Electromagnetic Theory II · Electromagnetic Theory II G. Franchetti, GSI CERN Accelerator...

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

-

-

-

-

+

+

+

+

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

1

-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

0

0.5

1

-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

-1

-0.5

0

0.5

1

-2 0 2 4 6 8 10

Poynting vector

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-1

-0.5

0

0.5

1

-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

1

-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

0

0.5

1

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

1

-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