CHAPTE DIPOLE AND QUADUPOLE MOMENTS. Intoducton τ p E FIGUE III. Consde body whch s on the whole electclly neutl, but n whch thee s septon of chge such tht thee s moe postve chge t one end nd moe negtve chge t the othe. Such body s n electc dpole. Povded tht the body s whole s electclly neutl, t wll expeence no foce f t s plced n unfom extenl electc feld, but t wll (unless vey fotutously oented) expeence toque. The mgntude of the toque depends on ts oentton wth espect to the feld, nd thee wll be two (opposte) dectons n whch the toque s mxmum. The mxmum toque tht the dpole expeences when plced n n extenl electc feld s ts dpole moment. Ths s vecto quntty, nd the toque s mxmum when the dpole moment s t ght ngles to the electc feld. At genel ngle, the toque τ, the dpole moment p nd the electc feld E e elted by τ p E... The SI unts of dpole moment cn be expessed s N m (V/m). Howeve, wok out the dmensons of p nd you wll fnd tht ts dmensons e Q L. Theefoe t s smple to expess the dpole monent n SI unts s coulomb mete, o C m. Othe unts tht my be encounteed fo expessng dpole moment e cgs esu, debye, nd tomc unt. I hve lso hed the dpole moment of thundeclouds expessed n klomete coulombs. A cgs esu s centmete-gm-second electosttc unt. I shll
descbe the cgs esu system n lte chpte; suffce t hee to sy tht cgs esu of dpole moment s bout.6 C m, nd debye (D) s 8 cgs esu. An tomc unt of electc dpole moment s e, whee s the dus of the fst Boh obt fo hydogen nd e s the mgntude of the electonc chge. An tomc unt of dpole moment s bout 8.78 9 C m. I emk n pssng tht I hve hed, dstessngly often, some such emk s The molecule hs dpole. Snce ths sentence s not Englsh, I do not know wht t s ntended to men. It would be Englsh to sy tht molecule s dpole o tht t hs dpole moment. It my be woth spendng lttle moe tme on the equton τ p E. The equton shows tht the esultng toque s, of couse, t ght ngles to the dpole moment vecto nd to the feld. Gven, fo exmple, tht the dpole moment s j 6k C m, nd tht the electc feld s j 5k N C, t s esy to clculte, fom the usul ules fo vecto poduct, tht the esultng toque s 9 8j k N m. It s eqully esy to vefy, fom the usul ules fo scl poduct, tht the ngles betweenτ nd E nd between τnd p e ech 9º. The ngle between p nd E s 7º. Wht bout the nvese poblem? Gven tht the electc feld s E, nd the toque s τ, wht s the dpole moment? One cn sccely sy tht t s E τ! In fct, we shll shotly see tht the soluton s not unque, nd shll undestnd the eson why. Let soluton be p o. Then λe s lso soluton, whee λ s ny constnt hvng SI unts C N m. p (We cn see ths by substtutng p λe fo p n the equton τ p E, nd notng tht E E. ) In effect the component of p pllel to E s not specfed by the equton. Expessed othewse, the component of dpole moment pllel to the electc feld expeences no toque. Let us then choose p to be the component of p tht s pependcul to E. Let us multply the equton τ p E by E, whle substtutng p fo p. E τ E (p E).. On mkng use of well-known (!) elton fo tple vecto poduct, nmely A ( B C) ( A C) B ( A B) C, we obtn E ) τ E p ( E p E.. E τ But E p nd so p.. E E τ nd the genel soluton s p λe. E..5
We cn llustte wht ths mens wth smple dwng, nd thee s no loss n genelty n plcng the dpole monent n the plne of the ppe nd the electc feld dected to the ght. p λe p λe It wll be geed tht the dpoles lbelled p nd p λe wll ech expeence the sme toque n the electc feld E. Let us etun to ou numecl exmple, n whch we found tht the toque on dpole ws τ 9 8j k N m when t ws plced n n electc feld E j 5k N C. Afte lttle wok, t s found tht p (6 j k ) C m. Ths s the component of p pependcul to the electc feld. The genel soluton s p (6 j k) λ( j 5 k ) C m. In ou ognl queston, the dpole 9 moment ws j 6k C m. Any of the thee equtons 6,, 5 λ λ λ 6 wll esult n λ C N m. In othe wods, 9 9 9 9 j 6k C m s ndeed possble soluton; t s the one fo whch λ C N m. 9 One lst smll thought befoe levng the equton τ p E. It my be thought tht ou devton of the genel soluton (equton..5) s dffcult. Afte ll, who would hve thought of pplyng the opeton E? And who mong us emembes the vecto dentty A ( B C) ( A C) B ( A B) C? But ede wth good physcl nsght mght esly ve t equton..5 wthout ny need fo mthemtcl legedemn. Suppose you e sked to solve the equton τ p E gven tht p s pependcul to E nd to τ. Ths mens tht p s n the decton of E τ. And ts τ E τ mgntude must be, nd so the dpole moment n vecto fom s p. And E E fom physcl nsght one would undestnd tht, f the dpole moment lso hd component n the decton of E, ths would not chnge the toque on t, nd so the genel soluton must be tht gven by equton..5. Sometmes t s ese to solve poblem f you e not vey good t mthemtcs thn f you e! 9
. Mthemtcl Defnton of Dpole Moment In the ntoductoy secton. we gve physcl defnton of dpole moment. I m now bout to gve mthemtcl defnton. Q Q O Q FIGUE III. Consde set of chges Q, Q, Q... whose poston vectos wth espect to pont O e,,... wth espect to some pont O. The vecto sum p Q s the dpole moment of the system of chges wth espect to the pont O. You cn see mmedtely tht the SI unt hs to be C m. If we hve just sngle chge Q whose poston vecto wth espect to O s, the dpole moment of ths system wth espect to O s just Q, Thee mpotnt execses!... Execse. Convnce youself tht f the system s whole s electclly neutl, so tht thee s s much postve chge s negtve chge, the dpole moment so defned s ndependent of the poston of the pont O. One cn then tlk of the dpole moment of the system wthout ddng the de wth espect to the pont O. Execse. Convnce youself tht f ny electclly neutl system s plced n n extenl electc feld E, t wll expeence toque gven by τ p E, nd so the two defntons of dpole moment the physcl nd the mthemtcl e equvlent. Execse. Whle thnkng bout these two, lso convnce youself (fom mthemtcs o fom physcs) tht the moment of smple dpole consstng of two chges, Q nd Q septed by dstnce l s Ql. We hve ledy noted tht C m s n cceptble SI unt fo dpole moment.
