Phys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 4 Total 40 Points. 1. Problem Points

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1 Phys. 56 Electricity nd Mgnetism Winter Prof. G. Rithel Problem Set Totl Points. Problem 9. Points ). In the long-wvelength limit, in the source nd its immedite vicinity electro- nd mgnetosttic equtions pply. Thus, with Eq. 5.5 the mgnetiztion density M is, using ˆr ẑ = sin θ ˆφ = sin θ( ˆx sin φ + ŷ cos φ), nd v = αc M = x J(x) = ( ) ( iv rˆr ˆr ) + ẑ z = ( ) ( iv rˆr ˆr + ẑ z = ( ) iv r z ˆφ sin θρ(r, θ) = ( ) i v ˆφ tn θρ(r, θ) ( i αc = (ˆx sin φ ŷ cos φ) ) ρ(r, θ) ) tn θρ(r, θ) = M, q.e.d. () In the clcultion of multipole moments t frequency ω, we my thus replce the current by the given effective mgnetiztion density nd set J =. (Note tht both M nd J crry time fctor exp( iω t), which is not shown.) Since M is of the form M = ˆφf(r, θ) with function f tht doesn t depend on φ, it is M =. In the long-wvelength limit, for the multipole moments Eqs to 9.7 pply. Thus, with J = nd M = both M lm = nd M lm =. There re no mgnetic multipoles. From the orthogonlity of the sphericl hrmonics, the only non-vnishing Q lm is Q = Ω r ( e 6 r exp( r )Y Y ) Y r drdω

2 = = e π 56 6π 8 e r exp( r )dr () To find the Q lm, we note (r M) = iαc (ˆr ˆφ)r tn θρ(r, θ) = iαc (ˆθr tn θρ(r, θ) = iαc r sin θ θ r sin θ tn θρ(r, θ) () Since the ngulr dependence of ρ is Y cos θ, this is (r M) = iαc sin θ sin θρ(r, θ) = iαc ρ(r, θ) From Eq. 9.7 nd the previous result on Q lm, it is seen tht the only non-vnishing Q lm is Q = kα Q = π λ αq Since the fctor on the rhs is of order moments, we cn sfely ssume 7 nd the rdited power behves s the squre of the multipole Q = b: Using Eq. 9.69, it is E (, ) = ck i Q. Also, the rdited power P = Z k E. Inserting the results of prt ), it is P = This cn be expressed in the required unit, yielding π Z c k e. 9 W P = ( ) 8 ( α ) c hω

3 c: The trnsition rte is Γ = P hω = ( ) 8 ( α ) c Numericlly, Γ = s = (.59ns) This equls the quntum mechnicl decy rte of the hydrogen P level. Note. The only non-zero multipole moment found in the clssicl clcultion conforms with quntum mechnicl selection rules explined in Chpter 9.8. First, in trnsition from n upper P level into lower S level the tomic ngulr momentum chnges from to (with spin neglected). Thus, only l = rdition cn occur. Further, the trnsition from the P level into S reverses the prity of the tomic stte, requiring n emission field mode with odd prity (tht is, odd mgnetic field). This only leves electric l = decy modes. Finlly, in the given exmple both the upper nd lower sttes hve zero z ngulr momentum. Thus, the emitted field cnnot crry ny z ngulr momentum. In summry, the only multipole field llowed by selection rules is the E (l =, m = ), s found bove. d: According to n erlier homework problem, for n elementry chrge orbiting in the xy-plne t rdius, the only rdition multipole moment for dipole rdition is Q = Q where Q is usul sphericl multipole evluted in the rotting frme. Here, Q = e with phse φ tht we my set to zero. Thus, 8π exp( iφ ) leding to rdited power of Q = 8π e ( ) ( α ) c P cl = 6 hω The rtio of this clssicl power nd the quntum power of prt b) is P cl = =.6 P qm

