Versin Date July 10, 011 1 The Creatin and Prpagatin f Radiatin: Fields Inside and Outside f Surces Stanislaw Olbert and Jhn W. Belcher Department f Physics Massachusetts Institute f Technlgy Richard H. Price Department f Physics University f Texas at Brwnsville Abstract We present a new algrithm fr cmputing the electrmagnetic fields f currents inside and utside f finite current surces, fr arbitrary time variatins in the currents. Unexpectedly, we find that ur slutins fr these fields are free f the cncepts f differential calculus, in that ur slutins nly invlve the currents and their time integrals, and d nt invlve the time derivatives f the currents. As examples, we give the slutins fr tw cnfiguratins f current: a planar slenid and a rtating spherical carrying a unifrm charge density. Fr slw time variatins in the currents, we shw that ur general slutins reduce t the standard expressins fr the fields in classic magnetic diple radiatin. In the limit f extremely fast turn-n f the currents, we shw that fr ur general slutins the amunt f energy radiated is exactly equal t the magnetic energy stred in the static fields a lng time after current creatin. We give three assciated prblem statements which can be used in curses at the undergraduate level, and ne prblem statement suitable fr curses at the graduate level. These prblems are f physical interest because: (1) they shw that current systems f finite extent can radiate even during time intervals when the currents are cnstant; () they explicitly display transit time delays acrss a surce assciated with its finite dimensins; and (3) they allw students t see directly the rigin f the reactin frces fr timevarying systems.
Versin Date July 10, 011 1 Intrductin There are almst n analytic slutins fr the cmplete electrmagnetic fields everywhere in space due t time-varying current systems with at least ne finite and nnzer spatial dimensin. In principle, such prblems can be slved numerically, with the standard apprach being first t slve the time-harmnic prblem, and then t find the slutin fr arbitrary time-dependence using inverse Furier transfrms [1]. Hwever, in the general case, this apprach invlves numerical cmputatins f substantial cmplexity. As a result, we avid in the intermediate level classrm discussins f the fields assciated with such prblems. And even in graduate curses, where such prblems may be cnsidered, the physical meaning f the slutins are usually lst in a maze f mathematical cmplexity (e.g., the slutins t the general prblem invlve vectr spherical harmnics). Hwever, we wuld like a student in intermediate and graduate level curses t be able t slve analytically prblems where an abrupt change in current in ne part f a current system causes changes in the lcal fields, and where these field changes can then be seen t prpagate at the speed f light t ther parts f the same system f currents, and interact with thse currents, as well as eventually prducing radiatin far away frm the surce. Amng many pedaggical advantages, such prblems directly illuminate the nature f reactin frces frm first principles, instead f deriving the frm fr thse frces frm energy cnsideratins. They als shw that finite current systems can radiate even during time intervals where the currents are nt changing in time, a result that cannt be understd fr a pint surce, e.g. classic magnetic diple radiatin frm a pint diple. We are thus mtivated t lk at the prperties f the cmplete analytic slutins t tw systems f the type described abve: a planar slenid with the distance between the infinite planes f current given by a; and a rtating, unifrmly-charged spherical with radius a. Fr bth f these prblems, we give the cmplete vectr ptential everywhere in space, fr arbitrary time dependence f the currents. When we g t the slw-mtin r diple apprximatin limit, ur cmplete slutins illuminate prperties f magnetic diple radiatin that are nt accessible in the classic magnetic diple radiatin slutins. Our cmplete slutins als yield new infrmatin which was previusly inaccessible abut the energy radiated fr instantaneus turn-n. We give three prblem statements fr the undergraduate level related t the prperties f these slutins. We give a furth prblem statement, fr a related spherical f current prblem, apprpriate fr a graduate level curse in electrmagnetism. We nte that the kinds f mdels we cnsider here, especially that f the spherical, were at the frefrnt f scientific research ne hundred years ag in the Abraham- Lrentz mdels f the electrn. Far frm being a mribund field f study, the Abraham- Lrentz mdels f the spinning electrn have recently been rediscvered as a fascinating dynamical system []. Hwever, ur purpse here is much mre prsaic than current research addressing these dynamics. Here we assume the mtins are given, and we