5. Oscllton of Dpole n n Electc Feld p θ E FIGUE III. Consde dpole osclltng n n electc feld (fgue III.). When t s t n ngle θ to the feld, the mgntude of the estong toque on t s pe sn θ, nd theefoe ts equton of moton s I & θ pe sn θ, whee I s ts ottonl net. Fo smll ngles, ths s ppoxmtely I & θ peθ, nd so the peod of smll osclltons s P I π... pe Would you expect the peod to be long f the ottonl net wee lge? Would you expect the vbtons to be pd f p nd E wee lge? Is the bove expesson dmensonlly coect?. Potentl Enegy of Dpole n n Electc Feld efe gn to fgue III.. Thee s toque on the dpole of mgntude pe sn θ. In ode to ncese θ by δθ you would hve to do n mount of wok pe sn θ δθ. The mount of wok you would hve to do to ncese the ngle between p nd E fom to θ would be the ntegl of ths fom to θ, whch s pe( cos θ), nd ths s the potentl enegy of the dpole, povded one tkes the potentl enegy to be zeo when p nd E e pllel. In mny pplctons, wtes fnd t convenent to tke the potentl enegy (P.E.) to be zeo when p nd E pependcul. In tht cse, the potentl enegy s
6 P.E. pe cos θ p E... Ths s negtve when θ s cute nd postve when θ s obtuse. You should vefy tht the poduct of p nd E does hve the dmensons of enegy..5 Foce on Dpole n n Inhomogeneous Electc Feld Q δx Q E FIGUE III. Consde smple dpole consstng of two chges Q nd Q septed by dstnce δx, so tht ts dpole moment s p Q δx. Imgne tht t s stuted n n nhomogeneous electcl feld s shown n fgue III.. We hve ledy noted tht dpole n homogeneous feld expeences no net foce, but we cn see tht t does expeence net foce n n nhomogeneous feld. Let the feld t Q be E nd the feld t Q be E δe. The foce on Q s QE to the left, nd the foce on Q s Q(E δe) to the ght. Thus thee s net foce to the ght of Q δe, o: de Foce p..5. dx Equton.5. descbes the stuton whee the dpole, the electc feld nd the gdent e ll pllel to the x-xs. In moe genel stuton, ll thee of these e n dffeent dectons. ecll tht electc feld s mnus potentl gdent. Potentl s scl functon, whees electc feld s vecto functon wth thee component, of whch V the x-component, fo exmple s E x. Feld gdent s symmetc tenso x V V hvng nne components (of whch, howeve, only sx e dstnct), such s,, x y z etc. Thus n genel equton.5. would hve to be wtten s
7 E E E x y z V V V xx xy xz V V V xy yy yz V V V xz yz zz p p p x y z,.5. n whch the double subscpts n the potentl gdent tenso denote the second ptl devtves..6 Induced Dpoles nd Polzblty We noted n secton. tht chged od wll ttct n unchged pth bll, nd t tht tme we left ths s lttle unsolved mystey. Wht hppens s tht the od nduces dpole moment n the unchged pth bll, nd the pth bll, whch now hs dpole moment, s ttcted n the nhomogeneous feld suoundng the chged od. How my dpole moment be nduced n n unchged body? Well, f the unchged body s metllc (s n the gold lef electoscope), t s qute esy. In metl, thee e numeous fee electons, not ttched to ny ptcul toms, nd they e fee to wnde bout nsde the metl. If metl s plced n n electc feld, the fee electons e ttcted to one end of the metl, levng n excess of postve chge t the othe end. Thus dpole moment s nduced. Wht bout nonmetl, whch doesn t hve fee electons unttched to toms? It my be tht the ndvdul molecules n the mtel hve pemnent dpole moments. In tht cse, the mposton of n extenl electc feld wll exet toque on the molecules, nd wll cuse ll the dpole moments to lne up n the sme decton, nd thus the bulk mtel wll cque dpole moment. The wte molecule, fo exmple, hs pemnent dpole moment, nd these dpoles wll lgn n n extenl feld. Ths s why pue wte hs such lge delectc constnt. But wht f the molecules do not hve pemnent dpole moment, o wht f they do, but they cnnot esly otte (s my well be the cse n sold mtel)? The bulk mtel cn stll become polzed, becuse dpole moment s nduced n the ndvdul molecules, the electons nsde the molecule tendng to be pushed towds one end of the molecule. O molecule such s CH, whch s symmetcl n the bsence of n extenl electc feld, my become dstoted fom ts symmetcl shpe when plced n n electc feld, nd theeby cque dpole moment. Thus, one wy o nothe, the mposton of n electc feld my nduce dpole moment n most mtels, whethe they e conductos of electcty o not, o whethe o not the molecules hve pemnent dpole moments. If two molecules ppoch ech othe n gs, the electons n one molecule epel the electons n the othe, so tht ech molecule nduces dpole moment n the othe. The two molecules then ttct ech othe, becuse ech dpol molecule fnds tself n the nhomogeneous electc feld of the othe. Ths s the ogn of the vn de Wls foces.
8 Some bodes (I m thnkng bout ndvdul molecules n ptcul, but ths s not necessy) e moe esly polzed tht othes by the mposton of n extenl feld. The to of the nduced dpole moment to the ppled feld s clled the polzblty α of the molecule (o whteve body we hve n mnd). Thus p α E..6. The SI unt fo α s C m (V m ) nd the dmensons e M T Q. Ths bef ccount, nd the genel ppence of equton.6., suggests tht p nd E e n the sme decton but ths s so only f the electcl popetes of the molecule e sotopc. Pehps most molecules nd, especlly, long ognc molecules hve nsotopc polzblty. Thus molecule my be esy to polze wth feld n the x- decton, nd much less esy n the y- o z-dectons. Thus, n equton.6., the polzblty s elly symmetc tenso, p nd E e not n genel pllel, nd the equton, wtten out n full, s px py pz α α α xx xy xz α α α xy yy yz α α α xz yz zz E E E x y z..6. (Unlke n equton.5., the double subscpts e not ntended to ndcte second ptl devtves; the they e just the components of the polzblty tenso.) As n sevel nlogous stutons n vous bnches of physcs (see, fo exmple, secton.7 of Clsscl Mechncs nd the net tenso) thee e thee mutully othogonl dectons (the egenvectos of the polzblty tenso) fo whch p nd E wll be pllel..7 The Smple Dpole As you my expect fom the ttle of ths secton, ths wll be the most dffcult nd complcted secton of ths chpte so f. Ou m wll be to clculte the feld nd potentl suoundng smple dpole. A smple dpole s system consstng of two chges, Q nd Q, septed by dstnce L. The dpole moment of ths system s just p QL. We ll suppose tht the dpole les long the x-xs, wth the negtve chge t x L nd the postve chge t x L. See fgue III.5.
