Radiating Systems. (Dated: November 22, 2013) I. FUNDAMENTALS

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1 Classical Electodynamics Class Notes Radiating Systems (Dated: Novembe 22, 213) Instucto: Zoltán Tooczkai Physics Depatment, Univesity of Note Dame I. FUNDAMENTALS So fa we studied the popagation of electomagnetic waves in vacuum and matte, without and with boundaies and we have consideed the fields given. In this chapte we study how the fields ae geneated by souces and thei popeties. Assumptions Souces ae descibed by the electic chage density ρ(, t) and the electic cuent density J(, t). The souces ae in vacuum (ɛ, µ ). µ ɛ = 1/c 2. We conside that the spatial extent of the souces is finite, in othe wods they ae confined to a finite volume of chaacteistic scale (linea size) d. In othe wods, they ae localized. d 3. The time scale chaacteistic of the tempoal vaiations of the souces is T. In paticula fo souce oscillations with angula fequency ω, T = 2π/ω. Fo hamonic vaiations of the souces in time the chaacteistic wave numbe is k = ω/c and the chaacteistic wavelength λ = 2π/k = 2πc/ω. Souce chaacteistic velocity is v. Fo hamonic behavio v = d/t = ωd/(2π). Non-elativistic situation v c. This tanslates fom above into: d λ, (1) o in wods, the spatial extent occupied by the souces is small compaed to the wavelength of the emissions. Thee ae no boundaies pesent othe than possibly those of the souces. Some notations and Definitions The position vecto of a point in the space outside the souces is denoted by, with magnitude = and unit vecto n = /. Fo points unning within the confinement volume of the souces we use pimed notation (,, n ). Usually this is an integation vaiable. The methods of appoximation intoduced below divide the space in 3 egions of inteest: NEAR zone: λ o equivalently k 1 Physics Depatment, Univesity of Note Dame Fall 27 1

2 I FUNDAMENTALS Class Notes FAR (o Radiation, o Wave) zone: λ, o k 1 INDUCTION (o intemediate) zone: λ, o k 1. Potentials and fields Fo efeence, let me ecall the Maxwell equations with souces in vacuum expessed in tems of the electic (E) and magnetic (H) fields: H E + µ t =, E = 1 ρ(, t) ɛ (2) E H ɛ = J(, t), t H = (3) Within Loenz gauge these 4 equations ae educed to only 2 decoupled second-ode equations, involving the scala (Φ) and vecto potentials (A): Φ(, t) = 1 ɛ ρ(, t) A(, t) = µ J(, t) whee is the D Alembetian, o the wave opeato: The Loentz gauge condition equies: = 1 c 2 2 (4) t (5) A + 1 c 2 Φ t =. (6) When no boundaies ae pesent, the solutions to the wave equations (4) ae: Φ(, t) = 1 ( d 3 dt ρ(, t ) 4πɛ δ t + ) t c A(, t) = µ ( d 3 dt J(, t ) 4π δ t + ) t c (7) (8) The following is anothe (equivalent) vesion: Φ(, t) = 1 d 3 ρ (, t /c) (9) 4πɛ A(, t) = µ d 3 J (, t /c). (1) 4π These ae the etaded solutions, which peseve causality between the geneation of the field at the souces and its appeaance somewhee outside the souces. The fields can be obtained Physics Depatment, Univesity of Note Dame Fall 213 2

3 I FUNDAMENTALS Class Notes fom the potentials via: H = 1 µ A (11) E = A t Φ (12) The above equations along with the field equations (11), (12) allow us to detemine the fields anywhee in the space with no boundaies pesent. Mathematically, this whole chapte is about appoximating the integals (7), (8) and the coesponding fields (H, E) unde the assumptions pesented above and in the vaious egions of space. Finally, let us ecall the continuity equation connecting the chage and cuent densities, expessing the consevation of local chage: Enegy, Radiated Powe ρ t + J =. (13) Fom a pactical point of view we will be inteested in calculating the enegy contained in the fields and the powe emitted into space by the souces. The enegy density u(, t) is given by: u(, t) = ɛ 2 E E + µ 2 H H (14) The Poynting vecto S(, t) is the electomagnetic powe (enegy pe unit time) adiated though the unit suface and it is given by: Enegy consevation, diffeential fom: S(, t) = E H. (15) u t + S + J E = (16) S da is the field enegy flowing though an oiented suface element da pe unit time. The suface element is centeed on and its nomal vecto coincides with n. We have fom (A7): Thus the powe adiated into the unit solid angle is: dp = S da = n S(, t) 2 dω (17) dp dω = n S(, t) 2 (18) The total powe adiated into the space is calculated by taking the integal of (17) ove an abitay suface enclosing the souces. Usually this is taken as a sphee. Physics Depatment, Univesity of Note Dame Fall 213 3