Versin Date July 10, 011 3 fcus n causality, reactin frces, and transit time effects in the cnsequent generatin f electrmagnetic fields. The remainder f this paper is rganized as fllws: The planar slutin is derived in Sec., and the slutin fr the rtating spherical is sketched in Sec. 3. These slutins are used in Sec. 4 t examine issues f prpagatin, radiatin withut acceleratin, what is hidden by the slw-mtin r diple apprximatin, energy cnsideratins, lcal reactin frces, and mre. Sec. 5 gives a summary f results and insights, and Sec. 6 presents prblems suitable fr curses in electrmagnetism. The planar slenid The prblem f the fields f an infinite parallel current sheet changing arbitrarily in time has been treated by many authrs [3]. We give a brief derivatin f thse fields here, using an apprach which can als be applied t the prblem f the rtating spherical. The differential equatin fr the vectr ptential A in the Lrentz gauge is 1 A A J (1) c t In the case f a current sheet in the y-directin lcated at x 0, ur current density is given by J x, t t x y () ˆ where t is the current per unit length and is an arbitrary functin f time. Fr this current density, the vectr ptential A has nly a y-cmpnent. With Eq. (), Eq. (1) becmes 1 Ayx, t ( t) x (3) x c t The general slutin t the hmgeneus equatin in Eq.(3) is f the frm G( t x/ c) H( t x/ c) (4) where G and H are any functins f a single variable. T find the particular slutin t Eq.(3) with the inhmgeneus surce term, we build that slutin ut f slutins t the hmgeneus equatin. Using causality, we assume a functin prpagating t the left fr x 0 and t the right fr x 0. Cntinuity at the rigin then requires that y, / A x t G t x c (5) T determine G(t), we nte that Eq. (3) implies that we must have a jump in the z- cmpnent f the magnetic field (r in the x derivative f A ) acrss x 0 given by y
Versin Date July 10, 011 4 Ay Ay dg( t) Bz Bz t x x x x c dt x x (6) We have derived the last term n the right f Eq.(6) using Eq.(5) t evaluate the left-mst expressin in Eq. (6). Thus we have where c (1) x Ay x, t tc c t (1) t tdt 0 (7) (8) We nte that ur slutin fr the vectr ptential depends n the first time integral f the current as a functin f time, and nt n any f its time derivatives. We will see a similar behavir in the rtating spherical case, except we will have terms up t the third time integral f the current as a functin f time in that case (cf. Eq. (19) belw). The electric and magnetic fields, given by the spatial and tempral derivatives f the vectr ptential, will thus be prprtinal t t and nt t any f its time derivatives [4]. Thus the current sheet can radiate energy t infinity even when the current per unit length t is cnstant in time. One might suspect that this strange behavir (radiatin withut acceleratin) is smehw a feature f the infinite planar gemetry, and wuld nt appear in mre realistic gemetries. Hwever, we will btain the same behavir in the case f a rtating spherical f charge, as we discuss belw in Sec. 4. In that discussin we explain why this behavir (radiatin withut acceleratin) is in fact understandable physically in bth cases. We nw use superpsitin t write dwn the slutin fr a planar slenid. By planar slenid we mean that we have tw current sheets in the yz plane, ne lcated at x a and ne lcated at x a, with the current sheet at x a given by y ˆ () t and the current sheet at x a given by y ˆ () t. The slutin fr this case is a superpsitin f the tw slutins f the type given in Eq. (7) and can be written in the frm c (1) xa (1) xa A( xt, ) y ˆ t t (9) c c
Versin Date July 10, 011 5 3 The rtating spherical f charge We turn t the prblem f a unifrmly-charged rtating f radius a. We assume that the radius f the sphere and the distributin f charge remain cnstant even as the sphere rtates. This is impssible, f curse, since there are n materials that are perfectly stiff and perfectly insulating, but this is nt relevant here since ur fcus is n causality in the generatin f electrmagnetic fields. Our rtating carries current in the azimuthal directin, with the current depending n the sine f the plar angle. Slutins t this prblem in a very different frm frm urs have been given by Dabul and Jensen, and als by Vlasv [5]. Our current density is given by ˆ J r,, t t ( ra)sin (10) If the sphere has charge per unit area and rtates with an angular speed t, then t a t. Fr future use, we nte that the magnetic diple mment f the current distributin given in Eq. (10) is 3 1 3 4 a m t (, t) d x = ˆ t r J r z (11) 3 If the sphere has a ttal charge Q and is rtating at a fixed angular rtatin rate we have that the static magnetic diple mment m is given by, then m 4 3 a 0Q a (1) 3 3 The vectr ptential A fr the current density given in Eq. (10) nly has a -cmpnent, and A satisfies the differential equatin 1 1 1 1 ra sin A A J r r r sin c t (13) Fr the sin dependence f J in Eq. (10), A (, r,) t can be written in separable frm as A (, r,) t A(,)sin r t (14) We find the differential equatin fr Art (, ) by inserting Eq. (14) int Eq. (13) and using Eq. (10), yielding 1 r r c t rarara at (15)