4 . Problem 9.6 Points ): In this problem, clcultion in crtesin coordintes is the most strightforwrd. The current density is J(x) = ẑiδ(x)δ(y) sin(kz) for z λ/. The rdition pttern is only relevnt in the rdition zone. Thus, we clculte A(x) = µ exp(ikr) π r = ẑ µ I exp(ikr) π i r = ẑ µ I exp(ikr) π i r = ẑ µ π = ẑ µ I π I i exp(ikr) k r exp(ikr) ikr source z=+λ/ exp( ik ˆn x )J(x )dxdydz z= λ/ exp( ikz cos θ) (exp(ikz) exp( ikz)) dz [ ik( cos θ) exp(ikz( cos θ)) + ik( + cos θ) [ ] sin(π( + cos θ)) sin(π( cos θ)) ( + cos θ) ( cos θ) ( ) sin(π cos θ) sin θ ] z=+λ/ exp( ikz( + cos θ)) z= λ/ In the rdition zone, H = ik µ ˆn A, nd with ˆr ẑ = sin θ ˆφ H = ˆφ I exp(ikr) π r ( ) sin(π cos θ) sin θ The rdition pttern is dp dω = r Z E E = r Z H H, yielding dp dω = I Z 8π ( sin ) (π cos θ) sin θ The result is exct in the rdition-zone limit, kr. For the plot, see Problem 9.7. b): The rdited power P P = I Z 8π = I Z π = I Z π sin (π cos θ) sin πd cos θ θ cos (πx) x dx.5578 Sine the rdition resistnce is defined vi P = R rd I, it is

5 R rd = Z.5578 = 9.6Ω π

6 . Problem 9.7 Points ): We use Eqs nd 9.68 to obtin multipole moments tht re NOT in the smll-source pproximtion. Since Eqs. 9.67f re processed most efficiently in sphericl coordintes, we use nd J(x) = ˆr I(r) [δ(cos θ ) + δ(cos θ + )] πr ρ(x) = di(r) iωπr [δ(cos θ ) + δ(cos θ + )] dr with I(r) = I sin(kr) for < r < λ/ nd zero otherwise. It is esily verified tht the continuity eqution, J = iωρ, holds. Since there is no intrinsic mgnetiztion M nd since t ll loctions r where there is current flowing it is r J =, the mgnetic moments ll vnish. From Eq we find the electric-multipole mplitudes k { ( ) d E (l, m) = i Yl,m cρ l(l + ) dr rj l(kr) + ik(r J)j l (kr) r drdω k { = i Yl,m [δ(cos θ ) + δ(cos θ + )] dω l(l + ) π { ( ) ( ) c di d iω dr dr rj l(kr) + ikri(r)j l (kr)dr k { ( ) ( ) di d = {Y l, (θ = ) + Y l, (θ = π) δ m, l(l + ) k dr dr rj l(kr) { ( ) ( ) k l + di d = δ m, δ l,even l(l + ) π dr dr rj l(kr) k ri(r)j l (kr)dr { ( ( k l + d = δ m, δ l,even rj l (kr) di )) rj l (kr) l(l + ) π dr dr = δ m, δ l,even k l(l + ) l + π { [ rj l (kr) di dr ] L L ( d I rj l (kr) dr + k I(r) kri(r)j l (kr)dr ( d ) I dr k ri(r)j l (kr)dr ) dr where the ntenn hlf-length L = λ/. We lso use the definition δ l,even = for even l nd δ l,even = for odd l. For the given I(r) = I sin(kr) it is d I dr + k I(r) =, nd Ik l + E (l, m) = δ m, δ l,even l(l + ) π [rj l(kr)k cos(kr)] λ/ Ik l + λ = δ m, δ l,even l(l + ) π j l(π)k π(l + ) = δ m, δ l,even Ikj l (π) l(l + ) ()