4 II FIELDS OF AN OSCILLATING POINT DIPOLE Class Notes Hamonic Behavio The most natual way of poducing electomagnetic fields is via oscillating souces. As a matte of fact the Fouie composition theoem allows us to compose any abitay tempoal behavio fom hamonics e iωt. Thus it is woth studying the case of oscillating souces: { ρ(, t) = ρ()e iωt J(, t) = J()e iωt (19) The physical quantities ae obtained fom the complex ones by taking thei eal pat. Most of the physical quantities, like the fields and the adiated powe will have a hamonic dependence. It is customay to expess the aveage ove a full cycle (T ) of these quantities. This is computed by the integal f = T 1 T f(t)dt fo a quantity f. In paticula, the Poynting vecto can be expessed in tems of the complex fields as: Thus the powe adiated in the unit solid angle: S = 1 2 Re [E H ]. (2) dp dω = 1 2 Re [n E H ] 2. (21) Pactically, one finds that calculating the scala potential Φ fom (7) o (9) and thus the electic field fom (12) is moe involved than calculating the vecto potential A. As a concete example see Section II. In case of hamonic dependence, E/ t = iωe and fom the cul equation (3) it follows that outside the souces: E = iz k H, (22) and thus we can avoid computing the scala potential in this case. Hee Z = µ /ɛ is the impedance of the vacuum. II. FIELDS OF AN OSCILLATING POINT DIPOLE Befoe studying geneal appoximative expessions, let us look at a specific case which can be calculated exactly (without any appoximations), that of an oscillating point dipole. We will compute the fields by computing both the scala and vecto potential and then employing (11) and (12). The validity of (22) can then be checked by diect calculation. We have leaned peviously of a dipole as being fomed by two chages q and q sepaated by a distance 2ε. The mid-point between the chages is consideed to be the oigin of the coodinate system and the unit vecto oiented fom the negative chage towads the positive chage is denoted by ˆp. The coesponding chage density is thus given by: ρ() = qδ( εˆp) qδ( + εˆp) (23) Assuming that ε is going to zeo we can expand the Diac-δ function into its Taylo seies Physics Depatment, Univesity of Note Dame Fall 213 4

5 II FIELDS OF AN OSCILLATING POINT DIPOLE Class Notes as δ( ± εˆp) = δ() ± εˆp δ() + O(ε 2 ). Theefoe (23) becomes: ρ() = 2qεˆp δ() + O(ε 2 ) p δ() + O(ε 2 ) (24) whee p = j q j j = (+q)(εˆp) + ( q)( εˆp) = 2qεˆp is the familia dipole moment. The chage density defined via (24) in the ε limit with p being kept as a non-zeo constant vecto is defined as the static point dipole. In ode to define an oscillating point dipole we intoduce the time dependence via: ρ(, t) = p δ()e iωt (25) Oscillating chages will necessaily induce a (complex) cuent density J. Since thee ae no othe cuents pesent by assumption, these two ae diectly connected via the continuity equation (13): J = iωp δ()e iωt = [ iωpδ()e iωt]. (26) This allows us to infe the cuent density simply as: J(, t) = iωpδ()e iωt. (27) In the following subsection we summaize the esults fo the potentials and fields and powe emitted followed by the subsection whee the specific calculations ae shown in detail. Potentials (Complex) A. Summay of esults fo the point dipole Potentials (Real) A(, t) = µ e iω(t /c) iωp 4π Φ(, t) = 1 ( 1 p n 4πɛ iω 2 c (28) ) e iω(t /c) (29) A(, t) = µ sin ω(t /c) ωp 4π Φ(, t) = 1 [ cos ω(t /c) p n 4πɛ 2 ω c ] sin ω(t /c) (3) (31) In the following we intoduce the etaded time shothand notation τ = t /c. Note that the scala potential (29) can be witten as: Φ(, t) = Φ N (, t) + Φ F (, t) (32) Physics Depatment, Univesity of Note Dame Fall 213 5

