Advanced EM - Master in Physics 2011-2012 1 Let us now investigate the next term in the series expansion. Multipole expansion of radiation fields: the

Advanced EM - Master in Physics 2011-2012

1

Let us now investigate the next term in the series expansion.

Multipole expansion of radiation fields: the second term

V

dVikcr

tkrie')')('(

)(rnrjA2

With a long calculation (see Schwartz, and shown in next page)

it is found that A2 can be written as the sum of two terms:

2q2m2 AArA t,called, respectively

Potential of MAGNETIC DIPOLE ( A2m ) “ “ ELECTRIC QUADRUPOLE ( A2q )

The “magnetic dipole” term of the potential turns out to be:

02m μnjrnA

rikdV

rc

ik tkritkri eeV

)()('''

2

where the magnetic moment of the current distribution is defined as:

'''2

1dV

c V jrμ0

Once A2m is known (i.e. once the magnetic moment is known)

the fields are computed by taking its curl and, of its curl,

considering only the terms in 1/r, which are the radiation field. Being it the field of a dipole, one can expect an angular

dependence of the fields similar to that of the electric dipole |S| ~ (sinθ)2

we recognize here the r’ of the second term in the expansion


2

To see how the first term (the magnetic quadrupole) comes about, we use the vector formula: )()()( baccabcba

which we turn around: )()()( baccbacab

And apply to the previous formula:

V

dVikcr

tkrie')')('(

)(rnrjA2

We write the function to be integrated as

)](')'([2

1)]'([

2

1

)](')'([2

1)](')'([

2

1)'(

njrrnjrjn

njrrnjnjrrnjrnj

Now the formula for A2 can be written as the sum of two

terms, of which the first is, as was written in the previous page, the magnetic dipole and the second is the electric quadrupole.

V

dVcr

iktkrie

'2

)'(')( rjrnA2m

Note that we have taken the vector n out of the integral since

it does not vary with r’ (in first approximation), and inverted

the order of r’ and j’ in the integral which caused the change of sign in the formulas.

V

dVcr

iktkrie

'2

)](')'([)( njrrnjA2e


3

Example: A CIRCULAR CURRENT LOOP

A loop of radius “d” with a current tie 0II

has a magnetic moment

c

dI

4

20

0

What is found – in much a similar way to how we found the fields for the electric dipole – is:

2m2m

2m

BnE

μB

)(2

tkrier

k

Which is very similar , but not equal to what we found for the electric dipole. In this case it is the magnetic field which is equal to the dipole component orthogonal to the observation direction (i.e. in the plane containing the line of sight and the dipole).

The average irradiated power is computed as for the electric dipole. One finds:

c

kdIW m 4

)( 420

2 To be compared with

c

kdIW

12

)( 20

1

At equal size d of the systems, the power irradiated by a magnetic dipole is a factor (kd )2 smaller that that of the electric dipole. It was to be expected, since the magnetic dipole comes from the second term in the series expansion.


4

The electric quadrupole

The second term of A2, called A2q, is the electric quadrupole. The

fields are products of a tensor (the quadrupole moment of the

charge distribution) times the unitary vector of the line of observation.

)(6

')'(]')'('3[6

)(

)(

3

'

23

nn

rnrnrnBV

2q

Qr

ik

dVrr

ik

tkri

tkri

e

e

where the second equation contains the definition of the quadrupole tensor of the charge density ρ (electric quadrupole). The derivation of this expression from the formula given in page 2 is long but not difficult, and is given in the book by Schwartz, pp. 283-284

The quadrupole tensor of the charge density is a 9-component quantity. In the case – that we examined when we discussed the multipoles in electrostatics – of a system symmetric for rotations around an axis, the quadrupole tensor takes the diagonal form

0

0

0

200

00

00

Q

Q

Q

Q

if we choose the symmetry axis of the system as the third (“z”) axis. is the quadrupole moment of the charge

distribution, that we had called B2. Note that the system is

symmetric for exchange “z” to “-z”. And as a consequence the

P2(cosθ) is – like all even polynomials – an even function of

cosθ.

0Q

2d

+q

+q

-2q


5

Let us now examine the term to be integrated, i.e.

)'(]')'('3[ 2 rnrnr r

in order to show that it is the product of a tensor times n. The product term is (has to be) a vector. Its components are linear

functions of the three components of vector n, and quadratic

functions of the components of r’ (x’,y’,z’). The vector

properties of this functions do not depend on ρ, so we study only the first part (between square brackets) of this formula.

]3)3()3([

]3)3()3([

]3)3()3([

])(3[

])(3[

])(3[

])(3[

2

2

2

2

)rz(yzxzn

yz)(ryxyn

xz)(xyrxn

rnznynxnz

rnznynxny

rnznynxnx

r

22z

22y

22x

zzyx

yzyx

xzyx

kji

kji

kji

k

j

i

nrnr

This last formula shows that the term under the integral is linear in

the components of n. The quantities which multiply them are the 9

integrals of products of the components of r’. A quantity that, multiplied by a vector gives another vector is a tensor. Therefore the 9 integrals are the components of such tensor.