7 In the long-wvelength pproximtion, we use Eq We lredy note tht the long-wvelength pproximtion cnnot be expected to be tremendously ccurte in the given cse, becuse the ntenn length is not smll compred with the wvelength. As before, ll moments vnish except the Q l, with even l. It is With E (l, m) = Q l,m = = = Y πiω Ik πiω l,mr l ρd x { = δ m, δ l,even Ik iω ckl+ i(l+)!! r l ( di(r) dr ) { dr { l,m [δ(cos θ ) + δ(cos θ + )] dω Y { L l + r l cos(kr)dr π π { l + L r l cos(kr)dr π δ m,δ l,even l+ l Q lm, the electric-multipole mplitudes re, in the long-wvelength limit, { Ik l+ l + l + L E (l, m) = δ m, δ l,even r l cos(kr)dr (l + )!! l π b): The exct lowest non-vnishing mplitude is Using only this moment, the rdited power is 5π E (, ) = Ikj (π) 6 P = Z k E(, ) = [ ] Z 5π 6 j (π) I = [ ] Z 5π 9 6 π I = [ 5Z π ] I The numericl vlue for the rdition resistnce (term in rectngulr brckets) is [ ] 5Z R rd = π = 9.Ω The rdition pttern follows from Eq. 9.5 nd Tble 9., dp dω = Z k E(, ) X, = R rd I 5 8π sin θ cos θ

8 The lowest non-vnishing mplitude in the long-wvelength pproximtion is Using only this moment, the rdited power is { E (, ) = Ik 5 λ/ r cos(kr)dr 5 8π { = Ik π π k x cos(x)dx = πik π = Ik π 5 P = Z k E(, ) = [ Z π 5 ] I The numericl vlue for the rdition resistnce (term in rectngulr brckets) is [ ] Z π R rd = = 57.8Ω 5 The rdition pttern follows from Eq. 9.5 nd Tble 9., dp dω = Z k E(, ) X, = R rd I 5 8π sin θ cos θ Discussion of 9.6 nd 9.7. The rdition resistnces found re R = R rd,exct = 9.6Ω R = R rd,,exct = 9.Ω R = R rd,,pprox = 57.8Ω It is R < R. This is to be expected, becuse the totl rdited powers of multipoles dd incoherently. Thus, by neglecting higher exct multipoles we will slightly underestimte the rdited power, which is equivlent to underestimting the rdition resistnce. In the given cse, from R nd R it follows tht by neglecting higher-order exct multipoles we underestimte the rdited power by.% (this is not so bd). It is R >> R. This is not unexpected, becuse by mking the smll-source pproximtion we essentilly neglect destructive interference of rdition originting from different portions of the source. The destructive interference reduces the rdition efficiency of sources tht re not much smller thn the wvelength. In the cse of lrge sources, neglecting this destructive interference cn led to gross overestimtes of the rdited power, s in our cse.

9 Figure : Rdition ptterns for the indicted cses. Bold nd solid: exct clcultion. Solid: Lowest exct multipole term (this term is due to E (, )). Dshed: Sme multipole term in the long-wvelength pproximtion.

10 . Problem 9. Points ): Electric-multipole modes = TM modes. We use Eq. 9. s strting point. Since the fields must be regulr t r =, we choose j l (kr) for ll rdil functions. The generic form of the field of T M lm -mode, with mplitude E (l, m) set to, then is H = j l (kr)x lm E = iz k H = iz k j l(kr)x lm The boundry conditions re tht t r = the electric field must only hve rdil component nd the mgnetic field must be trnsverse. The second condition is utomticlly stisfied becuse of the trnsverslity of the X lm. To mtch the first, we use X l,m = ( ˆLYl,m = ˆφ θ l(l + ) l(l + ) i ˆθ ) sin θ φ Y l,m to first write out the H field components, H r = H θ = m j l (kr) l(l + ) sin θ Y l,m H φ = i l(l + ) j l(kr) θ Y l,m Then, the electric-field components follow from E = iz k (ˆθHθ + ˆφH φ ), E r = iz k = iz k = Z k r sin θ [ θ sin θh φ φ H θ ] i j l (kr) l(l + ) r sin θ j l (kr) l(l + ) r j l (kr) = Z l(l + ) kr r rrh φ Z = l(l + ) kr E θ = iz k E φ = iz k r rrh θ = iz m l(l + ) kr [ θ sin θ θ + im φ [ sin θ θ sin θ θ Y l,m [ ] d dr rj l(kr) [ θ Y l,m ] [ ] [ ] d dr rj l(kr) sin θ Y l,m sin θ m sin θ ] Y l,m ] Y l,m