6 II FIELDS OF AN OSCILLATING POINT DIPOLE Class Notes with: Φ N (, t) = 1 4πɛ Φ F (, t) = 1 4πɛ p n 2 e iωτ (33) iω c p n e iωτ (34) o with thei eal countepats: Φ N (, t) = 1 4πɛ Φ F (, t) = 1 4πɛ p n 2 sin ωτ (35) ω c p n cos ωτ In the amplitude of (33) (o (35) ) we ecognize the expession fo the static dipole field. This tem of the two is dominant fo small values and thus it epesents the NEAR field egion. Since /c is vey small, the fields follow the dipole s time vaiation pactically instantaneously. In the FAR zone Φ F is the dominant tem, the othe vanishes. One can see fom its fomal expession that it is in fact a spheical wave solution decaying as 1/ with distance fom the souce and having no angula dependence (thus intensities decay as 1/ 2 and since the suface of the sphee gows as 2 this keeps the adiated enegy conseved). Fields (Complex) H(, t) = ck2 4π ( 1 1 ik E(, t) = 1 {k 2 e iωτ 4πɛ ) e iωτ (36) (n p) (37) [n (n p)] + [3(n p)n p] ( 1 ik ) } e iωτ 3 2. (38) One can see that the magnetic field is always pependicula to the adius vecto, but the electic field can have both pependicula and paallel components. The eal expessions of the fields ae easily obtained fom above. Radiated Powe dp dω = c2 Z k 4 32π 2 (n p) n 2. (39) P = c2 Z k 4 12π p 2. (4) Physics Depatment, Univesity of Note Dame Fall 213 6

7 II FIELDS OF AN OSCILLATING POINT DIPOLE Class Notes B. Deivations fo the point dipole, II A Fist we calculate the vecto potential A fom (8): A(, t) = µ ( d 3 x dt J(, t ) 4π δ t + ) t = c = iωµ p dt e iωt d 3 x δ( ) δ(t + /c t) = 4π = iωµ p dt e iωt δ(t + /c t) = iωµ p e iω(t c), (41) 4π 4π equivalent to (28). Next we compute the scala potential fom (7). In this case the spatial integal ove is: I p (, t, t ) d 3 x p δ( ) δ(t + /c t). (42) Hee f( ) means that the nabla opeato is defined though patial deivatives with espect to the components of. We notice that this integal is of type: d 3 x p δ( ) f( ), whee p is a constant vecto. In ode to establish a ule fo computing this, let us go back to the egulaized fom (23): p δ( ) = qδ( εˆp) qδ( + εˆp) + O(ε 2 ) (43) in the small ε limit. Inseting it in the integal above we easily obtain: d 3 x p δ( ) f( ) = qf(εˆp) qf( εˆp) + O(ε 2 ) Afte expanding in seies and taking the ε limit with p = 2qεˆp kept non-zeo and constant, the integation ule is found: d 3 x p δ( ) f( ) = p f( ). (44) = Note that fomally the same esult is obtained if one would use integation by pats. Now we need to compute the integal (42). Note that in ou case f( ) = g( ) with g(s) = δ(t t + s/c)/s. Then: f( ) = dg(s) (A15) = ds [ δ(t t + /c) 1 2 c δ (t t + /c) ]. Physics Depatment, Univesity of Note Dame Fall 213 7

8 III APPROXIMATIONS Class Notes Hee δ (s) = dδ(s)/ds. Thus: [ δ(t I p (, t, t t + /c) ) = (p n) 1 2 c ] δ (t t + /c) (45) Finally, the scala potential is: Φ(, t) = 1 4πɛ = p n 4πɛ dt e iωt I p (, t, t ) = dt e iωt [ δ(t t + /c) 2 1 c ] δ (t t + /c) (46) Using the identity (A16) to calculate the integal containing the total deivative of the delta function, one obtains (29): Φ(, t) = 1 ( 1 p n 4πɛ iω ) e iω(t /c). (47) 2 c The fields ae obtained fom using Eqs. (11) and (12) with (28) and (29). III. APPROXIMATIONS In this subsection we look at the expessions of potentials and fields in the nea and fa zone fo a geneal time dependence, not necessaily just hamonic. The equations can eadily be adapted to hamonic dependence. Nea zone In the nea zone /c is small and we can expand the chage density and the cuent density fom Eqs. (9) and (1) in powes of this quantity. In geneal f(t /c) = f(t) f(t) /c + f(t) 2 /2c If the time scale ove which f changes is on the ode of T then the second tem is on the ode of /(T c) = /λ. Keeping explicitly the fist tem of this expansion, only we obtain: Φ N (, t) = 1 [ ( )] d 3 ρ(, t) 1 + O (48) 4πɛ λ (49) A N (, t) = µ [ ( )] d 3 x J(, t) 1 + O (5) 4π λ The expessions of the fields above ae fomally identical to those fo static fields found in chaptes 4 and 5 of Jackson. This means that the fields ae essentially of static natue and the time delay effects (caused by finite velocity popagation) ae negligible. Since these types of integals have been extensively dealt with aleady in pevious chaptes, we shall not epeat any of those hee. Instead we will focus on adiative effects, which can be measued in the fa zone. Physics Depatment, Univesity of Note Dame Fall 213 8

