10-7 ?0
(permittivity of free space) 8.854 10-12 c
(speed of light) 2.99792458 108 m/s
8
Maxwells 1st Equation
Equivalent to Gauss Flux Theorem The flux of
electric field out of a closed region is
proportional to the total electric charge Q
enclosed within the surface. A point charge q
generates an electric field
Area integral gives a measure of the net charge
enclosed divergence of the electric field gives
the density of the sources.
9
Maxwells 2nd Equation
Gauss law for magnetism The net magnetic
flux out of any closed surface is zero. Surround
a magnetic dipole with a closed surface. The
magnetic flux directed inward towards the south
pole will equal the flux outward from the north
pole. If there were a magnetic monopole source,
this would give a non-zero integral.
Gauss law for magnetism is then a statement
that There are no magnetic monopoles
10
Maxwells 3rd Equation
Equivalent to Faradays Law of Induction (fo
r a fixed circuit C) The electromotive force
round a circuit          is proportional
to the rate of change of flux of magnetic field,
             through the circuit.
Faradays Law is the basis for electric
generators. It also forms the basis for inductors
and transformers.
11
Maxwells 4th Equation
Originates from Ampères (Circuital) Law
Satisfied by the field for a steady line
current (Biot-Savart Law, 1820)
12
Need for Displacement Current

• Faraday vary B-field, generate E-field
• Maxwell varying E-field should then produce a
  B-field, but not covered by Ampères Law.

13
Consistency with Charge Conservation

• Charge conservation
• Total current flowing out of a region equals the
  rate of decrease of charge within the volume.

• From Maxwells equations Take
  divergence of (modified) Ampères equation

Charge conservation is implicit in Maxwells
Equations
14
Maxwells Equations in Vacuum

• In vacuum
• Source-free equations
• Source equations

• Equivalent integral forms (useful for simple
  geometries)

15
Example Calculate E from B
16
Lorentz Force Law

• Supplement to Maxwells equations, gives force on
  a charged particle moving in an electromagnetic
  field
• For continuous distributions, have a force
  density
• Relativistic equation of motion
• 4-vector form
• 3-vector component

17
Motion of charged particles in constant magnetic
fields
No acceleration with a magnetic field
18
Motion in constant magnetic field
Constant magnetic field gives uniform spiral
about B with constant energy.
19
Motion in constant Electric Field

• Solution of

is
Constant E-field gives uniform acceleration in
straight line
20
Relativistic Transformations of E and B

• According to observer O in frame F, particle has
  velocity v, fields are E and B and Lorentz force
  is
• In Frame F?, particle is at rest and force is
• Assume measurements give same charge and force,
  so
• Point charge q at rest in F
• See a current in F?, giving a field
• Suggests

Rough idea
21
Potentials

• Magnetic vector potential
• Electric scalar potential
• Lorentz Gauge

Use freedom to set
22
Electromagnetic 4-Vectors
23
Relativistic Transformations

• 4-potential vector
• Lorentz transformation
• Fields

24
Example Electromagnetic Field of a Single
Particle

• Charged particle moving along x-axis of Frame F
• P has
• In F?, fields are only electrostatic (B0), given
  by

Observer P
Origins coincide at tt?0
25

• Transform to laboratory frame F

26
Electromagnetic Energy

• Rate of doing work on unit volume of a system is
• Substitute for j from Maxwells equations and
  re-arrange into the form

Poynting vector
27
Integrated over a volume, have energy
conservation law rate of doing work on system
equals rate of increase of stored electromagnetic
energy rate of energy flow across boundary.
electric magnetic energy densities of the fields
Poynting vector gives flux of e/m energy across
boundaries
28
Review of Waves

• 1D wave equation is with
  general solution
• Simple plane wave

29
Phase and group velocities
Superposition of plane waves. While shape is
relatively undistorted, pulse travels with the
group velocity
30
Wave packet structure

• Phase velocities of individual plane waves making
  up the wave packet are different,
• The wave packet will then disperse with time

31
Electromagnetic waves

• Maxwells equations predict the existence of
  electromagnetic waves, later discovered by Hertz.
• No charges, no currents

32
Nature of Electromagnetic Waves

• A general plane wave with angular frequency ?
  travelling in the direction of the wave vector k
  has the form
• Phase 2? ? number of waves and
  so is a Lorentz invariant.
• Apply Maxwells equations

33
Plane Electromagnetic Wave
34
Plane Electromagnetic Waves
Reminder The fact that is
an invariant tells us that is a
Lorentz 4-vector, the 4-Frequency vector. Deduce
frequency transforms as
35
Waves in a Conducting Medium

• (Ohms Law) For a medium of conductivity ?,
  
• Modified Maxwell                                 
                     
• Put

36
Attenuation in a Good Conductor
copper.mov water.mov
37
Charge Density in a Conducting Material

• Inside a conductor (Ohms law)
• Continuity equation is
• Solution is

So charge density decays exponentially with time.
For a very good conductor, charges flow instantly
to the surface to form a surface charge density
and (for time varying fields) a surface current.
Inside a perfect conductor (???) EH0
38
Maxwells Equations in a Uniform Perfectly
Conducting Guide
Hollow metallic cylinder with perfectly
conducting boundary surfaces
Maxwells equations with time dependence exp(iwt)
are
g is the propagation constant
Can solve for the fields completely in terms of
Ez and Hz
39
Special cases

• Transverse magnetic (TM modes)
• Hz0 everywhere, Ez0 on cylindrical boundary
• Transverse electric (TE modes)
• Ez0 everywhere, on cylindrical
  boundary
• Transverse electromagnetic (TEM modes)
• EzHz0 everywhere
• requires

40
A simple model Parallel Plate Waveguide
Transport between two infinite conducting plates
(TE01 mode)
41
Cut-off frequency, wc

• wltwc gives real solution for g, so attenuation
  only. No wave propagates cut-off modes.
• wgtwc gives purely imaginary solution for g, and a
  wave propagates without attenuation.
• For a given frequency w only a finite number of
  modes can propagate.

For given frequency, convenient to choose a s.t.
only n1 mode occurs.
42
Propagated Electromagnetic Fields

• From

43
Phase and group velocities in the simple wave
guide
44
Calculation of Wave Properties

• If a3 cm, cut-off frequency of lowest order mode
  is
• At 7 GHz, only the n1 mode propagates and

45
Waveguide animations

• TE1 mode above cut-off ppwg_1-1.mov
• TE1 mode, smaller ? ppwg_1-2.mov
• TE1 mode at cut-off ppwg_1-3.mov
• TE1 mode below cut-off ppwg_1-4.mov
• TE1 mode, variable ? ppwg_1_vf.mov
• TE2 mode above cut-off ppwg_2-1.mov
• TE2 mode, smaller ppwg_2-2.mov
• TE2 mode at cut-off ppwg_2-3.mov
• TE2 mode below cut-off ppwg_2-4.mov

46
Flow of EM energy along the simple guide
Fields (wgtwc) are
Time-averaged energy
47
Poynting Vector
Electromagnetic energy is transported down the
waveguide with the group velocity