9 E θ θ P E y Q Q L L FIGUE III.5 Let us fst clculte the electc feld t pont P t dstnce y long the y-xs. It wll be geed, I thnk, tht t s dected towds the left nd s equl to Q L E cos θ E cos θ, whee E nd cos. E θ / πε ( L y ) ( L y ) QL p Theefoe E. / / πε ( L y ) πε ( L ).7. y Fo lge y ths becomes E p..7. πε y Tht s, the feld flls off s the cube of the dstnce. To fnd the feld on the x-xs, efe to fgue III.6.
x Q L L Q E P E FIGUE III.6 It wll be geed, I thnk, tht the feld s dected towds the ght nd s equl to Q. ( ) ( ) E E E.7. πε x L x L Q Ths cn be wtten, ( / ) ( / ) nd on expnson of ths by πε x L x L x the bnoml theoem, neglectng tems of ode ( L / x) nd smlle, we see tht t lge x the feld s p E. πε x.7. Now fo the feld t pont P tht s nethe on the xs (x-xs) no the equto (y-xs) of the dpole. See fgue III.7. P (x, y) Q Q L θ L φ FIGUE III.7
It wll pobbly be geed tht t would not be ptcully dffcult to wte down expessons fo the contbutons to the feld t P fom ech of the two chges n tun. The dffcult pt then begns; the two contbutons to the feld e n dffeent nd wkwd dectons, nd ddng them vectolly s gong to be bt of hedche. It s much ese to clculte the potentl t P, snce the two contbutons to the potentl cn be dded s scls. Then we cn fnd the x- nd y-components of the feld by clcultng V / x nd V / y. Q Thus V. / / {( ) } {( ) }.7.5 πε x L y x L y To stt wth I m gong to nvestgte the potentl nd the feld t lge dstnce fom the dpole though I shll etun lte to the ne vcnty of t. At lge dstnces fom smll dpole (see fgue III.8), we cn wte x y, P(x, y) θ FIGUE III.8 nd, wth L <<, the expesson.7.5 fo the potentl t P becomes
V Q Q (( / ) / ( / ) / ). / / ( ) ( ) Lx Lx πε Lx Lx πε When ths s expnded by the bnoml theoem we fnd, to ode L/, tht the potentl cn be wtten n ny of the followng equvlent wys: V QLx πε px πε p cos θ πε p πε..7.6 Thus the equpotentls e of the fom c cos θ,.7.7 p whee c. πε V.7.8 px Now, beng n mnd tht x y, we cn dffeentte V wth πε espect to x nd y to fnd the x- nd y-components of the feld. Thus we fnd tht E x p x pxy nd E. 5 5 y.7.9,b πε πε We cn lso use pol coodntes to fnd the dl nd tnsvese components fom V V p cos θ E nd Eθ togethe wth V to obtn θ πε p cos θ p sn θ p E, E nd cos θ. θ E.7.,b,c πε πε πε The ngle tht E mkes wth the xs of the dpole t the pont (, θ) s θ tn tn θ. p Fo those who enjoy vecto clculus, we cn lso sy E, fom whch, πε fte lttle lgeb nd qute lot of vecto clculus, we fnd ( p ) p E. 5.7. πε
Ths equton contns ll the nfomton tht we e lkely to wnt, but I expect most edes wll pefe the moe explct ectngul nd pol foms of equtons.7.9 nd.7.. Equton.7.7 gves the equton to the equpotentls. The equton to the lnes of foce cn be found s follows. efeng to fgue III.9, we see tht the dffeentl equton to the lnes of foce s d FIGUE III.9 dθ E θ E E dθ dθ Eθ sn θ tn, d cos θ θ.7. E whch, upon ntegton, becomes sn θ..7. Note tht the equtons c cos θ (fo the equpotentls) nd sn θ (fo the lnes of foce) e othogonl tjectoes, nd ethe cn be deved fom the othe. Thus, dθ gven tht the dffeentl equton to the lnes of foce s tn θ wth soluton d sn θ, the dffeentl equton to the othogonl tjectoes (.e. the d equpotentls) s tn θ, wth soluton c cos θ. d θ In fgue III., thee s supposed to be tny dpole stuted t the ogn. The unt of length s L, hlf the length of the dpole. I hve dwn eght electc feld lnes (contnuous), coespondng to 5, 5,,,, 8, 6,. If s
Q expessed n unts of L, nd f V s expessed n unts of, πε L the equtons.7.7 nd cos θ.7.8 fo the equpotentls cn be wtten, nd I hve dwn seven V equpotentls (dshed) fo V.,.,.,.8,.6,.,.6. It wll be notced fom equton.7.9, nd s lso evdent fom fgue III., tht E x s zeo fo θ 5 o '. 9 FIGUE III. V. 8 y/l 7 6 5 V. 6 8 x/l At the end of ths chpte I ppend (geophyscl) execse n the geomety of the feld t lge dstnce fom smll dpole. Equpotentls ne to the dpole These, then, e the feld lnes nd equpotentls t lge dstnce fom the dpole. We ved t these equtons nd gphs by expndng equton.7.5 bnomlly, nd neglectng tems of hghe ode thn L/. We now look ne to the dpole, whee we cnnot mke such n ppoxmton. efe to fgue III.7.