9 III APPROXIMATIONS Class Notes Fa zone Let us fo now focus on the vecto potential, only. Using the fist two tems in expansion (A13): t τ + 1 c c n. (51) Fo the denominato in (1) we substitute. One finds that in the fa zone with good appoximation: A F (, t) = µ d 3 J(, τ + n /c) (52) 4π Fo this expession we have not used the non-elativistic small-speed limit d λ and thus it is a geneal expession. To take into account this non-elativistic condition (equivalent to λ), we expand the vecto potential in Taylo seies: J(, τ + n /c) = J(, τ) + 1 c (n ) J +... (53) t t=τ Inseting this in (52) we find: A F (, t) = µ [ 1 4π d 3 J(, τ) + 1 c τ ] d 3 (n ) J(, τ) (54) Let us now ecall a few definitions used in electo- and magneto- statics: The electic dipole moment vecto p(, t) is defined via: p(t) = d 3 ρ(, t). (55) The magnetic dipole moment vecto is defined as: m(t) = 1 d 3 [ J(, t)]. (56) 2 The electic quadupole moment tenso is defined though: ( ) Q α,β (t) = d 3 3x α x β 2 δ αβ ρ(, t) (57) whee α, β {1, 2, 3} coesponding to the x, y, z coodinates of the vecto. We will also need the vecto Q(, t) defined by its components: Q α (t) = 3 Q α,β (t) n β, (58) β=1 whee n β ae the components of the unit vecto n. Physics Depatment, Univesity of Note Dame Fall 213 9

10 III APPROXIMATIONS Class Notes and It can be shown that the following elations hold: ṗ(t) = dp dt = d 3 J(, t) (59) d 3 (n ) J(, t) = m(t) n Q(, t) n d dt Inseting (59) in (54) we find the vecto potential to be of the fom: A F (, t) = µ w(, τ) 4π d 3 2 ρ(, t). (6), τ = t /c. (61) whee w only depends only on the unit vecto n of (since Q depends on the diection n only, fom (58) ) and it is: w(, τ) = w(n, τ) = ṗ(τ) + 1 c ṁ(τ) n + 1 6c Q(n, τ). (62) The dots and double dots mean patial deivative with espect to the etaded time τ. As you can notice, we have omitted the last tem, popotional to n fom (6) fo the eason that its contibution to the fa zone fields (and thus to the adiated powe) is zeo. To show that let us conside that the vecto potential has a tem of the fom nf(t) whee f(t) is some function. Then the contibution of this tem to the field is: [ ] f(τ) [nf(τ)] = (A8) = [ ] f(τ) (A11) = d d [ ] f(τ) n =. Next we show that fo vecto potentials of the geneal fom (61) the fields can be obtained simply (fo simplicity of the notation we will dop the fa subscipt) as : H = 1 n ẇ(n, τ) 4πc (63) We have (using also (A12) ): [ ] 1 w(n, τ) = n w w. (64) Now we calculate w, up to O( 1 2 ) tems, taking into account that w depends on also though τ = t /c. We obtain: w(n, τ) = 1 c (n ẇ) + O ( 1 2 ) (65) Inseting this in (64) and odeing the tems in invese powes of, we get: [ ] 1 w(n, τ) = n ẇ ( ) 1 + O c 2 (66) Physics Depatment, Univesity of Note Dame Fall 213 1

