*58 for general multiples (2023)


We carried out electrical dipole, magnetic dipole and electrical radiation: quadrupol. What higher contributions? The techniques used so far in the really appropriate sand - - We have to be more systematic.Spherical vector harmonic, that appear in other areas, the dynamics of exposure fluids. We see that this Abelaw.jackson continues to find energy and angle impulse (Wedon does not do this here).Brown presented, theCommentThe"Jackson's algebra is more than one bit." These superior multipoles are not taken into account in Landaus Buch or Likharev Stonybrooks notes and they optionally opened. They were used more often in core physics than now.

*58 for general multiples (1)

However, for the highest multipoles we are interested in inspection fronts (but it is not symmetrically spherical!). In the distant radiation zone we will have waves that seem very similar to the waves of the family level.

It does not conclude that the mi , So, H The fields have no components to spread - - It is fair that these components are getting smaller and smaller (factor 1/Riñonal ) With regard to the transverse components during the wave, and these components play an important role in radiation of the angle impulse.

For more information on possible field configurations, we have to solve Maxwell's equations in spherical polar coordinates.

Scalar wave

As a heating exercise (which turns out to be very relevant), we start with the spherical wave equation for AclimbCampos Fi X, So,Y, So,z, So,T Before you switch to the most difficult vestocuations for electrical and magnetic fields

2 Fi X, So,Y, So,z, So,T - - 1 C 2 2 Fi X, So,Y, So,z, So,T T 2 =0

in spherical polar coordinates.

As a dependence on the usual time mi - -UeVayaT , So,

Fi X, So,Y, So,z, So,T =Fi X, So,Y, So,z mi - -UeVayaT , So,

The wave equation is

2 + + k 2 Fi X, So,Y, So,z =0, So,k=Vaya/C.

Of course, this is identical to the equation that we have solved for the guides, but now we will use Riñonal, So,º, So,ϕ Coordinates.

However, it should still be familiar: 2 In the spherical polar it is exactly the equation for the problem of hydrogen atom in the elementary quantum mechanics, which is dissolved by variable separation and the same trick works here: the standard ball extension

Fi X, So,Y, So,z = , So,Metro F Metro Riñonal Y Metro º, So,ϕ

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Where (remember) Y Metro º, So,ϕ = 2+ +1 4Pi - -Metro ! + +Metro ! Pag Metro ribº mi UeMetroϕ .

Radial wave equation: Bessel spherical functions

The radial function fulfills (regardless of Metro )

D 2 D Riñonal 2 + + 2 Riñonal D DRiñonal + + k 2 - - + +1 Riñonal 2 F Riñonal =0.

At that time Jackson replaces F Riñonal = Of Riñonal / Riñonal , So, To find that Of Riñonal The beetle equation fulfills for a total average value Norte=+ + 1 2 :

D 2 D Riñonal 2 + + 1 Riñonal D DRiñonal + + k 2 - - + + 1 2 2 Riñonal 2 Of Riñonal =0.

So Of Riñonal Fulfills the equation of Bessel - - except that It is replaced by + + 1 2 . Remember that the usual Bessel equation arises from the 2 Operator expressed incylindricalWe only derive coordinates the same equation, but with + + 1 2 von 2 EmsphericalCoordinates. The solutions for this new version are mentioned, not surprising thatsphericalFunctions of Bessel (Neumann, Hankel), written with small letters J Riñonal etc. and can be derived from the functions of Bessel (cylindrical) (Express series form) as follows:

J Riñonal = Pi/2Riñonal J + + 1 2 Riñonal , So, Norte Riñonal = Pi/2Riñonal Norte + + 1 2 Riñonal , So, H (1, So,2) Riñonal = Pi/2Riñonal J + + 1 2 Riñonal ±Ue Norte + + 1 2 Riñonal .

That means we can find J Riñonal The expansions of the infinite series of functions of the Bessel theory, for example

J Norte Riñonal = Riñonal 2 Norte J=0 - -1 J J!C J+ +Norte+ +1 Riñonal 2 2J

and use C 1 2 = Pi , So,C 3 2 = 1 2 Pi , So,C Norte = Norte- -1 C Norte- -1 We found:

J 0 Riñonal = sinRiñonal Riñonal , So, J 1 Riñonal = sinRiñonal Riñonal 2 - - ribRiñonal Riñonal , So,...

Surprise: You are much easier than the functions of Bessel's theory!

ÖThe spherical functions of Bessel are finite polynomesEm 1/Riñonal With coefficient sinRiñonal, So,ribRiñonal. The only justification to derive it, as Wejet did, is to show why they are called Bessel functionsremovedeasier to derive them by writing F Riñonal = Of Riñonal /Riñonal In the original differential equation as an unexpected text of the introductory quantum mechanics!In particular, I indicate complete treatment (based on the representation of Landau) in my quantum mechanicsHere.

Asymptotic for great Riñonal, So,

J Riñonal 1 Riñonal sin Riñonal- - Pi 2 , So, Norte Riñonal - - 1 Riñonal rib Riñonal- - Pi 2 , So, H 1 Riñonal - -Ue + +1 mi UeRiñonal Riñonal .

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That is, Hankel's spherical function H 1 Riñonal It is the function that corresponds to the starting waves. We are interested in the inertia, so we are looking for solutions in the form , So,Metro A Metro kRiñonal Y Metro º, So,ϕ . Ö Y Metro (Cube is expressly close to the beginning of this section) are the angle family of the angular pulse operator Ue = 1/Ue Riñonal × familiar with the quantum mechanics (of course without ) Essentially the gradient operator on the spherical surface and of course, of course, of course, Riñonal Ue =0.