5 We cn wte.7.5 s Q V ( x, y),.7. πε whee ( x L) y nd ( x L) y. If, s befoe, we expess Q dstnces n tems of L nd V n unts of, πε L the expesson fo the potentl becomes V ( x, y),.7.5 y whee ( x ) y nd ( x ). One wy to plot the equpotentls would be to clculte V fo whole gd of (x, y) vlues nd then use contou plottng outne to dw the equpotentls. My computng sklls e not up to ths, so I m gong to see f we cn fnd some wy of plottng the equpotentls dectly. I pesent two methods. In the fst method I use equton.7.5 nd endevou to mnpulte t so tht I cn clculte y s functon of x nd V. The second method ws shown to me by J. Vsvnthn of Chenn, Ind. We ll do both, nd then compe them. Fst Method. To ntcpte, we e gong to need the followng: A ( x y ) x B,.7.6 B ( x y ),.7.7 A nd [( x y ) x ] ( B ),.7.8 whee A x.7.9 nd B x y..7. Now equton.7.5 s V. In ode to extct y t s necessy to sque ths twce, so tht nd ppe only s nd. Afte some lgeb, we obtn
6 [ V V ( )]..7. Upon substtuton of equtons.7.6,7,8, fo whch we e well peped, we fnd fo the equton to the equpotentls n equton whch, fte some lgeb, cn be wtten s qutc equton n B: B B B B,.7. whee A( V ),.7. A V,.7. A V,.7.5 A,.7.6 V nd..7.7 V The lgothm wll be s follows: Fo gven V nd x, clculte the qutc coeffcents fom equtons.7.-7. Solve the qutc equton.7. fo B. Clculte y fom equton.7.. My ttempt to do ths s shown n fgue III.. The dpole s supposed to hve negtve chge t (, ) nd postve chge t (, ). The equpotentls e dwn fo V.5,.,.,.,.8.
7 FIGUE III..5 V.5 y/l.5 V..5.5 V.8 V. V. 5 6 x/l Second method (J. Vsvnthn). In ths method, we wok n pol coodntes, but nsted of usng the coodntes (, θ), n whch the ogn, o pole, of the pol coodnte system s t the cente of the dpole (see fgue III.7), we use the coodntes (, φ) wth ogn t the postve chge. Fom the tngle, we see tht Fo futue efeence we note tht φ L L cos..7.8 L cos φ..7.9 Povded tht dstnces e expessed n unts of L, these equtons become cos φ,.7. cos φ..7.
8 If, n ddton, electcl potentl s expessed n unts of gven, s befoe (equton.7.5), by Q, the potentl t P s πε L V (, φ )..7. ecll tht s gven by equton.7., so tht equton.7. s elly n equton n just V, nd φ. In ode to plot n equpotentl, we fx some vlue of V; then we vy φ fom to π, nd, fo ech vlue of φ we hve to ty to clculte.ths cn be done by the Newton- phson pocess, n whch we mke guess t nd use the Newton-phson pocess to obtn bette guess, nd contnue untl successve guesses convege. It s best f we cn mke fly good fst guess, but the Newton-phson pocess wll often convege vey pdly even fo poo fst guess. Thus we hve to solve the followng equton fo fo gven vlues of V nd φ, f ( ) V,.7. beng n mnd tht s gven by equton.7.. By dffeentton wth espect to, we hve f '( ) cos φ,.7. nd we e ll set to begn Newton-phson teton: f / f '. Hvng obned, we cn then obtn the ( x, y) coodntes fom x cos φ nd y sn φ. I ted ths method nd I got exctly the sme esult s by the fst method nd s shown n fgue III.. So whch method do we pefe? Well, nyone who hs woked though n detl the devtons of equtons.7.6 -.7.7, nd hs then ted to pogm them fo compute, wll gee tht the fst method s vey lboous nd cumbesome. By compson Vsvnthn s method s much ese both to deve nd to pogm. On the othe hnd, one smll pont n fvou of the fst method s tht t nvolves no tgonometc functons, nd so the numecl computton s potentlly fste thn the second method n whch tgonometc functon s clculted t ech teton of the
9 Newton-phson pocess. In tuth, though, moden compute wll pefom the clculton by ethe method ppently nstntneously, so tht smll dvntge s hdly elevnt. So f, we hve mnged to dw the equpotentls ne to the dpole. The lnes of foce e othogonl to the equpotentls. Afte I ted sevel methods wth only ptl success, I m gteful to D Vsvnthn who ponted out to me wht ought to hve been the obvous method, nmely to use equton.7., whch, n ou (, φ) coodnte dφ Eφ system bsed on the postve chge, s, just s we dd fo the lge d dstnce, smll dpole, ppoxmton. In ths cse, the potentl s gven by equtons.7. nd.7.. (ecll tht n these equtons, dstnces e expessed n unts of L Q nd the potentl n unts of.) The dl nd tnsvese components of the feld πε L V V e gven by nd Eφ, whch esult n φ E E E cos φ.7.5 sn φ nd E. φ.7.6 Q Hee, the feld s expessed n unts of, lthough tht hdly mttes, snce we πεl dφ Eφ e nteested only n the to. On pplyng to these feld components we d obtn the followng dffeentl equton to the lnes of foce: sn φ dφ d..7.7 / ( cos φ) ( cos φ) Thus one cn stt wth some ntl φ nd smll nd ncese successvely by smll ncements, clcultng new φ ech tme. The esults e shown n fgue III., n whch the equpotentls e dwn fo the sme vlues s n fgue III., nd the ntl ngles fo the lnes of foce e º, 6º, 9º, º, 5º. E
FIGUE III..5.5 y/l.5.5 5 6 x/l Hee s yet nothe method of clcultng the potentl ne to dpole, fo those who e fml wth Legende polynomls. Fo those who e not fml wth them, hee s quck ntoducton: / The expesson ( x) tuns up qute often n vous geometcl stutons n physcs. Unsupsngly (thnk of the Cosne ule n solvng plne tngle) t often tuns up n context whee x s of the fom x cos θ. Tht s, we hve to del wth n expesson of the fom [ ] / ( ) ( ) cos θ / / ( cos θ). Ths cn be wtten Wth n effot (some mght sy wth consdeble effot) ths cn be expnded by the bnoml theoem s powe sees n ( ), n whch the successve coeffcents e functons of (n fct polynomls n) cos θ. Thus [ ( ) ( ) cos ] / θ P (cos θ) P (cos θ)( ) P (cos θ)( )... nd of couse, [ ( ) ( )cos ] / θ P (cos θ) P (cos θ)( ) P (cos θ)( )... The fst few of these coeffcents, whch e clled Legende Polynomls, e