11 III APPROXIMATIONS Class Notes Since we ae in the fa zone, one can neglect the O(1/ 2 ) tem and finally ecove (63). The electic field outside of the souces can be obtained fom the cul equation (3): E t = 1 H = 1 [ ] n ẇ ɛ 4πɛ c. (67) The cul can be evaluated in a vey simila fashion to the above case, only moe complicated due to the exta vectoial poduct in the agument. Howeve, keeping only the leading tem, we find: [ ] n ẇ n [n ẅ(n, τ)] = (68) c Thus: E t = 1 n [n ẅ(n, τ)] 4πɛ c 2 This equation can now be integated, and also using c 2 = 1/(µ ɛ ), one obtains: E(, t) = µ n [n ẇ(n, τ)] 4π (69) = µ c (n H) = Z (n H). (7) In summay, fo the geneal dependence (61) the coesponding fields in the fa zone will be given by Eqs. (63) and (7). One can obseve that independently of the explicit expession fo w, we have that both the magnetic and electic fields ae pependicula to the diection of popagation, n, and that the two fields ae also pependicula on each othe. This expesses the fact that the fa fields ae spheical tansvesal waves. The fields ae also in phase and that the magnitude of the electic field is c times stonge than the magnitude of the magnetic induction E = c B. The fa zone Poynting vecto can be calculated based fom (15) and using identity (A1): S(, t) = µ c H 2 n = Z H 2 n = µ n ẇ(n, τ) 2 n. (71) 16π 2 c 2 This shows that the electomagnetic powe tavels in the adial diection. adiated in the unit solid angle (18) is given by: dp dω = µ n ẇ(n, τ) 2 16π 2 c The enegy Note that the above expession has no dependence on except though the etaded time τ. Futhe calculations ae only possible if we know the angula dependence of the coss poduct. We will need to specialize the calculations accoding to the 3 tems of w given by (62). The fist tem is the leading tem and the coesponding adiation ceated by it is called the electic dipole adiation. The othe two tems have compaable magnitude and they induce the so-called magnetic dipole and electic quadupole adiations. Sections I and deal with each case sepaately. (72) Physics Depatment, Univesity of Note Dame Fall

12 MAGNETIC-DIPOLE AND ELECTRIC-QUADRUPOLE RADIATION Class Notes I. ELECTRIC DIPOLE RADIATION Keeping only the leading tem in (62) the vecto potential (61) becomes: The fields fom (63) and (7) ae: A ed (, t) = µ ṗ(τ) 4π H ed (, t) = 1 n p(τ) 4πc E ed (, t) = µ n [n p(τ)] 4π, τ = t /c. (73) The Poynting vecto and powe adiated into the unit solid angle ae: The total powe emitted: (74). (75) S ed (, t) = µ n p(τ) 2 n, (76) 16π 2 c 2 dp ed dω = µ n p(τ) 2. (77) 16π 2 c P ed (τ) = µ 16π 2 c dω n p(τ) 2 Fo each given τ value choose the z-axis such as to coincide with p(τ). Then we have n p(τ) = p(τ) sin θ. The angula integal can easily be pefomed, since dω sin 2 θ = 2π dφ π dθ sin3 θ = 8π/3. Thus: (78) P ed (τ) = µ 6πc p(τ) 2. (79) Once can easily see that these expessions ae genealizations of the paticula ones obtained peviously fo the fa zone in the case of an electic oscillating point dipole.. MAGNETIC-DIPOLE AND ELECTRIC-QUADRUPOLE RADIATION If in paticula p(τ) = identically, then the next ode tems in (62) become impotant. The vecto potential coesponding to the magnetic dipole tem: A md (, t) = µ 4πc n ṁ(τ). (8) Physics Depatment, Univesity of Note Dame Fall

13 MAGNETIC-DIPOLE AND ELECTRIC-QUADRUPOLE RADIATION Class Notes The coesponding fields fom (63) and (7) ae : H md = 1 n [n m(τ)] 4πc 2 E md = µ n m(τ) 4πc Fo the electic field we had to calculate the tiple coss poduct n [n [n m(τ)]] epeatedly using (A1). The powe adiated by the magnetic dipole into the fa zone depends on the poduct: n ẇ(n, τ) 2 = 1 c n [n 2 m(τ)] 2 = 1 c m (n m) 2 n 2 = 1 m c 2 α m β (δ αβ n α n β ) whee n α ae the components of n. Thus, fom (72): dp md dω = µ 16π 2 c 3 α,β α,β (81) (82) m α m β (δ αβ n α n β ) (83) The double-dot tems ae independent of n and thus the angula integal will only affect the last tem, the poduct n α n β. The following angula integals can be shown to hold: n α n β dω = 4π 3 δ αβ (84) n α n β n γ n η dω = 4π 15 (δ αβδ γη + δ αγ δ βη + δ αη δ βγ ) (85) Using the fist identity fom hee we can calculate the total adiated powe as: P md (τ) = µ 6πc 3 m(τ) 2. (86) We can now compae the powe emitted by an electic dipole with that emitted by a magnetic dipole fo slow moving souces using the following ode of magnitude estimates: p ρd 4, p ρd 4 /T 2, m Jd 4 ρd 4 v. This leads to: P md P ed ( m 2 v ) 2 c 2 p 2 1. (87) c The vecto potential coesponding to the electic quadupole tem: A eq (, t) = µ 24πc Q(n, τ). (88) Physics Depatment, Univesity of Note Dame Fall

