We carried out electrical dipole, magnetic dipole and electrical radiation: quadrupol. What higher contributions? The techniques used so far in the really appropriate sandWe 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.
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 theThe fields have no components to spreadIt is fair that these components are getting smaller and smaller (factor) 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.
As a heating exercise (which turns out to be very relevant), we start with the spherical wave equation for AclimbCamposBefore you switch to the most difficult vestocuations for electrical and magnetic fields
in spherical polar coordinates.
As a dependence on the usual time
The wave equation is
Of course, this is identical to the equation that we have solved for the guides, but now we will useCoordinates.
However, it should still be familiar: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
Radial wave equation: Bessel spherical functions
The radial function fulfills (regardless of)
At that time Jackson replacesTo find thatThe beetle equation fulfills for a total average value
SoFulfills the equation of Besselexcept thatIt is replaced byRemember that the usual Bessel equation arises from theOperator expressed incylindricalWe only derive coordinates the same equation, but withvonEmsphericalCoordinates. The solutions for this new version are mentioned, not surprising thatsphericalFunctions of Bessel (Neumann, Hankel), written with small lettersetc. and can be derived from the functions of Bessel (cylindrical) (Express series form) as follows:
That means we can findThe expansions of the infinite series of functions of the Bessel theory, for example
and useWe found:
Surprise: You are much easier than the functions of Bessel's theory!
ÖThe spherical functions of Bessel are finite polynomesEmWith coefficientThe only justification to derive it, as Wejet did, is to show why they are called Bessel functionsremovedeasier to derive them by writingIn 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
That is, Hankel's spherical functionIt is the function that corresponds to the starting waves. We are interested in the inertia, so we are looking for solutions in the formÖ(Cube is expressly close to the beginning of this section) are the angle family of the angular pulse operatorfamiliar with the quantum mechanics (of course without) Essentially the gradient operator on the spherical surface and of course, of course, of course,
Maxwell's equations in Espaço Empio (over time),
And the same equations keep it forThese equations can be dissolved for the three in standard fashionCartesian Coordinates,ButWe are interested in spherical starting waves and there isTrivial separationof these vectors in spherical coordinatesHow do we go on?
Jackson uses an intelligent solution from Bouwkamp and Casimir (1954): The equations for theclimbAmountsThis copies the standard approach for inwaveguides waves, whereby the approach should first solve the field component for spreading.Ö
In this spherical case,
and in the same way tooThen from, mi
SoEsA solution for the wave equation! (How is it)
The general starting solution is a series in spherical harmonious with Hankel functions that are accompanied.
To see how this works, we will select a certain multipolus.
Magnetic (and electrical) field fields
After Jackson we followdefineA multipolar magnetic field ofFor conditions
(Then this is analogous to a wave guide mode) whereFor 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 patternAs 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,
Then withDescribes that this is a magnetic multipolus)
But when we expand itIn spherical harmonious of the operatorswill create other neighboring
So that cannot be correctunless It is a properties of:
Remember that the magnetic field standard can determine
ÖelectricThe multipolar fields are defined in the same way, whereby the fields are replaced:
The electrical multipolar fields are
Spherical vector harmonic
It is useful to introduce a little more annotation, the spherical harmonious ones that are called:
These have simple orthogonality properties:
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
WoThese are the amplitudes of the electrical and magnetic multipolar fields.
Coefficients can be determined ifmiYou are known
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.