P (cos θ) P (cos θ) cos θ P (cos θ) P (cos θ) P (cos θ) P (cos θ) 5 8 8 (cos (5cos (5cos (6cos θ ) θ cos θ) 5 cos 7cos θ ) θ 5cos θ) Extensve tbles of these, s well s othe popetes of the Legende polynomls cn be found n vous plces. I lst some of them n Secton. of my ste www.sto.uvc.c/~ttum/celmechs/celm.pdf Now bck to the dpole: P Q θ FIGUE III. Q The potentl t P s gven by πε V Q / / ( cos θ) ( cos θ) Q [( ( )cos ( ) ) / θ ( ( )cos θ ( ) ) / ]. Expnd ths:
πε V Q [ P (cosθ) P (cosθ)( ) P (cosθ)( )... [ P (cosθ) P (cosθ)( ) P (cosθ)( )...]] [ P (cosθ)( ) (cosθ)( ) (cosθ)( )...] P P Q V 5..7.8 5 πε Fo lge only the fst tem contbutes ppecbly nd the expesson then becomes Q cosθ p cosθ V, whee p Q s the dpole moment. Fo smlle (close to πε πε the dpole, we dd hghe-ode tems. We cn convet to ctesn components f needed, nd we cn fnd the feld components by ptl dffeentton n the ppopte dectons..8 A Geophyscl Exmple Assume tht plnet Eth s sphecl nd tht t hs lttle mgnet o cuent loop t ts cente. By lttle I men smll comped wth the dus of the Eth. Suppose tht, t lge dstnce fom the mgnet o cuent loop the geomety of the mgnetc feld s the sme s tht of n electc feld t lge dstnce fom smple dpole. Tht s to sy, the equton to the lnes of foce s sn θ (equton.7.), nd the d dffeentl equton to the lnes of foce s (equton.7.). dθ tn θ Show tht the ngle of dp D t geomgnetc lttude L s gven by tn D tn L..8. The geomety s shown n fgue III.. The esult s smple one, nd thee s pobbly smple wy of gettng t thn the one I ted. Let me know (jttum t uvc dot c) f you fnd smple wy. In the mentme, hee s my soluton. I m gong to ty to fnd the slope m of the tngent to Eth (.e. of the hozon) nd the slope m of the lne of foce. Then the ngle D between them wll be gven by the equton (whch I m hopng s well known fom coodnte geomety!) tn D m m..8. m m
The fst s esy: o m tn(9 ). θ.8. tn θ Fo m we wnt to fnd the slope of the lne of foce, whose equton s gven n pol coodntes. So, how do you fnd the slope of cuve whose equton s gven n pol coodntes? We cn do t lke ths: x cos θ,.8. y sn θ,.8.5 dx cos θd sn θdθ,.8.6 dy sn θd cos θdθ..8.7 Fom these, we obtn dy dx sn θ cos θ d dθ d dθ cos θ. sn θ.8.8 d In ou ptcul cse, we hve dθ nto equton..8 we soon obtn tn θ (equton.7.), so f we substtute ths sn θcos θ m..8.9 cos θ Now put equtons.8. nd.8.9 nto equton.8., nd, fte lttle lgeb, we soon obtn tn D tn L..8. tn θ
y D m m South L θ x Noth Equto FIGUE III. Hee s nothe queston. The mgnetc feld s genelly gven the symbol B. Show tht the stength of the mgnetc feld B(L) t geomgnetc lttude L s gven by L B ( L) B() sn,.8.
5 whee B() s the stength of the feld t the equto. Ths mens tht t s twce s stong t the mgnetc poles s t the equto. Stt wth equton.7., whch gves the electc feld t dstnt pont on the equto p of n electc dpole. Tht equton ws E. πε y In ths cse we e delng wth mgnetc feld nd mgnetc dpole, so we ll eplce the electc feld E wth mgnetc feld B. Also p /( πε ) s combnton of electcl qunttes, nd snce we e nteested only n the geomety (.e. on how B ves fom equton to pole, let s just wte p /( πε ) s k. And we ll tke the dus of Eth to be, so tht equton.7. gves fo the mgnetc feld t the sufce of Eth on the equto s k B ( )..8. In sml ven, equtons.7.,b fo the dl nd tnsvese components of the feld t geomgnetc lttude L (whch s 9º θ) become k sn L k cos L B ( L) nd B ( L). θ.8.,b θ And snce B B B, the esult mmedtely follows..9 Second moments of mss If we hve collecton of pont msses, we cn clculte numbe of second moments. Lkewse, f we hve collecton of pont chges, we cn clculte numbe of second moments. Thee e mny smltes between the two stutons - but thee e few dffeences. The dffeences se ptly fom tdton, but they mostly se fom the ccumstnce tht thee s only one sot of mss, whees thee e two sots of chges - postve nd negtve. In ths secton we befly evew the concepts of second moments of mss (wth whch we ssume the ede s ledy fml). In the next secton we ntoduce second moments of chge, so tht the ede cn compe nd contst the two. If we hve collecton of pont msses - m t ( x, y, z), m t ( x, y, z), etc., we cn clculte sevel second moments. We cn clculte the second moments wth espect to the yz-, zx- nd xy-plnes: m x, m y, m z
6 As f s I know these ptcul moments don t hve much of ole n dynmcl theoy. We cn clculte the second moments wth espect to the x-, y, nd z-xes. These do ply vey lge ole n dynmcl theoy, nd, physclly, they e espectvely the ottonl nets bout the x-, y, nd z-xes -.e. the to of n ppled toque to esultnt ngul cceleton. They e usully gven the symbols A, B nd C: A m ( y z ), B m ( z x ), C m ( x y ).9. We cn clculte the mxed second moments. They, too, ply lge ole, nd e usully gven the symbols F, G nd H. F m y z G m z x, F m x y..9., It s often useful to gthe these togethe n mtx known s the net tenso: A H G I H B F.9. G F C Ths tenso ppes n the elton between the ngul momentum L nd the ngul velocty ω of sold body: L Iω. One futhe second moment tht cn be clculted s the second moment bout pont (pehps bout the ogn of coodntes). I use the symbol I fo ths. It s known s the geometc moment of net (n contst to the dynmc moment of net), nd t s defned by ( I m m x y z ) ( A B C) TI.9. Wth ll these second moments vlble, the tem moment of net s open to some mbguty, nd t my not lwys be used n dffeent contexts o by dffeent wtes to men exctly the sme thng. When we hve system of pont msses, t s often convenent to efe to pont ( x, y, z) known s the cente of mss nd defned by whee M m. M x m x, My m y, Mz m z,.9.5