Versin Date July 10, 011 6 The reader shuld cmpare Eq. (15) fr the spherical case t Eq. (3) fr the planar case. The general slutin t the hmgeneus equatin in Eq.(15) is f the frm cg( t r / c) ch ( t r / c) r A(,) r t G( tr/ c) H( tr/ c) (16) r r As in the planar case, we find the particular slutin t Eq.(15) by assuming the frm in Eq. (16), but nw we use causality t assume prpagatin bth inward and utward fr r a and nly utward fr r a, and we require regularity f the slutin at r = 0. We then require cntinuity f A(,) rt at r = a as well as the jump in the r-derivative f A(,) rt at r = a implied by the delta functin in the surce term in Eq. (15). As in the planar case, these requirements suffice t cmpletely determine ur particular slutin. We can als find the slutin t Eq. (15) by Laplace transfrming and inverting. Since the details f either prcess are invlved, and we are mainly interested in the prperties f the slutins in any case, we relegate the details f bth the time-dmain and the Laplace transfrm apprach t a technical nte [6], and nly state the results here. We use Eq. (11) t write ur slutin in terms f mt instead f t, t facilitate cmparisn with the classic magnetic diple slutins, which invlve mt and its time derivatives. If we define and r min( r, a) r max( r, a) (17) r r t t c c (18) then we can write the slutin fr the vectr ptential f ur spherical, which nly has a cmpnent, as (1) (1) c () () m ( t) m ( t ) m ( t ) m ( t ) 3 sin r c (,,) 8 ar c () () c (3) (3) m ( t) m ( t) m ( t) m ( t) r r A r t (19) where we have defined the successive integrals f the diple mment 0 mt by 3 (0) 4 a m t m t t (0) 3 t ( n 1) ( n m t m ) tdt n0,1,... (1)
Versin Date July 10, 011 7 Nte that even fr a cnstant magnetic diple mment, the expressin fr the vectr ptential in Eq. (19) appears at first glance t be a functin f time, because the successive time integrals f a cnstant mt in Eq. (1) are functins f time, even if the magnetic diple mment itself is nt. In fact, this is nly a prblem at first glance, and a clser examinatin f Eq. (19) reveals the fllwing. Suppse that the magnetic diple mment mt is zer fr t 0, then varies in the time interval 0 t T in an arbitrary manner, and then reaches a cnstant value m fr t T. Then it can be shwn that [6] ra 0 t c (,,) msin r r a t T 4 a rr c A r t () At very lng times, this is the crrect slutin fr the vectr ptential f a unifrmlycharged spherical spinning at a cnstant rate, as we expect. We nly see time changes in the vectr ptential at a given radius r in the time interval r a r a t T (3) c c Fr instantaneus spin-up (T = 0), if r athe bserver will nly see fields changing in time ver a time a/ c, and if r a, the bserver will nly see fields changing in time ver a time r/ c. Fr future use, we als define dm 1,... (4) dt n ( n) m t n n We will switch back and frth between the ntatin in Eq. (4) and the ntatin in which we use dts ver the functin t dente time differentiatin, fr example dmt ( ) mt m t (5) dt The assciated electric and magnetic fields can be fund by taking the negative time derivative and the curl f A, respectively. We give the explicit frms fr these fields in [6]. We als shw in [6] that the slutin abve in Eq. (19) is identical t the ne given by the authrs in [5], althugh upn first examinatin they lk very different.
Versin Date July 10, 011 8 4 Discussin f the cmplete field slutins The remarkable thing abut ur slutin fr the spherical ptential in Eq. (19) is that it des nt invlve any time derivatives f the magnetic diple mment mt ( ). Instead, the vectr ptential and the assciated fields are prprtinal t mt ( ) itself and its time integrals, evaluated at tw different times. In the cntext f ur experience with the classic thery f magnetic diple radiatin, which we review belw, this is a ttally unexpected result. There are three bvius questins that arise. (a) Can we recver the classic magnetic diple radiatin frmulae frm ur cmplete spherical slutin when we g t the slw-mtin r diple apprximatin limit, where we assume that the speed f light transit time acrss the sphere is negligibly small cmpared t the time scale fr variatin in the surce? (b) Can ur cmplete slutin t the spherical case when cnsidered in the slwmtin r diple apprximatin limit, illuminate features f magnetic diple radiatin that we cannt answer in the classic slw-mtin r diple apprximatin slutin? (c) Is the planar slenid slutin we give abve cntained in the spherical slutin in sme limit? In particular, can we find situatins in the spherical case, as we d in the planar slenid case, where the sphere radiates even thugh it is spinning at a cnstant rate? If s, what d these radiatin withut acceleratin slutins mean physically? We cnsider each f these three questins in turn. First, thugh, we review the standard develpment leading t classic magnetic diple radiatin. In that develpment, we begin with the general slutin fr the vectr ptential using the free space Green s functin. We then assume that J vanishes utside f a sphere f radius a and that we are far away frm that regin. We als assume that if T is the time scale fr changes in J, then T a/ c (6) We call this apprximatin the slw-mtin r diple apprximatin. It is als called the lng wavelength apprximatin, since in this apprximatin the wavelength f the radiatin generated is much larger than the dimensins f the surce. We assume that J is well behaved, in that we can use a Taylr series expansin fr the time dependence f J abut the time t r/ c. If we als assume that the diple mment m t is always alng the z-axis, then t first-rder in the small quantities a/ rand a/ ct, the vectr ptential nly has a cmpnent, which can be shwn t be [7] classic A r,, t / / sin m t r c m t r c 4 r rc (7)