6

The irradiated power has an angular dependence of the type

2)sin(cos~ 2qd

dW

while the total emitted power –for equal current, system size and oscillation frequency- is of the same order of magnitude as the magnetic dipole.

CONCLUSIONS

The vector potential in the case of a charge and current distributions oscillating harmonically in time, with a wavelength of the generated radiation much longer than the physical dimensions of the system, can be expanded in a series of multipolar radiation fields, of emitted power decreasing as

powers of (d/λ)2.

Each term of the series is characterized by the angular distribution of the emitted power, and the amplitude of each

multipole is determined by the multipole expansion of the ρ

and j.


7

Interaction of radiation with matter ( i.e. with charges)

Let the radiation be a plane wave, with its frequency, its energy density and its Poynting vector. In its path there is a free electron at rest. What happens?

The wave has its transverse electric field E; the electron will be accelerated by the Lorentz force

m

e

m

EFa

And, being accelerated, but with speed near zero, will emit radiation in the form of spherical wave. The total power is given by the Larmor formula.

32

24

3

22

3

2

3

2

cm

Ee

c

aeW

On the other hand the incident radiation has its Poynting’s

vector S which is its energy flow.

The ratio

ThomsonS

2042

0

4

2

32

24

3

8

3

8

4

3

2

rcm

e

Eccm

EeW

has the dimensions of a surface, and is called the

THOMSON CROSS SECTION


8

The absorption and re-emission of radiation are condensed in one parameter, called the Thomson cross section: It is as though in the way

of the radiation was an absorbing disk, of area σ, that absorbs the incident radiation and reemits it with its own angular distribution. The

radius r0 of that virtual disk is called the ELECTRON CLASSICAL RADIUS.

Its value is fermicm

cm

er

e

82.21082.2 132

2

0

The next step – in the study of the interaction of radiation with matter – is the study of absorption and re-emission of radiation by the electrons orbiting around a nucleus.

To calculate the interaction of radiation with matter we will employ a very simplified model of matter; more precisely, a simple model of how the charged particles –in practice, only the electrons- which constitute matter act under the force exerted on them by the radiation. In this model only the electrons are considered because their mass is ~2000 times smaller than that of the protons; and in the Thomson cross section the mass is in the denominator of the formula for the acceleration. In most materials – the exception being the conductors – the electrons are not free, but bound to the nuclei. In the model it is this binding of electrons that is described in an approximate way: they are considered to be in a sort of equilibrium position, with an elastic force to bring them back to it. rrF 2

0mk where the resonant frequency of the electrons has been inserted explicitly.

A remark on the expression classical electron radius.With these words we do not mean to say that the electron is a uniformly charged sphere, of radius 2.82 fermi. This expression is derived from classical theories (before the advent of quantum mechanics) in which the mass of the electron was interpreted as due to its electrostatic energy. All things that required the electron to have a finite volume. While now the electron size is known to be no larger than 10-16cm.


9

This approximation can seem incongruous if applied in a situation dealing with atoms and bound electrons, realm of quantum mechanics. But, if we apply it to an em radiation of wavelength of the order of visible light or longer, we are talking of wavelengths three orders of magnitude larger than the atom size. The atom sees basically a uniform electric field, rising and decreasing so slowly that it can be considered constant. The electron displacement we are discussing here can be interpreted -now that we know about quantum mechanics - as the displacement of the whole orbit under the effect of a very slowly varying external electric field. This model and its results are usually referred to as the classical electron theory of matter.

We shall now study in more detail the scattering of radiation by these electrons. The incident radiation will be a simple linearly polarized plane

wave, of angular frequency ω, and electric field E [E0cosωt, 0,0] . The

electron’s equation of motion is:

We have added to the forces in equilibrium also a dissipative force to keep the equation still finite at and near a resonant frequency. The solution of that differential equation is a

driven oscillation with angular frequency ω and amplitude:

dt

dx

m

)(

202

02

2titi ee

m

eEx

dt

dx

mdt

xd

)cos()( 22222

0

0

tm

eEx

Where the phase φ is: 220

)tan(

NOW, we have the electron’s equation of motion ;

•We derive it twice wrt time and have the acceleration

•We replace the acceleration in Larmor’s formula, average it over an oscillation period and obtain the total irradiated power

•We divide that power by the power per unit area of the incident plane wave and obtain the scattering cross section.


10

])[(22222

0

42

0.)(3

8

r

Wradscatt

inS

Remark how this formula becomes equal to that for the free electron (Thomson cross section) in case the electron is free

(γ=ω0=0). Remark also that a consequence of the electrons being

bound is that the cross section now depends on the driving frequency (it was independent for a free electron). Now, in the classical electron theory of matter the radiation wave which propagates through matter is the SUM (for the principle of superposition) of the incident plane wave PLUS the wave radiated by the electrons in their driven oscillatory motion.