7 If nsted of collecton of pont msses we hve sngle extended sold body whose densty ρ ves fom pont to pont ccodng to ρ ρ( x, y, z), the vous moments e clculted, f we e wokng n ctesn coodntes, by ntegls such s A ( y z ) ρ( x, y, z) dxdydz.9.6 etc. The lmts of ntegton extend fom the ogn to the sufce of the body, o, snce the densty outsde the body s pesumbly zeo, the lmts mght s well be fom to.. Second moments of chge If we hve collecton of pont chges - Q t ( x, y, z), Q t ( x, y, z), etc., we cn clculte sevel second moments. We cn clculte the second moments wth espect to the yz-, zx- nd xy-plnes:, Q y, Q z Q x... In contst to descbng system of pont msses, these do ply ole n electosttc theoy, nd I denote them espectvely by q q, q. Note tht I m usng the symbol xx, yy zz Q fo electc chge, nd symbols such s q xx fo elements of wht we shll come to know s the qudupole moment tenso. We cn lso clculte the second moments wth espect to the x-, y, nd z-xes: Q ( y z ), Q ( z x ), Q ( x y ), but s f s I know these do not ply sgnfcnt ole n electosttc theoy, so I shn t gve them ny ptcul symbols, no shll I gve them wth n equton numbe. We cn clculte the mxed second moments. q yz Q y z q Q z x, q Q x y..., zx xy It s often useful to gthe these togethe n mtx known s the qudupole moment mtx. qxx qxy qzx q q xy qyy qyz.. qzx qyz qzz
8 Ths s symmetc mtx, becuse q xy s obvously the sme s q yx. I hve chosen to wte the subscpts n cyclc ode n ech of these mtx elements, egdless of ow o column. One futhe second moment tht cn be clculted s the second moment bout pont (pehps bout the ogn of coodntes. Q Q ( x y z ) T q.. If we hve sngle extended body n whch the chge densty ρ ves fom pont to pont ccodng to ρ ρ( x, y, z) C m, we cn clculte the vous moments n sml wy to tht descbed n secton. fo msses. Unlke n the cse of pont msses, thee s lttle pont n defnng some pont ( x, y, z) by some such equton s Q x x - ptcully n such fequent cses n whch the totl chge s zeo. It s possble, howeve, (nd sometmes useful) to fnd pont such tht, when used s ogn of coodntes, the tce (sum of the dgonl elements) s zeo. Pehps ths s best seen wth numecl exmple. Let us suppose tht we hve system of pont chges s follows. Howeve unelstc t my be, I ll suppose tht the chges e n coulombs nd the coodntes n metes, n ode to keep the numecl clculton smple n SI unts. Q x y z 8 6 8 9 8 7 5 6 The totl chge on the system s zeo. The dpole moment mtx s 58 9 8 q 9 C m..5 8 88 Now let us mgne movng the ogn of the efeence xes (wthout otton), nd s we do so the numbes n the mtx chnge. Soone o lte we my fnd tht, when the ogn of the xes eches some pont (, b, c), the numbes n the mtx e such tht
9 the tce (sum of the dgonl elements) s zeo. Ths couldn t hppen wth set of pont msses, ll of whch hve postve mss. When the ogn s t the pont (, b, c) the coodntes of the chges e. Q x y z 8 6 b 8 c b 9 c 8 7 b c 5 b 6 c The dgonl elements of the qudupole moment mtx e now q q xx yy 8 ( ) 8 ( 6 b) ( ) ( b) ( 8 ) (7 b) (5 ) ( b) 58 b q zz 8 (8 c) (9 c) ( c) ( 6 c) 88 7c The tce s T q 8 b 7c. Thus, f we choose the ogn of coodntes to be ny pont (, b, c) tht les n the plne x 5 y 8z 95,..6 the tce of the qudupole moment mtx wll be zeo. Tht s, Q. Fo exmple, we mght choose the pont,, 95 ) ) s ou ogn of coodntes. In tht ( 8 cse the coodntes of the fou pont chges e Q x y z 8 6 7 8 65 8 8 7 8 8 5 5 8 nd the qudupole moment tenso n ts mtx epesentton becomes
58. 9. 69.& q 9.. 8.7& C m..7 69. & 8.7& 568. Its tce s zeo. Now, the thn bodly movng the ogn fom one plce to nothe wthout otton, let us keep the ogn of coodntes fxed, but we ll otte the system of coodnte xes (keepng them othogonl, of couse). As we do so, we ll see the elements of the mtx chngng. Howeve, egdless of whethe we use the ognl ogn (whch gve us nonzeo tce) o the new ogn (whch gves us zeo tce), we ll notce tht dung the otton the tce (sum of the dgonl elements) does not chnge. Ths s not n the lest supsng, becuse the tce s Q nd none of the chnge s long s we keep the ogn fxed. We my lso fnd tht thee s one oentton of the xes whch esults n mtx whose off-dgonl elements e ll zeo. You d be lucky to stumble upon ths specl oentton by ccdent o by tl nd eo. Some edes wll know how to dgonlze symmetc mtx by fndng ts egenvlues nd ts egenvectos. I gve detled exmple of how to do t n Secton.8 of Chpte n my notes on Clsscl Mechncs (http://oc.phys.uvc.c/~ttum/clssmechs/clss.pdf ), so I ll just do t quckly hee fo the mtx gven n equton..7. Some wll wnt to skp ths, nd just see the nswes. Fst we hve to fnd the egenvlues of the mtx, whch e gven by the soluton of the chctestc equton: 58. λ 9. 69.& 9.. λ 8.7&..8 69. 8.7 568. & & λ Ths s cubc equton, nd vewes who e new to ths my fnd t stghtfowd but exceedngly tedous. Howeve, once you hve done t, nd f you ntcpte hvng to do ths sot of thng often, you wte compute pogm fo t so tht n futue you cn do t nstntly wthout hvng to emembe ll the detls of how to do t. I de sy some pogms, vlble onlne, such s Wolfm Alph, cn do t fo you. In ny cse the thee solutons of the bove equton e λ 97. 56.7 6.9 (Moe ccutely they e 97.685, 56.68988, 6.8576. In the text tht follows, I hve pnted only few sgnfcnt fgues; the ctul clcultons wee done by compute n double pecson.)