Versin Date July 10, 011 9 The electric field and magnetic fields f the classic magnetic diple can be fund frm Eq. (7) by taking the negative time derivative f and the curl f the vectr ptential, respectively. We give explicit frms fr these fields in [6]. Nte that in the classical slw-mtin r diple apprximatin fr the ptential, we have shrunk the radius a f the sphere t zer. The nly infrmatin we have abut the spatial structure f the current distributin inside a is its spatial mments, thrugh m () t, but we cannt peer int the structure itself and we have n idea hw the fields behave fr r less than r n the rder f a. The crrespnding ttal rate at which energy is radiated t infinity int all slid angles in classic magnetic diple radiatin is given by lim E B r r classic classic ˆ r sindd m 6 c 3 (8) Nw let us turn t ur three questins abve, first cnsidering questin (a) (can we recver the classic magnetic diple radiatin frmula frm ur cmplete spherical slutin?). It is by n means bvius at first glance that we can d this, since the cmplete spherical slutin invlves mtand its first, secnd and third time integrals, whereas the classic magnetic diple radiatin invlves mtand its time mt are well behaved, derivatives. But let us assume that all f the time derivatives f and that we can expand mt in a Taylr series. If we lk at Eq. (19) and expand abut the time tt r / c, then we can shw using the standard Taylr series expansin that this equatin can be written as r 3c sin r r ar k c r c k k cm t k c 1k (9) A (, r,) t m t 4 k 1 1! Eq. (9) hlds bth inside and utside the spherical. If we assume that r a, then we have r 3c sin k a r ar k c r c k k cm t c 1k (30) A (, r,) t m t ra 4 k 1 1! Keeping nly the first tw terms in Eq. (30), we have A r t a sin mt mt 1 mt mt (,, )... ra 4 r rc 10c r rc (31)
Versin Date July 10, 011 10 where nw tt r/ c. In the limit that a ges t zer, we see frm Eq. (31) that the vectr ptential fr ur cmplete spherical slutins reduces t the familiar expressin fr the ptential assciated with classic magnetic diple radiatin, as given in Eq. (7). Nw let us turn t questin (b) (can ur cmplete slutin t the spherical case illuminate features f magnetic diple radiatin that we cannt answer in the classic slw-mtin r diple apprximatin slutins?). The answer t this questin is a resunding yes. Mst imprtantly, in ur cmplete slutins, we still retain a finite and nn-zer value f a even when the slw-mtin r diple apprximatin hlds. T make clear what this means, if we assume that a/ ct 1, and that mt ( ) can be expanded in a Taylr series, then ur expressin fr the electric field evaluated at r a can be written (setting r a in Eq.(31) and taking the negative time derivative) as E a t a sin mt mt 1 mt mt (,, )... 4 a ac 10c a ac (3) where nw t t a/ c becmes. If expand mt a/ c in a Taylr series abut t, Eq. (3) (,,) () () ()... (33) 4 a 5c 3c 3 sin a 1a E a t m t m t m t Eq. (33) is an expressin fr the electric field at the surface f the spherical that we have n access t in the classic pint magnetic diple radiatin slutin, since there we have taken the limit that a is zer. A very imprtant cnsequence f ur "nn-classical" diple result in Eq. (33) is related t reactin frces, that is, the frces that resist the spinning up f the. In the classical apprach these radiatin reactin frces must be inferred frm energy cnservatin. But ur slutin in Eq.(33) describes the fields acting n the spherical, and therefre gives us the reactin fields directly. T spin up the sphere, the external agents must prvide a frce which exactly cunterbalances the electric field reactin frces given in Eq. (33). With Eq. (33), we can calculate explicitly the rate at which wrk is being dne against this reactin frce by the external agents spinning up the sphere. This rate is given by the vlume integral f J E, where this quantity is zer everywhere except n the surface f the sphere. Explicitly, using Eq. (10) and Eq. (33), we have du dt wrk J E rd 3 a a ( ) ( ) ( ) ( ) ( ) ( )... 3 mtmt mtmt mtmt 3 a 5 c 3c (34)
Versin Date July 10, 011 11 Suppse the magnetic diple mment has been zer fr t < 0. If the diple mment then ges frm zer t a cnstant value in a finite time span, and if the first thrugh third derivatives f mt ( ) are zer at 0 and at T, then the first term n the far right in Eq. (34) will yield the energy stred in the static magnetic field (see Eq. (36) belw), the secnd term will integrate t zer, and the third term will give the ttal energy radiated t infinity (when integrated by parts twice). That is, the ttal wrk dne by the external agent spinning up the sphere is the sum f the energy carried away t infinity by the radiatin and the energy stred in the static field after spin-up. By directly using the reactin frces, and cmputing the wrk dne against them, we have discvered smething new and interesting: the "reactin" frces are nt nly thse necessary t supply the energy radiated, but als t establish the energy stred in the final static magnetic field f the rtating. In additin t giving us new access t the reactin frces when the slw-mtin r diple apprximatin hlds, ur cmplete spherical slutins