ELECTROMAG THEORY 1
ELECTROMAG THEORY 1 PHY 6346
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Date Created: 09/18/15
Chapter 8 Bessel Functions ln cylindrical coordinates p7 1572 Laplace7s equation is 6ltIgt 1 62 62 V2lt1gt 81 p 3p p2 dab 322 This equation has separable solutions of the form ltIgt RpQ Zz The one dimensional equations in the 1572 coordinates are of the simplest type dZZ i 2 0 82 dzz 7 dZQ 2 W V Q 07 83 and all the complications are in the radial equation dZR 1 dB V2 kzi R0 84 dp2pdplt 92gt Putting z kp7 this becomes the standard Bessel equation dZR 1 dB V2 wgalt1i gt1 0 85 In this chapter we deal only with solutions in the full azimuthal domain 0 S b S 27139 Consequently the index V takes on integer values and corresponds precisely to the index m of the spherical harmonics However7 we use the notation V because non integer values will be encountered later When k 0 the solutions do not depend on 2 and the problem is in effect two dimensional We already know that in this case the radial functions are simply pquot and p for V 7E 07 and they are 1 and lnp for V 0 For k 7E 0 there are solutions that resemble sinkz and coskx7 known as the Bessel functions Jkp7 and solutions that resemble expz k and exp7z39kx7 known as the Han kel functions H 1kp and H 2kp The Hankel functions will be used later7 in radiation problems Thus all we need for now are the Bessel functions for integer m 63 81 Bessel functions of integer order The functions Jmkp are the analogues of the associated Legendre functions Pl cos t97 except that k is a continuous index unlike l7 if the domain of de nition is 0 S p S 00 As in the case of Pfquot and me7 negative values of m give nothing new and we can choose Jmkp 71mekp The functions Jm have series expansions and recursion relations given later7 but there is no Rodrigues formula for them7 since they are not reducible to polynomials lnstead7 there is the very useful integral representation 1 27 d Jmx exp in cos p 7 imltp i7 86 Zm 0 27139 that can be written in many other ways7 including 7 d Jmx cos x cos p 7 imp 87 0 7r Closely related to these formulas through Fourier7s theorem is the generating function expansion expikp cos p 2 im eimWJmkp 88 The useful lore about Bessel functions includes a large number of special integrals that we will discuss as needed We note here only the orthogonality and normalization integral over the domain 07 oo 0 JmltkpgtJmltk pgt pdp i 6a e la 89 Successive Jm functions have interlacing zeros and maxima7 like the Legendre func tions It is important to remember that Jmx behaves like 1 for small 7 so it is regular at the origin For a given in2 there is also the second solution77 of the Bessel differential equation7 which for small a behaves like quot for m gt 0 and like mm for m 0 Its standard form is known as the Neumann function7 or the Bessel function of the second kind7 and is denoted with Nm by some authors7 Ym by others It is discussed further here below It is used in problems where p 0 is not part of the domain We close this section by writing down the general interior solutions of the Laplace equations in cylindrical polar coordinates They are7 for k gt 07 eikz eii W Jmkp7 810 and represent standing waves in the radial direction7 evanescent waves in the z direction If instead of the Laplace equation we had solved the Helmholtz equation7 V21 7wCZi7 the solutions would be I I eilkzz eimw Jm kpp 811 with k kg LuC 82 General Bessel and Hankel functions Many properties of the Bessel functions hold for any value of the index V and are sometimes easier to discuss for non integer V 0 Series expansions For a given V27 the two solutions W i 83912 n0 JVz 7V i 2n 813 n0 are linearly independent7 except when V is an integer note that the P function of a negative integer7 or zero7 is in nite To avoid this dif culty7 one de nes the Neumann function J J Aftz Ax COS VTF 7 1x7 814 Sin V which remains nite in the limit V a integer o Hankel functions are de ned as H 1gtz Jyx mm 815 H 2gtx Jl i NV 816 Compare expiz z cos x i z sin x o Recursion relations that hold for JV as well as for NV7 H197 H52 are 2 JV1 JV1 11V 817 x d JV 71 2 V 818 1 1 d9 lt gt They are useful to compute integrals7 but are numerically unstable as step up relations 0 The behavior for small x can be read from the series see J389 90 Note however the special case 2 z N0z a ln 5 ygt7 819 where y 2 05772 is the Euler Mascheroni constant 0 The asymptotic behavior for m gtgt V is Jlx N NV1z N 0048 7 g i E 820 65 0 Of special interest for elds con ned inside a conducting cylinder are the zeroes of They are tabulated by Jackson The orthogonality and normalization relations over a nite domain are also given These relations are of interest to people who play round drurns They will also be useful to discuss the propagation of waves in cylindrical bers and waveguides when we get to it 83 Modi ed Bessel functions If we are going to make virtual photons out of the solutions of the wave equation aim aim Jmkpp7 821 all we have to require in the zero frequency limit is that k k 0 Clearly7 in addition to the option kz z k7 kp k7 we have the option kz k7 kp z k7 which gives the electrostatic solutions gill aim Imkp7 822 where7 by de nition7 Im i meQw 823 is a modi ed Bessel function 0x resernbles cosh 7 1 resernbles sinh x Every Im is well behaved at the origin7 but not at in nity thus these functions are to be used for