The qudupole mtx when efeed to the new xes s : 97. q 56.7 C m..9 6.9 Ths tells us wht the dgonl elements e when efeed to the new xes, but t doesn t tell us the oentton of the new xes. By the oentton I men the decton cosnes of the new xes when efeed to the old xes. Let us cll the decton cosnes of l one of the new xes m. It must stsfy n 58. 9. 69.& l l 9.. 8.7& m 97. m... 69. 8.7 568. & & n n Ths looks lke thee lne equtons n the thee decton cosnes l, m, n. Unfotuntely the thee equtons e not ndependent, the thd beng lne combnton of the othe two. We need n ddtonl equton, whch s povded by the theoem of Pythgos, whch tells us tht The fst two equtons mplct n.. e l m n... 6.8 9 l n l n 9 69.&.8 m n m 8.7& n.... Combned wth equton.., these esult n l ±.798 m ±.6875 n.9575.. m You my choose ethe sgn. The xs cn go n ethe of these two opposte dectons. It should be vefed, s check, tht these lso stsfy the thd equton mplct n.., nmely 69.& l 8.7& m 665...5 n n
(I hve been wokng to double pecson n the compute, but I hve pnted the numbes to only few sgnfcnt fgues.) In sml mnne you cn fnd the decton cosnes of the othe two xes fom 58. 9. 69. & 9.. 8.7& 69.& l 8.7& m 568. n l 56.7m n...6 nd 58. 9. 69. & 9.. 8.7& 69.& l 8.7& m 568. n l 6.9 m n...7 The esults e: l ±.676 m m.7 n.678..8 m l ±.86 m m.5 n.98..9 ± Whle you e fee to choose ethe sgn fo l nd l, you e not qute fee to do so fo l. The ognl xes wee pobbly ght-hnded system, nd you pobbly wnt the new xes, fte otton, lso to be ght-hnded. Fst, you should check tht l l l m m m n n n ±, n whch the vetcl lnes ndcte the detemnnt of the mtx... Ths checks tht you thee new xes e stll othogonl (t ght ngles to ech othe). If equton.. s not stsfed, you hve mde n thmetc mstke somewhee. In Secton.8 of Chpte n my notes on Clsscl Mechncs (http://oc.phys.uvc.c/~ttum/clssmechs/clss.pdf ) I show how you cn use the popetes of n othogonl mtx to locte vey quckly whee the mstke s. Then, f you wnt to mke sue tht you hve not chnged the chlty (hndedness) of the xes, you choose the sgns so tht the mtx s, not. In ou pesent cse we fnd tht
.798.676.86.6875.7.5.9575.678.98. If you wsh to convet these decton cosnes to Eule ngles, efe to the bove undelned blue lnk.. Qudupole Moment. Potentl ne n bty chge dstbuton. We noted n Secton.9 tht thee e so mny second moments of mss tht the mee unqulfed tem moment of net s open to some mbguty, nd not ll wtes my men the sme thng. We mke mentl note lwys to mke sue n ou own wtng tht ou edes wll undestnd wht we men n ptcul context. Lkewse wth dstbutons of electc chge, thee e mny second moments of chge, nd the mee unqulfed tem qudupole moment s subject to sml mbguty. We e gong to look n ths secton t the poblem of clcultng the electc potentl n the vcnty of n bty dstbuton of chge. It cn be shown (we shll shotly do so) tht the electc potentl n the vcnty of n bty dstbuton of chges cn be wtten s sees of the fom: cp (cos θ) cp (cos θ) cp (cos θ) cp (cos θ) πε V..... Hee the P e the Legende polynomls. (obvously some knowledge f these wll be needed n wht follows. See, fo exmple, Secton. of http://oc.phys.uvc.c/~ttum/celmechs/celm.pdf ) The c e coeffcents tht e detemned by the chge dstbuton nd geomety of the system. They hve dmensons s follows: [c ] Q [c ] QL [c ] QL [c ] QL. We shll see shotly tht c s dentfed wth the net chge Q of the system. Some wtes wll dentfy c s the dpole moment of the system, nd c (o some smple multple of t) s the qudupole moment, nd c s the octupole moment. In ths secton we ll see wht c nd c elly e.
z P (, θ, φ) Q (, θ, φ ) ω y x FIGUE III.5 Let us mgne tht we hve chge Q t sphecl coodntes (, θ, φ ). See fgue III.5. We shll m to nswe the queston: Wht s the electcl potentl V t pont P(, θ, φ), expessed n tems of Q nd the sphecl coodntes of the chge nd the pont P? The potentl V t P s, of couse, gven by Q πε V,.. so the poblem s just the geometcl one of expessng ι n tems of, θ, φ ) nd (, θ, φ). Fom the cosne ule of plne tgonomety we hve (
5 ω cos,.. nd fom the cosne ule of sphecl tgonomety we hve ) cos( sn sn cos cos cos φ φ θ θ θ θ ω,.. so we hve now been ble to expess n tems of the coodntes. We cn wte equton.. s / cos ω πε Q V..5 We mght f we wsh choose ou coodnte system so tht the pont P s on the z-xs, n whch cse ω s meely θ. In ny cse we cn expnd the expesson n equton..5 s sees of Legende polynomls. See equton.. of www.sto.uvc.c/~ttum/celmechs/celm.pdf. /... ) (cos ) (cos ) (cos ) (cos ω ω ω ω πε P P P P Q V..6 If we hve mny pont chges, not just one, the potentl s clculted fom /... ) (cos ) (cos ) (cos ) (cos ω ω ω ω πε Σ P P P P Q V..7 O f we hve sngle extended body, n whch the the chge densty (C m ) s functon of poston, t s clculted fom /... ) (cos ) (cos ) (cos ) (cos τ ω τ ω τ ω τ ω ρ πε d P d P d P d P V..8 Hee dτ s the elementl volume dxdydz n ctesn coodntes, o φ θ θ d dd sn n sphecl coodntes. The ntegton s to be ced out thoughout the volume of the