als illuminate prperties f radiatin frm a sphere when the sphere is spun-up s rapidly that the slwmtin r diple apprximatin des nt hld. The classic magnetic diple slutins have nthing t say in this regime, since t derive these slutins it is assumed that ur spin-up is slw in the sense that T a/ c. In cntrast, ur cmplete spherical slutins d have significant statements t make, since they are derived with n cnstraint n the spin-up time. T illustrate what this means, we cmpute the ttal energy radiated in ur cmplete spherical slutins fr any value f a/ ct, and cmpare it t the amunt radiated in the classic magnetic diple radiatin slutins (cf. Eq. (8)). We chse a frm fr the time dependence f mtwhich has well-behaved time derivatives t all rders, s that we can cmpute the energy radiated in the classic magnetic diple radiatin expressin in Eq. (8). We take m smth m t ( t) arctan 1 T /5 (35) T calculate the ttal energy radiated in the spherical slutin, we need nly keep the inverse radius terms in the electric and magnetic fields assciated with A in Eq. (19), and integrate the assciated Pynting flux ver time and ver the surface area f a sphere at infinity. We thereby numerically cmpute an expressin fr the ttal energy radiated by the spherical during the smth spin-up fr any value f a/ ct. We chse t nrmalize the ttal energy radiated by the energy cntained in ur static fields fr cnstant rtatin rate fr the spherical case, which is given by the frmula U mag m (36) 3 4 a The nrmalized radiated energy fr the smth turn-n fr ur cmplete spherical slutin is pltted as a functin f the rati a/ ct in Figure 1, and labeled Smth. Fr cmparisn, the dashed straight line labeled Diple in Figure 1 is the
Versin Date July 10, 011 1 nrmalized radiated energy using the usual expressin fr the magnetic diple radiatin rate in the slw-mtin r diple apprximatin, given by Eq. (8). If we cmpute m using the time dependence given in Eq. (35) and integrate the rate at which energy is radiated t infinity in Eq. (8) ver all time, we easily see that the classic diple ttal energy radiated scales asa/ ct 3. At lw values f the rati a/ ct, ur spherical slutin fr the ttal energy radiated, the Smth curve in Figure 1, shws this behavir, and is essentially identical with the classical result fr magnetic diple radiatin. Hwever, as the rati a/ ct becmes cmparable t and much greater than unity, ur numerical result fr the energy radiated fr the spherical slutin appraches the cnstant value U. mag Figure 1: The nrmalized ttal energy radiated by the spherical as a functin f a/ ct fr the smth spin-up with characteristic time T. Ramp spin-up is discussed at the end f Sec. 4. The curve labeled Diple is the ttal nrmalized energy radiated by a pint magnetic diple fr the smth spin-up time functin. That is, if we spin up the spherical in a time T shrt cmpared t a/ c, we find that the energy radiated away in this prcess is exactly equal t the energy stred in the magnetic field after the currents reach their terminal values. This is true bth fr ur planar slenid slutins and fr ur spherical slutins. This als turns ut t be true in the spherical case nt nly fr current distributins n the sphere prprtinal t sin, but fr any axially -symmetric current distributin with any dependence n the sphere (see Sample Prblem 6.4 belw). In cntrast, the energy radiated in the classic diple case as predicted by Eq. (8) increases withut limit as a/ ct increases. Of curse this estimate has n meaning nce a/ ct becmes cmparable t ne, since the derivatin f the classic equatins begins with the assumptin that a/ ct 1. Thus ur cmplete spherical slutins illuminate features f the radiatin frm the spin-up f a spherical that are nt accessible in the classic magnetic diple radiatin slutins. This is because we can g t
Versin Date July 10, 011 13 the limit f instantaneus turn-n in ur cmplete slutins, whereas the classic magnetic diple slutins are intrinsically limited t slw turn-n. Finally, let us turn t questin (c) (is the planar slenid slutin cntained in sme sense in the spherical slutin?). T answer this questin, we must let the radius f the sphere a and the distance frm the rigin r g t infinity in such a way that the difference x r a is finite. In this limit, if we are sitting in the equatrial plane where sin 1, at a distance x ra a frm the spherical surface, the spherical surface lks planar. T g t this limit we essentially are invking the reverse f the slw-mtin r diple apprximatin in Eq. (6). That is we are assuming T a/ c, rather than the ther way arund, since we are letting a g t infinity hlding T fixed. We can never apprach this limit in the classic magnetic diple radiatin slutin, because there we have shrunk the sphere away t a pint, and thus we can hardly return t the limit where its radius is finite. But we have n prblem taking this limit with ur spherical slutin and the assciated fields given in Eq. (19). Cnsider the equatrial plane at /, and let r and a g t infinity with x ra a finite. It is straightfrward t see that in this limit, the nly term t survive in the equatin fr (which is the negative time derivative f the expressin in Eq. (19)) is E 3c c (, /,) ( / ) ( / ) (37) 