interior problems The standard second solution is the modi ed Hankel function Kmm gWHH m 824 This blows up at the origin and vanishes at in nity thus it is to be used for exterior problems The above de nitions are valid for any V7 not just for integer values m The prefactors are chosen to make Iva and KVQ real for real V and x Chapter 9 Electric Multipoles In many cases we are interested in the exterior electric eld of a charge distribution px that is contained in a sphere of radius Pi centered at x 0 If the charge distribution is spherically symmetric the potential ltIgt for r gt P is just q47T60r where q px d 91 is the total charge Quite generally it is 1 PX 3 ltigt d 92 X 47TEO lx 7 x l z Using the expansion 1 47139 l Y 6 7 93 Xixq g2l1r l 7 Tl1 7 we obtain 4 Y 6 5 7T lm 7 ltlgt m 94 x gm T lt gt where the spherical multipole moments qlm are by de nition qlm pX TlYlm67 dSz39 For 7 gtgt Pi this multipolar expansion is very good and often only the rst non vanishing term is kept It is also used for nuclei atoms and molecules where the charge falls off exponentially In this case the multipolar expansion is only an asymptotic approximation because it handles incorrectly the charge outside the sphere of radius 7 However it is often an excellent approximation making a percentage error exp7ra where a is of the order of 10 8 cm for atoms The multipolar expansion in spherical harmonics is convenient for formal manipula tions but the moments qlm as de ned by Jackson carry unwieldy normalization coef cients 67 that are hard to remember When only dipole and quadrupole moments are kept it is more convenient to use the Taylor expansion 1 W 1 1 1 1 where 3 ddxi for short lnserted into Coulomb7s integral this gives ltIgtx eo g m Qu 97 where p px d 98 are the Cartesian components of the dipole moment p and Q 7 26 px dgx 99 are the Cartesian components of the quadrupole moment tensor Note that the quadrupole tensor is not the second moment tensor MM fxgx jpb d but only the traceless part of it apart from the factor of 3 that is for convenience Any tensor T has trace T ZiTZi and can be written as the sum of an isotropic tensor T lj and a traceless tensor Ti 7 T lj The simple reason why we can discard the isotropic part of M without altering ltIgtx is that 25765371 vzl 0 for r gt Pi 910 T T The deeper reason has to do with the symmetries of the multipole elds as we now discuss 91 Tensors and multipoles We see that are the components of a vector are components of a second rank tensor and so on The connection between and the interior dipole elds77 T Y1mt9 15 is obvious and was discussed in Lecture 7 The connection between and the interior quadrupole elds77 r 212m0 15 was also given there but is a little subtler because there are 6 compo nents of x and only 5 harmonics Yme However the combination 2 7 does not 1 appear in the Taylor expansion of 1 lx 7 x l and we can write 1 1 1 1 1 1 67 zlz 7 r 26gt 36 911 lx7x l r lr2lt17 3 W 17r ln tensorial jargon 2 is the trace of the tensor and 7 26 is its traceless part Just as r 212m6 15 corresponds to the traceless part of so in general the 2l1 quantities r lYlm6 correspond to the traceless part of the rank l tensor xgls x il 68 The general symmetric tensor of rank l has E i l1l22 components Till2w7 but the zero trace requirement imposes the conditions Eh Till aw 0 the number of distinct pairs chosen from Z objects is lll 7 2121 ll 7 127 which is the number of conditions imposed by haVing zero trace One can check that the numbers are right z1z2 4171 21 1 912 92 Multipolar expansion of energy and forces Not only the eld external to a charge distribution7 but also the forces acting on a charge distribution can be expressed in terms of its multipoles Consider a distribution localized near x0 and write its density at the point x as px 7 x0 Expand the external potential acting on it as 1 X0X x0 6iltlgtx0 Z 7 grad 6i 37 X0 7 913 where we have played the usual trick of keeping only the traceless part lnserting in the expression for the energy Wx0 ltlgtxpx 7 x0 dgx x0 x px dgx 914 we obtain 1 which can also be written 1 WltX0gt qqX0 7 p 39 EltX0gt 7 E EjltX0gt 39 39 39 ii In words7 the total charge couples to the potential7 the dipole moment to the eld7 the quadrupole moment to the eld gradient7 and so on Differentiating with respect to x07 we nd that the force acting on the distribution has components 1 FkX0 qakwxo Zpi aiakqxo 6 2 Q1761 ajakqxo 39 39 397 917 139 ij or7 in vector notation7 1 i ij 93 Reduction of multipolar components The dipole p like any vector has a magnitude and a direction By a suitable rotation of coordinates the direction can be chosen to be along the z axis for instance Thus essentially the magnitude of the dipole moment is all we need to know about the shape of the charge distribution at this level just one number The tensor Q like any symmetric tensor has principal axes yz which can be found by the usual methods The corresponding principal values are QM ny Q Because Q is traceless Qm ny Q 0 Thus for instance Q and Q11 7 ny are all we need to know about the shape of the charge distribution at this level just two numbers Often the distribution is suf ciently symmetrical about the z axis to make Qm ny Then all we need to know is Q which is simply called the quadrupole moment The other principal values Qm and ny are each equal to i sz
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