Electromagnetic Theory II
G. Franchetti, GSI
CERN Accelerator – School
2-14 / 10 / 2016
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Dielectrics
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Dielectrics and electric field
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E
electric dipole moment
s
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Polarization
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For homogeneous isotropic dielectrics
dielectric susceptibility
polarization number of electric dipole momentper volume
electric field “in” the dielectric
Example
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free ch
arges
bo
un
de
d ch
argesTotal Electric field
Field internal to the capacitor
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relative permettivity
Material
Vacuum 1
Mica 3-6
Glass 4.7
water 80
Calcium copper titanate
250000
Electric Displacement
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Field E0 depends only on free charges
We give a special name: electric displacement
first Maxwell equation
Free charges
Bounded current
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Suppose that Eis turned on in the time
The polarization changes with time
Bounded current
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The polarization changes with time
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Suppose that Eis turned on in the time
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Density of current due to bounded charges
It has to be included in the Maxwell equation
= number of dipole moments Per volume
It is already in the definition of
Single electric dipole moment
Magnetic field in matter
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As there are no magnetic charges
Magnetic phenomena are due to “currents”
Magnetic moment
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torque acting on the coil
Magnetic moment
Example
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The effect of the magnetic field is to create a torque on the coil
I
A
Magnetic moments in matters
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v
r
I
e-
Orbits of electrons
Intrinsic magnetic moments: ferromagnetism
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Spin of electrons
Without external magnetic field
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Random orientation (due to thermal motion)
Without external magnetic field
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dipoles moment of atoms orientates according to the external magnetic field
Magnetization
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B is the macroscopic magnetic field in the matter
This surface current produces the magnetic field produced by magnetized matter
sheet current
= magnetic susceptibility
Non uniform magnetization
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Mz Mz + DMz Mz + 2 DMz
Δy
DI DI DI
Δz
Non uniform magnetization
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Mx
Mx + DMx
Mx + 2 DMx
DI
DI
DI
Δy
Δz
Free currents and bounded currents
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The bounded currents are given by
This current should be included in Ampere’s Law
Define
Magnetic susceptibility
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this field depends on all free and bounded currents: NOT PRACTICAL
this field dependsonly on the currentthat I create
relative permeability
Material
Vacuum 0 1 4π × 10−7
water −8.0×10−6 0.999992 1.2566×10−6
Iron (pure) 5000 6.3×10−3
Superconductors -1 0 0
Maxwell equation in matter
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Summary of quantities
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Boundary conditions
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1
2
Gauss
Boundary conditions
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1
2
Stokes
Summary boundary conditions
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1
2
Waves
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At t=0
sin(x)
-1
-0.5
0
0.5
1
-10 -5 0 5 10
xwave number
Waves
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x
sin(x-3)
-1
-0.5
0
0.5
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-10 -5 0 5 10
At time t
the wave travels of
wavevelocity
Waves
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At fixed x
y oscillates with period
sin(x)
-1
-0.5
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0.5
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-10 -5 0 5 10
t
Wave equation
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The vector gives the direction of propagation of the wave
the velocity of propagation is
Electromagnetic waves
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Maxwell equations in vacuum
speed of light !!
Planar waves
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From 1st equation
The electric field is orthogonal to the direction of wave propagation
Starting ansatz
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From 3rd equation
This satisfy the 2nd equation, in fact
Integrating over time
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The 4th equation is satisfied too
Planar wave solution
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Sinusoidal example
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with
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0.5
1
-2 0 2 4 6 8 10
Poynting vector
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-1
-0.5
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0.5
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-2 0 2 4 6 8 10What is the flux of energy going through the surface A ?