8 ar E r t m t x c t x c 3 where we have used Eq. (11) t replace mt ( ) by 4 a ( t)/3, and we have assumed that mt () is zer in the far distant past. We thus see that in this limit we recver exactly the strange behavir f the planar slenid case, that is, radiatin fields which are nt zer even when the sphere is rtating at a cnstant rate. This result is truly surprising. Even thugh radiatin frm a planar sheet mving at cnstant speed has been discussed previusly by a number f authrs [3], we are encuntering it here fr the first time in a mre realistic (spherical) gemetry. Because we find radiatin in this circumstance t be s nvel, we revisit the issue f hw this can be physically. When the surface is mving at a cnstant speed, the fields in the vicinity f the surface are nt changing in time, yet at and near the surface there is a radiatin field E ; this is incntrvertibly a radiatin field in that this field prduces a radiatin reactin. Even mre surprising is the fact that this lcally static radiatin reactin field at the is assciated with changes in the electrmagnetic field that are happening at a distance, pssibly a large distance, frm the, the distance t the "wavefrnt" f the changes that were started with the spin-up prcess began. One way f understanding (r at least accepting) that E is really a radiatin field is t return t the mre transparent case f the planar sheet f charge and t cnsider Lrentz transfrmatins fr mtin in the y-directin. If the sheet f charge has never been mving, then in the frame in which the sheet is statinary the electric field has nly
Versin Date July 10, 011 14 an x cmpnent E x. A Lrentz transfrmatin in the y directin des nt change this fact. In a frame in which the sheet is mving in the y directin with sme velcity (and has always been mving in the y directin with that same velcity) there is still n E y cmpnent. The E y cmpnent is nt the result f bserving in ne frame r anther. This insight can be applied t the case f a sheet that starts frm rest in the "lab frame," and then settles dwn t sme cnstant nn-zer speed in that frame. The Ey cmpnent cannt be remved by ging t a frame c-mving with the sheet at late time. That E y is irrevcably, irrefutably, and unremvably present. The E y field, a true radiatin field, "knws" nt nly abut the mtin f the sheet, but als abut its histry! These cunterintuitive features f the radiatin underscre the subtlety f the relatinship f radiatin and its surces. T gain additinal insight int the physics, let us examine this frm anther pint f view based n mmentum transfer fr a charged sheet. In the fllwing discussin, we keep nly terms t first rder in v / c, but neglect terms f higher rder. In the planar case, a single current sheet can be thught f as a mving unifrmly-charged sheet with psitive charge per unit area, mving at velcity v t vt y ˆ, with the current per unit length given by t v t. In the discussin f the planar sheet in Sec., we assumed that the mving psitive charge is exactly balanced by an statinary negative charge, s that there is n net electrstatic field (see [4] fr ther pssibilities). But cnsider the case where there is n statinary negative charge, s that we have an electrstatic field. The unbalanced psitive charge per unit area prduces an electrstatic field f magnitude / perpendicular t the sheet in the x -directin. Suppse that at t = 0, we bring this sheet f charge instantaneusly frm zer speed t cnstant speed v. Then at time t = T > 0, fr x ct we will simply see the electrstatic field in the x-directin, and fr x ct we will see bth that field and the radiatin electric and magnetic fields. The rati f the magnitude f the radiatin electric field in the y-directin t the electrstatic field in the x-directin fr x ct is given by (cf. Eq. (37)) E E c v / c y x (38) This rati is exactly what we expect frm simple gemetry. The ft f the electrstatic field line is rted in the charges making up the sheet, and the parts f the field line fr x ct are mving alng with the ft in the y-directin at speed v after t =0. Thus we expect frm gemetry alne t see the rati given in Eq. (38). This is exactly like waves n a string, and we wuld have an exactly analgus situatin if a
Versin Date July 10, 011 15 pst supprting a string is suddenly set in mtin with speed v perpendicular t the string at t =0, except that the speed c wuld be replaced by the speed f waves n the string. Thus the reasn that the unbalanced psitive sheet f charge cntinues t radiate even after it has been brught up t cnstant speed is that even thugh the external agents fr a time just greater than zer have already put in the energy required t get the sheet itself up t cnstant speed, they have t cntinue t d wrk fr t > 0 t bring the electrstatic fields assciated with the unbalanced charges in the sheet up t speed. The rate at which they cntinue t add mmentum in the y-directin per unit area in the yz plane is just the frce that they must exert t cunterbalance the reactin electric field, that is E c E y v x0 c (39) Where des the mmentum prvided by the external agents g? The ttal electrmagnetic mmentum in the y-directin per unit area in the yz plane at time t = T > 0 is EE y E0 ct E B ct