11

Radiation propagation in matter:

the case of a transparent foil

One of the main concepts (principles) in electromagnetism is the principle of superposition. It states that fields are the vector sum of all the fields generated by all charges, near or far away. And, another principle (of relativity, this one) is that the velocity of light in vacuum is “c”.

There are no problems with these principles in vacuum; there are however in matter:

•In transparent materials, the light travels at a different speed: c/n, where n is the index of refraction, which also turns out to be a function of the light frequency.

•In most materials, which are not transparent, the light does simply not travel, i.e. gets absorbed. Not only; in some materials (conductors), it even is reflected.

Now, how can all this be compatible with the previous principles? After all, matter consists of many charges in vacuum.

To these questions gives an answer the classical electron theory.

We have already seen the basic interaction of radiation – which

we have taken to be a simple harmonic function, i.e. a sin(ωt) - with a free electron and with an elastically bound electron. The next step is to consider matter as a dense –to the extent that it may be treated as a continuous charge distribution – a dense collection of charges, be they light electrons (bound or, in same cases, free) or heavy nuclei, whose motion under the effect of a field is however very slow owing to their large mass and can be neglected.


12

To study the interaction of light with dense matter we shall study a well defined physical system, but greatly simplified.

•We consider radiation to impinge at right angle on a thin sheet (foil) of material, causing the electrons in it to oscillate and as a consequence to generate a radiation.

•The incident radiation we consider is a plane wave. Since it hits the sheet at right angle, all the electrons in the sheet see the same value for the fields at any given time.

•The charges, although discrete, are so many and so close and evenly distributed that they can be treated as a continuous charge density (NB: this condition may not be applicable to gases. Example: the high atmosphere).

•The material considered is in the form of a thin sheet. How thin? And… why is it thin? The answers: the first is is that the electrons in the matter are subject to an oscillating field with a finite wavelength. Which field? The field incident on the material plus the field generated by all other electrons; and at this point things get complicated: better to start with a simple system, a very thin sheet, so that we do not have to consider the effect of the other electrons. The second answer is that all the electrons in the sheet

must be subject (for simplicity) to the same field. Then Δ<<λ.


13

The incident radiation is a linearly polarized plane wave, with its electric field aligned along the “x” axis, while the wave propagates along the “z” axis. The sheet contains a charge surface density of

N·e·Δ, where N is the number of electrons per unit of volume. All

electrons are subjected to the incident field Es=iE0exp(iωt) (where i is the unitary vector along the “x” axis, and Es is the “source incoming radiation).

REMARK that at this point we spend the request that the sheet of material be very thin: we approximate the real field acting on the charges with the incident field only.

Under the influence of the electric field the charges will all move

-with equations given in the previous pages for free and bound electrons

-all electrons will move coherently (i.e. in phase).

As a consequence, all electrons will have the same acceleration at the same time and will irradiate the same electric field directed, for symmetry reasons of course, along the “x” axis.


14

The total electric field which emerges forward will be directed

along the “x” axis and be the sum of Es and Ef (generated by the

foil of material).

Now… WHAT DO WE EXPECT –from 1st grade physics-TO HAPPEN WHEN THE FOIL IS MADE OF A TRANSPARENT MATERIAL?

The light which has gone through a glass foil of refraction index

“n” and thickness Δ is retarded –wrt the incident wave- of the

phase δφ )1( nk(it still oscillates with angular frequency ω, but through the foil

with speed c/n). Therefore, if we take

1. the incident wave at the foil entrance to be ~ exp(iωt)

2. the incident wave (unaltered by the material) at the foil exit is

~ exp(-i[kΔ-ωt])

3. the wave measured at the exit of the foil is exp(-i[nkΔ-ωt])=

])[exp()]1[exp(

)]1[exp(

tkikni

tiikkni

i.e., we expect –from experimental classical optics- the wave at

the foil’s exit to be multiplied by a phase factor exp(-i[n-1]kΔ).

On the other hand, we know that Eout=ES+EL: the sum of two

fields, not the product of the incoming wave with a phase factor. But,the refracted wave can also be written as a sum (Taylor series expansion of the phase factor):

knikni )1(1]1[exp( And the exiting wave becomes:

ssout EkniEE )1(


15

Real

Im

-ikΔESEout

ssout EkniEE )1(

Which is what optics tells us. We therefore expect – from the experimental knowledge of the effect of the passage of light through a thin layer of transparent foil of matter- that the wave at its exit will be the incident wave slightly delayed in phase or –whichamounts to the same- a wave that has passed through the foil unaffected plus a small wave retarded in phase by 90 degrees.

Let us now see what our approximated model calculates! We already know the motion of the charges, i.e. the motion of the whole sheet of charges:

im

eEAAx tie

)(

122

0

ES

Documents

Advanced EM - Master in Physics 2011-2012 1 Let us now investigate the next term in the series expansion. Multipole expansion of radiation fields: the