A
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-1
-0.5
0
0.5
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-2 0 2 4 6 8 10
A
Electric field density energy
Magnetic field density energy
Energy through A in time Dt
Energy flux: Poynting vector
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Energy flux
But for EM wave
Poynting vector
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Interaction with conductors
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-1
-0.5
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0.5
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0 2.5 5 7.5 10 12.5 15
x
y
z
EM wave in a conducting media
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Similar relation is found for H
Ohm’s Law
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Starting ansatz
Wave propagation
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It depends on
If then
wave is un-damped
Bad conductor
if then
Good conductor
Skin depth
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-1
-0.5
0
0.5
1
0 2.5 5 7.5 10 12.5 15
x
y
z
(drawing for not ideal conductor)
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consider copper which has an electricalconductivity , and
f δ
60 Hz 8530 μm
1 MHz 66.1 μm
10 MHz 20.9 μm
100 MHz 6.6 μm
1 GHz 2.09 μm
Transmission, Reflection
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E0i
H0i
E0rH0r
MaterialVacuum
E0t
H0t
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At the interface between the two region the boundary condition are
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Perfect conductor
Perfect dielectric
Perfect dielectric Perfect conductor
Snell’s Law
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1
2
EM in dispersive matter
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s
Response to “external” electromagnetic field needs “time”
Electric field Magnetic field
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Wave velocity depends on The relation for k and
becomes more complicated becausev depends on omega
Waves at different frequencies travels with different velocity they “spread”
Usually a pulse of electromagnetic wave is composed by several waves of differentfrequency -
Phase velocity and group velocity
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A general wave can be decomposed in sum of harmonics
If is independent from the wave does not get “dispersed”
If is peaked around k0 then can be expanded around
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Fast wave Slow wave modulating thefast wave
Speed
Example
-1
-0.5
0
0.5
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-20 -10 0 10 20
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15 harmonics: each with different wave number, and wavelength
Wave packet
is the “group velocity”. The speed of the wave packet
-0.6
-0.4
-0.2
-0
0.2
0.4
0.6
-20 -10 0 10 20
vg
Phase velocity
Waveguides
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Walls: Perfect conductor
Inside the guide: Perfect dielectric
Boundary condition at the walls x
y
z
Ex(0,x,z) = 0Ex(x,a,z) = 0
Ey(0,y,z)=0 Ey(b,y,z)=0
a
b
In the perfect dielectric
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Maxwell equations Workingansatz
Dispersion relation
In the perfect dielectric
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Only E0z, and H0z are in the partial derivatives: special solutions
Transverse electric wave TE E0z = 0 Transverse magnetic wave TM H0z = 0
TE waves
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Equations
If you know H0z, then you know everything
These eqs. +
Automatically
Is satisfied
Boundary conditions: modes
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Ex(0,x,z) = 0Ex(x,a,z) = 0
Ey(0,y,z)=0 Ey(b,y,z)=0
Search for the solution
Boundary conditions
Cut-off frequency
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Only if k > 0 the wave can propagate without attenuation
Speed of wave
Cut-off frequency
Given the fix frequency of a wave, only a certain number of modes can exists in the waveguide
Dispersionrelation
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If a > b consider the TE10 mode
0
2
4
6
8
10
0 2 4 6 8 10
Forbidden region
cut-off
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If a > b consider the TE10 mode
0
2
4
6
8
10
0 2 4 6 8 10
cut-off
visually TE
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E0y
H0x
kH0z
kx
E0y
ky
E0x
H0z
Standing wave
Standing wave
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Cavity
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Walls: Perfect conductor
Inside the guide: Perfect dielectric
Boundary condition at the walls x
y
z
In every wall the tangent electric field Is zero
a
b
c
Cavity (rectangular)
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Standing wave
Standing wave
Boundary condition
Normal modes are only standing waves
Electromagnetic standing waves
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Dispersion relation
Final Observations
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Potential vector was here presented for static field
However one can also re-write the Maxwell equation in terms of the potential vector, and find electromagnetic wave of ”A”
Potential vector
Internal degree of freedom: Gauges
Electric potential
( A is defined not In unique way)
( V is not unique)
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Reference frame ?
I
v
FI
v
F
v
The particledoes not move..Is there a force F ?
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Reference frame ?
v
Maxwell equationstells me the speed is c
E0i
H0i
Maxwell equationstells me the speed is c ( but I move, mm)
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References:
(1) E. Purcell, Electricity and Magnetism (Harvard University)(2) R.P. Feynman, Feynman lectures on Physics, Vol2. (3) J.D. Jackson, Classical Electrodynamics (Wiley, 1998 ..) (2) L. Landau, E. Lifschitz, Klassische Feldtheorie, Vol2. (Harri Deutsch, 1997) (4) J. Slater, N. Frank, Electromagnetism, (McGraw-Hill, 1947, and Dover Books, 1970) (5) Previous CAS lectures