EBzcT T v y c c c (40) Thus we see frm Eq. (40) that the electrmagnetic mmentum in the y-directin per unit area in the yz directin is cntinuusly increasing fr t = T > 0, as the radiatin fields mve utward at the speed f light, and that the rate at which the electrmagnetic mmentum is increasing is exactly the rate at which the external agents are prviding mmentum at x = 0 (see Eq. (39)). Even lng after the unbalanced charged sheet itself is up t speed, the sheet cntinues t radiate, and that radiatin is carrying mmentum utward at a rate sufficient t get the mre and mre distant electrstatic fields up t speed. Fr a single sheet this radiatin never stps, because there is an infinite amunt f electrstatic energy t get up t speed. Fr ur planar slenid, there is nly a finite amunt f electrstatic energy per unit area in the yz-directin, and thse fields are up t speed after a time a/ c, s that the sheets cease t radiate after that time. We expect exactly this behavir in spinning up ur spherical f charge. Frm this discussin, it is clear that the situatin in which we find radiatin withut acceleratin fr the spherical case will ccur when the time T in which we spin the sphere up is shrt cmpared t the speed f light transit time acrss the radius f the sphere, and mrever that this radiatin withut acceleratin shuld cease after a time a/ c. This is sme sense bvius. Suppse we take any finite distributin f current and turn the currents n instantaneusly. Far away frm this event we wuld expect t see a burst f radiatin that lasts a time f rder the finite dimensins f the system
Versin Date July 10, 011 16 divided by the speed f light. Thus the current distributin must be radiating after the currents are n lnger changing in time, since the time interval when the currents are changing is zer (r at least arbitrarily small). But even thugh this is bvius in sme sense, we have in Eq. (19) fr the first time a cmplete analytic slutin which allws us t examine the prcess in exact detail, and we explre these details belw. It helps t lk at this situatin numerically. Cnsider a value f a/ ct which is large cmpared t 1, but nt infinity. Fr this purpse, we chse a ramp turn-n fr the time dependence f the magnetic diple mment, that is 0 fr t 0 t mramp () t m fr 0t T T 1 fr t T (41) The ttal nrmalized energy radiated during this ramp spin-up functin is shwn in Figure 1 as a functin f a/ ct (this is the curve labeled Ramp ). In Figure, we take a value a/ ct 10, r T 0.1 a/ c. We plt E, the cmpnent f the electric field f the divided by c, and B, the cmpnent f the magnetic field f divided by, at a time t 0.5 a/ c after this ramp turn-n. Bth f the field cmpnents are evaluated in the equatrial plane at /. Figure is ne frame f a cmplete mvie that can be fund nline [6]. Figure : The nrmalized and cmpnents f the electric and magnetic field f the in the equatrial plane fr a ramp turn-n with T 0.1 a/ c. We shw the fields at a time t 0.5 a/ c after a turn-n starting at t 0.
Versin Date July 10, 011 17 What we see in Figure are apprximately the fields we wuld see arund a single infinite plane f current lcated at x = a with a turn-n f the current per unit length given by ramp turn-n f Eq. (41). In the planar case, at t 0.5 a/ c, the nrmalized electric field wuld be cnstant at a value f 1 in the interval 0.6a x 1.4a, linearly decreasing frm that value t zer at x 0.5a and x 1.5a. Similarly, in the planar case at t 0.5 a/ c, the magnitude f the nrmalized magnetic field wuld be cnstant at a value f 1 in the interval 0.6a x 1.4a, reversing sign acrss the current sheet at x a, and then linearly decreasing t zer at x 0.5a and x 1.5a, respectively. The departures frm this planar behavir in Figure 1 are caused by the spherical gemetry, but the similarities are abundantly clear. We als want t give sme feel fr the verall tplgy f the field at the time shwn in Figure. Rather than simply pltting the cmpnents f the fields in the equatrial plane, we als shw in Figure 3 a line integral cnvlutin representatin [8] f the magnetic field at this time. The line integral cnvlutin methd f displaying fields has an advantage ver field line r vectr field array displays, in that it shws the structure f the field at a reslutin clse t that f the display. In this display, the streaks are parallel r anti-parallel t the directin f the magnetic field. At this time, the magnetic field is nly nn-zer in the interval 0.5a r 1.5a. This is cnsistent with the fact that the infrmatin that the spherical surface at r = a has started spinning prpagates away frm the surface f the sphere at the speed f light, beginning at time t 0, and we are lking at a later time t 0.5 a/ c. We have clr-cded Figure 3 s that the red znes crrespnd t fields assciated with the times at which the currents in the sphere are in the prcess f turning n, and the darker blue znes are the fields assciated with the times at which the currents in the spherical are cnstant in time. Nte that mst f the fields created by the spin-up f the sphere are assciated with times after the sphere is rtating at cnstant angular speed and is n lnger accelerating. Figure is ne frame f a cmplete mvie that can be fund nline [6].