That is,

Ue 2 Y Metro = + +1 Y Metro Ue + + Y Metro = - -Metro + +Metro+ +1 Y Metro+ +1 , So, Ue - - Y Metro = + +Metro - -Metro+ +1 Y Metro- -1 , So, Ue z Y Metro =Metro Y Metro .

Also, Ue × Ue =Ue Ue .

Multipolarer Formalismus

Maxwell's equations in Espaço Empio (over time mi - -UeVayaT , So,Vaya=Ck ),

× mi =Uek Z 0 H , So, × H =- -Uek mi / Z 0 mi =0, So, H =0

Follow that

2 + + k 2 mi =0, So, mi =0, So,

And the same equations keep it for H . These equations can be dissolved for the three in standard fashionCartesian X, So,Y, So,z Coordinates,ButWe are interested in spherical starting waves and there isTrivial separationof these vectors in spherical coordinates Riñonal, So,º, So,ϕ . How do we go on?

Jackson uses an intelligent solution from Bouwkamp and Casimir (1954): The equations for theclimbAmounts Riñonal mi , So, Riñonal H . This copies the standard approach for inwaveguides waves, whereby the approach should first solve the field component for spreading. mi z Ö H z .

In this spherical case,

2 Riñonal mi = Riñonal 2 mi + +2 mi , So,

and in the same way too H , So, Then from mi =0, So, H =0 , mi 2 mi =- - k 2 mi , So,

2 + + k 2 Riñonal mi =0, So, 2 + + k 2 Riñonal H =0.

So Riñonal mi EsA solution for the wave equation! (How is it Riñonal H . )

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The general starting solution is a series in spherical harmonious with Hankel functions that are accompanied.

Fi Riñonal = , So,Metro A Metro 1 H 1 kRiñonal + + A Metro 2 H 2 kRiñonal Y Metro º, So,ϕ .

To see how this works, we will select a certain multipolus.

Magnetic (and electrical) field fields

After Jackson we followdefineA multipolar magnetic field of , So,Metro For conditions

Riñonal H Metro METRO = + +1 k Gramm kRiñonal Y Metro º, So,ϕ , So, Riñonal mi Metro METRO =0, So,

(Then this is analogous to a wave guide mode) where Gramm kRiñonal = H 1 kRiñonal For the starting waves. We know that this is a solution to the wave equation, and we can visualize the component of the magnetic field, which points vertically on a spherical surface with an angle pattern Y Metro º, So,ϕ As a wave that stands in a spherical balloon (and for larger spherical surfaces that drops to a size like the inverter).

The electrical field is only tangential, but has the same standard: from Maxwell'Sefations,

Z 0 k Riñonal H =- -Ue Riñonal × mi =- -Ue Riñonal × mi = Ue mi .

Then with METRO Describes that this is a magnetic multipolus)

Ue mi Metro METRO Riñonal H Metro METRO Y Metro º, So,ϕ .

But when we expand it mi In spherical harmonious of the operators Ue will create other neighboring

So that cannot be correctunless mi º, So,ϕ It is a properties of Ue :

mi Metro METRO = Z 0 Gramm kRiñonal Ue Y Metro º, So,ϕ , So, H Metro METRO =- - Ue/k Z 0 × mi Metro METRO , So,

Then Ue mi Metro METRO Ue 2 Y Metro º, So,ϕ .

Remember that the magnetic field standard can determine

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Riñonal H Riñonal × mi Riñonal × Ue Y Metro = Riñonal × Ue Y Metro = Ue 2 Y Metro = + +1 Y Metro .

ÖelectricThe multipolar fields are defined in the same way, whereby the fields are replaced:

Riñonal mi Metro mi =- - Z 0 + +1 k F kRiñonal Y Metro º, So,ϕ , So, Riñonal H Metro mi =0.

The electrical multipolar fields are

H Metro mi = F kRiñonal Ue Y Metro º, So,ϕ , So, mi Metro mi = Ue Z 0 k × H Metro mi .

Spherical vector harmonic

It is useful to introduce a little more annotation, the spherical harmonious ones that are called:

X Metro º, So,ϕ = 1 + +1 Ue Y Metro º, So,ϕ .

These have simple orthogonality properties:

X ' Metro ' X Metro = D ' D Metro Metro ' , So, X ' Metro ' Riñonal × X Metro DOh=0.

At this point, Jackson states (but not proves) that these two types of waves form a complete set of vector solutions for Maxwell's equations at the origin of the origin. This is the general solution

H = Metro A mi , So,Metro F kRiñonal X Metro - - Ue k A METRO , So,Metro × Gramm kRiñonal X Metro mi = Z 0 Metro Ue k A mi , So,Metro × F kRiñonal X Metro + + A METRO , So,Metro Gramm kRiñonal X Metro

Wo A mi , So,Metro , So, A METRO , So,Metro These are the amplitudes of the electrical and magnetic multipolar fields. Fi Riñonal = , So,Metro A Metro 1 H 1 kRiñonal + + A Metro 2 H 2 kRiñonal Y Metro º, So,ϕ .

Coefficients can be determined if Riñonal H mi Riñonal mi You are known

A METRO , So,Metro = k + +1 Y Metro * Riñonal H DOh

A METRO , So,Metro Gramm kRiñonal = k + +1 Y Metro * Riñonal H DOh , So, Z 0 A mi , So,Metro F kRiñonal =- - k + +1 Y Metro * Riñonal mi DOh .

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With the result

In the rest of Chapter 9, Jackson uses the impressions of vonxes to find the energy and angle motif of the multipolar radiation, the strangers, the seeds and a linear antenna that is fed by the center. We will not appeal to this material.

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