14 MAGNETIC-DIPOLE AND ELECTRIC-QUADRUPOLE RADIATION Class Notes The fields: H eq = 1... n Q(n, τ) 24πc 2 E eq = µ n 24πc [n... Q(n, τ) To calculate the powe we fist need to evaluate: n ẇ(n, τ) 2 = 1 6c n Q(n,... τ) = 1 36c 2 α,β,γ,η = 1 [ ] 3 n Q(n, τ) 24πc 2 τ 3 ] 2 = µ 3 n [n Q(n, τ)] 24πc τ 3 = 1 36c 2 ( n Q... ) (n Q... ) = Q αγ Q βη (δ αβ n γ n η n α n β n γ n η ) (89) (9) This leads to: dp eq dω = µ 576π 2 c 3 α,β,γ,η Q αγ Q βη (δ αβ n γ n η n α n β n γ n η ) (91) Tho calculate the total adiated powe, we need to calculate the angula integals, which can be pefomed with the aid of fomulas (84) and (85). The final esult is: P eq (τ) = µ... Q 72πc 3 αβ 2. (92) Obseving that in odes of magnitude: Q αβ ρd 5,... Q αβ ρd 5 /T 3 pv. Thus, a compaison between the powe adiated by an electic quadupole and that of an electic dipole: P eq P ed m 2 p 2 α,β ( v c ) 2 1. (93) Eqs. (87) and (93) show that the magnetic-dipole and electic-quadupole adiations ae of the same ode of magnitude. To calculate the total adiated powe fom the moments consideed above we need to add all the contibutions: [ P (τ) = µ p(τ) πc c 2 m(τ) Q 12c 2 αβ (τ) ( ) ] 2 v 4 + O. (94) c 4 α,β Physics Depatment, Univesity of Note Dame Fall

15 A FORMULA BASKET Class Notes Appendix A: Fomula Basket 1. Coodinate Systems The Catesian coodinates (x, y, x) of a point = xˆx + yŷ + zẑ in 3D space can be tansfomed into: Spheical Pola Coodinates: (x, y, z) (, θ, φ) whee: x = sin θ cos φ, y = sin θ sin φ, z = cos θ (A1) with = x 2 + y 2 + z 2, ( z ) ( y θ = accos, φ = actan x) (A2) in the domain [, ), θ [, π], φ [, 2π]. The line, suface and volume elements in spheical coodinates ae: The solid angle element is: It follows: ds = n + sin θdφ ˆφ + dθ ˆθ da = n 2 sin θ dθdφ d = 2 sin θ dθdφd. dω = sin θ dθdφ da = n 2 dω. (A3) (A4) (A5) (A6) (A7) 2. ectoial Identities Most of these identities can be checked to hold by diect calculations on thei components. (ψa) = ψ a + ψ a. = n. (A8) (A9) In paticula: a (b c) = (a c)b (a b)c f() = df df = d d = df d n (A1) (A11) ( ) 1 = = n 3. (A12) 2 3. Expansion fomulas Notation: { =, = n. > = max{, }, < = min{, } Physics Depatment, Univesity of Note Dame Fall

16 A FORMULA BASKET Class Notes We have = n + 2 (n ) 2 2 [ ( ) ] 3 + O (A13) The thid tem on the hs is of ode O[( /) 2 ]. The above follows fom witing: = ( ) = 1 2(n n ) 2 + and expanding in Taylo seies. Expansion in spheical hamonics Y lm (θ, φ): 1 = 4π l l= m= l 1 2l + 1 < l > l+1 Y lm (θ, φ)y lm(θ, φ ) (A14) whee (θ, φ) and (θ, φ ) ae the angula coodinates fo and. This is called the addition theoem fo spheical hamonics. 4. Diffeentials and Integals = which can be checked diectly. [Use = (x x ) 2 + (y y ) 2 + (z z ) 2.] (A15) The following identity follows fom patial integation fo functions f with compact suppot: dx δ (x)f(x) = df. (A16) dx x= Physics Depatment, Univesity of Note Dame Fall