6 body, o thoughout ll of spce f t s ssumed tht the chge densty s zeo outsde the body. Now P (cos ω). Theefoe the fst tem n equtons..7 nd..8 s meely Q, whee Q s the totl net chge of the system of pont chges o of the extended body. The coeffcent c of equton.. s dentfed wth Q. Futhe, P (cos ω) cos ω. efeence to fgue III.5 should convnce the ede tht ΣQ cos ω, o ρ cos ωdτ, s the component of the dpole moment p of the system n the decton to P. I ll cll ths component p. Thus the second tem n equtons..7 nd.8 s p, nd the coeffcent c of equton.. s dentfed wth p. We hve, then, so f; Q p πε V plus hghe-ode tems. Fndng the next tem s lttle moe dffcult, but t cn be mde much ese, wth no loss of genelzton, by efeng eveythng to set of coodnte xes such tht the pont P s on the z-xs. Ths wll, of couse men, expessng the coodntes of ech pont chge to these xes, whch my need some pelmny wok. Wth ou now pefeed xes, equtons..7 nd...8 become πε V Q pz Q (cos θ )... P..9 nd πε V Q pz ρ P (cos θ) dτ.....9b (efeed to these xes, wht we clled p s now p z ) Now P (cos θ) (cos θ ). Theefoe the thd tem s equton..9 s Q cos Q θ Σ Q z (q T ( zz q) q zz qxx q yy ). ΣQ.. Hee q s the dgonlzed qudupole moment mtx. Thus the potentl (whethe of system of chges o sngle extended body) t pont P stuted on the z-xs t dstnce fom the ogn s gven by
7 Q pz πεv (q ) K. zz qxx qyy.. The coeffcent c of equton.. s now dentfed s (q q q ). I beleve the quntty q q q s efeed to by some wtes s the qudupole zz xx yy moment, though t s hd to tell f n utho does not explctly efne hs o he tems. zz xx yy. Two smple qudupoles Q Q Q Q Q Q () (b) FIGUE III.6 Nethe of the two systems shown n fgue III.6 hs net chge o net dpole moment, nd nethe wll expeence foce o toque n unfom feld. Both of them hve qudopole (nd hghe-ode) moments, nd both my (dependng on the geomety of the feld) expeence foce o toque s nonunfom feld. Let us look t system (), nd we ll stt by puttng the system long the z-xs of the ctesn coodnte system, wth the Q chge t the ogn. Thee wll be no loss of
8 genelty f we put the pont P n he zx-plne. The ctesn coodntes of P e ( x,, z) nd ts sphecl coodntes e (, θ,). See fgue III.7. P(, θ, ) Q z θ Q x Q FIGUE III.7 The potentl t P s gven (exctly) by: πε V Q Q Q Q ( ) cos θ ( ) Q ( Q ) cos θ ( )... Tht expesson s exct fo the potentl t pont (, θ) nd stghtfowd to clculte - so why would one eve wnt to fnd n ppoxmte expesson? Well, suppose we wnted to dw the equpotentls (nd the lnes of foce, whch e othogonl to the equpotentls). We would wnt to e-cst equton.. n the fom ( θ; V ). If you ty ths, you wll see the dffculty. One could pobbly wte t s cos θ s functon of nd V, whch would enble you to dw the exct equpotentls, but unless you wnt to do ths vey close to the lne qudupole, n ppoxmton my well suffce.
9 To obtn n ppoxmton fo the potentl t lge dstnce, expnd ths s f s ( ) (o futhe f you wsh). Then V s gven ppoxmtely by πε V Q Q [( P (cos θ) P (cos θ)( ) P (cos θ)( )... P (cos θ) P (cos θ)( ) P (cos θ)( )...] [ P (cos θ) P (cos θ)( ) ] Mke use of P ( x), P ( x) (x ), nd ths becomes just.. Q πεv (cos θ )... At lge dstnce fom the qudupole, the equton to the equpotentls s Q Q (cos θ ) (cos θ ).. πε V πε V Ths s shown n fgue III.8 fo V.5,.5,.,. nd. tmes Q. πε To fnd the equton to the lnes of foce we dffeentte clculte d dθ d dθ 6sn θ cos θ..5 Then eplcement of d dθ wth d θ wll gve the dffeentl equton to the d othogonl tjectoes -.e. to the lnes of foce: dθ 6sn θ..6 d cos θ The ntegl of ths equton,.e. the equton to the lnes of foce, s / [sn θ( cos θ)]..7
In fgue III.9 we see, n blck, the equpotentls fo V.5,.5,.,. nd. tmes Q nd, n ed, the lnes of foce fo,,, 8. πε 8 6 z - - -6-8 - - x FIGUE III.8
8 6 z - - -6-8 -6 - - 6 x FIGUE III.9 edes wll undestnd tht these cuves e ppoxmtons fo lge /. To dw the exct equpotentls close to the dpole, we need equton... It my not be esy to wte ths equton expessng explctly n tems of θ nd V. It my be less dffcult to wte t expessng cos θ n tems of nd V. If you cn do ths you should be ble to plot the equpotentls. If ny ede mnges to do ths, I would be hppy to ncopote the esult, wth cknowledgment, of couse, n these notes. You cn fnd me t jttum t uvc dot c It should be possble to epet the nlyss fo the system (b) of fgue III.6. Thee wll be fou dstnces to be expessed n tems of nd θ. These cn be expnded n Legende polynomls. When ll fou e dded, the fst two tems should be zeo, nd the thd tem wll be the equed equton to the equpotentls. Agn, f nyone succeeds n dong ths, let me know nd we cn ncopote t n these notes.
. Octupole Moments Look t the thee-dmensonl system of fgue III.. Q Q Q Q Q Q Q Q FIGUE III. It hs no net chge, no dpole moment, nd no qudupole moment. It does, howeve hve n octupole moment, nd hghe-ode moments. I suppose one mght use the symbol o fo octupole moment. It would be tenso of the thd nk, nd ts mtx epesentton would be thee-dmensonl mtx, whch would eque thee sheets of ppe pled one upon nothe. Its 7 elements (not ll of whch e dstnct) would ech hve thee subscpts, of the knds xxx, xyz, zzx, etc. One could fnd the potentl t pont by the methods of the pevous secton - t would just be bt moe tedous. I m not
gong to pusue tht futhe hee. As S Isc Newton sd, of qute dffeent but even moe dffcult poblem: It doth mke my hed ke. Just one emk. Fgue III. mght look s though t epesents unt cell of sodum chlode cystl. It does to some extent, but you must be n mnd tht the d of the electon clouds of the sodum nd the chlode ons e compble n sze to the sze of the cube. The outemost electons n effect fll up most of the cube. Howeve, to the extent tht the potentl t pont extenl to spheclly-symmety chge dstbuton s the sme s f ll the chge wee concentted t pont, fgue III. mght, fo the pupose of clcultng the potentl t some dstnce fom the cube, be esonble epesentton of sodum chlode.