11 The cvity frequencies follow from the requirement E θ = E φ = t r =. The frequencies cn be obtined from the trnscendentl eqution [ ] [ ] d d dr rj l(kr) = r= dx xj l(x) = x=k d Denoting the n-th root of dx (xj l(x)) with x ln, it is k = ω lmn c = x ln. The resonnce frequencies thus re ω lmn = x ln c Note tht l = does not exist, nd tht the frequencies re degenerte in m, i.e. for given l nd n there re l + TM-modes with the sme frequency. Mgnetic-multipole modes = TE modes. We use Eq. 9. s strting point. The generic form of the field of T E lm -mode, with mplitude M (l, m) set to, then is H = i k j l(kr)x lm E = Z j l (kr)x lm Comprison with the nlogous eqution for TM-modes shows tht the fields of the TE-modes re obtined by replcing the former H with E/Z nd the former E with Z H. Thus, for TE-modes it is E r = E θ = Z m j l (kr) l(l + ) sin θ Y l,m E φ = Z i l(l + ) j l(kr) θ Y l,m nd H r = l(l + ) j l(kr) kr Y l,m [ ] d H θ = l(l + ) kr dr rj l(kr) [ θ Y l,m ] [ ] [ ] im d H φ = l(l + ) kr dr rj l(kr) sin θ Y l,m The conditions of vnishing trnsverse electric nd vnishing norml mgnetic field t r = re stisfied vi the trnscendentl eqution

12 j l (k) = Denoting the n-th root of j l (x) with x ln, it is k = ω lmn c = x ln. The resonnce frequencies thus re ω lmn = x lnc Agin, l = -modes don t exist, nd for given l nd n there re l + TE-modes with the sme frequency. b): (required for TE-modes only). From ω lmn = πc λ lmn = x lnc we see tht λ lmn = π x ln Numericlly we find the lowest roots of sphericl Bessel functions to be x =.9, x = 5.76, x = nd x = The lowest four TE-modes therefore re: λ lmn l n Figure : Lowest sphericl Bessel functions nd their roots. c):

13 The lowest TE-modes re the degenerte T E l=,m=,n=, T E l=,m=,n= nd T E l=,m=,n= -modes. To obtin their fields, use the bove generl equtions for the TE-fields to obtin: l =, m = : E r = E θ = Z 8π j ( x r) exp(iφ) E φ = Z i 8π j ( x r) cos θ exp(iφ) H r = j ( x r) x sin θ exp(iφ) 8π H θ = 8π H φ = i 8π r x r x r [ d dr rj ( x [ d dr rj ( x r) ] r) ] cos θ exp(iφ) exp(iφ) l =, m = : E r = E θ = E φ = Z i π j ( x r) sin θ H r = j ( x r) x π cos θ H θ = π H φ = r x r [ d dr rj ( x ] r) sin θ l =, m = : E r = E θ = Z 8π j ( x r) exp( iφ) E φ = Z i 8π j ( x r) cos θ exp( iφ) H r = j ( x r) x 8π r sin θ exp( iφ) H θ = [ d 8π x r dr rj ( x ] r) cos θ exp( iφ) H φ = i [ d 8π x r dr rj ( x ] r) exp( iφ)