Versin Date July 10, 011 18 Figure 3: A line integral cnvlutin representatin f the magnetic field tplgy fr a ramp turn-n in a time T 0.1 a/ c. We shw the field at a time t 0.5 a/ c after the turn-n starting at t 0. The sphere is indicated by the silver circle. T fllw this prcess f an abrupt ramp turn-n t its end, we shw in Figure 4 a plt f the same quantities as in Figure, fr the same fast ramp-up time scale f T 0.1 a/ c, except nw at a much later time t 3 a/ c. At this time, the electric field f the is zer except in the spatial interval 1.9a r 4a. Inside f the distance 3 r 1.9a, we see zer electric field and a static magnetic field which falls ff as 1/r fr ar 1.9 a and fr r a is cnstant with a value f B /3 in the equatrial plane (crrespnding t an upward cnstant field f z ˆ /3). As expected frm Ampere s Law, we see a jump in B acrss r a f, with the sign f the jump cnsistent with a cnstant current in the psitive -directin. The reasn that the electric field is zer fr r 1.9ain Figure 4 is that at t 3 a/ c, if r 1.9a, the electric field generated by the far side f the rtating sphere has had time t prpagate acrss the sphere and exactly cancel ut the electric field generated by the near side f the rtating sphere. We thus see a burst f radiatin prpagating t infinity with a spatial extent f abut a, crrespnding t a time duratin
Versin Date July 10, 011 19 f abut a/ c, just as we wuld expect. At the time shwn in Figure 4, the external agents respnsible fr spinning up the sphere are n lnger ding wrk, since the electric field at the surface f the sphere is zer. These agents nly d wrk in the time interval 0t.1 a/ c, and in that time they d wrk sufficient t bth establish the static magnetic field and t prvide the energy carried ff t infinity by the burst f radiatin. Fr T 0.1 a/ cthese tw energies (radiated and stred) are apprximately equal, and in the limit that T ges t zer (instantaneus turn-n), they becme exactly equal. Figure 4: The nrmalized and cmpnents f the electric and magnetic field f the in the equatrial plane fr a ramp turn-n with T 0.1 a/ c, shwing the fields at a time t 3 a/ c turn-n starting at t 0 after a T give sme feel fr the verall tplgy f the field at the time shwn in Figure 4, we again shw in Figure 5 a line integral cnvlutin representatin f the magnetic field at this time. We have clr-cded this figure s that the red znes crrespnd t fields assciated with the times when the currents in the sphere are in the prcess f turning n, the darker blue znes are the fields assciated with the times when the currents are cnstant in time but the agents spinning up the sphere are still ding wrk, and the light blue znes are the static magnetic field regins, assciated with times when the external agents are n lnger ding wrk. At the pint in time at which Figure 5 applies, the agents respnsible fr spinning up the sphere are n lnger ding wrk. The energy required t finish establishing the static magnetic field everywhere in space and t fuel the burst f radiatin ging t infinity is being transprted utward (and lcally depsited) by the fields in the red and dark blue regins f Figure 5.
Versin Date July 10, 011 0 Figure 5: A line integral cnvlutin representatin f the magnetic field tplgy fr a ramp turn-n in a T a c t a c t. time 0.1 /. We shw the field at a time 3 / after the turn-n starting at 0 Finally, cnsider the ttal radiated energy versus a/ ct fr the ramp spin-up f Eq. (41). The functin in Eq. (41) is, f curse, nt expandable in a Taylr series, and the classic expressin fr the radiated pwer in the diple limit, prprtinal t the square f the secnd time derivative f the magnetic diple mment, cannt be evaluated. Hwever there is n prblem in evaluating ur cmplete spherical expressins using Eq. (41) t cmpute the fields, because we can easily integrate the ramp successive times unambiguusly. We shw that numerical calculatin fr the ttal energy radiated in the ramp spin-up in Figure 1, in the curve labeled Ramp. The behavir at lw values f a/ ct is nw prprtinal t ( a/ ct ), and this is understandable analytically (with sme effrt). The behavir at high values f a/ ct is the same as fr the Smth functin, appraching the same cnstant, as we expect, since in this limit bth mramp ( t ) and m () smth t apprach a step functin at t = 0. 5 Summary Studies f the electrmagnetic fields everywhere in space fr current distributins with arbitrary time dependence and at least ne finite and nn-zer spatial dimensin are difficult. Pedaggically, hwever, even in intermediate level curses, we wuld like a student t be able t bserve an abrupt change in current in ne part f such a system
Versin Date July 10, 011 1 causing changes in the lcal fields, and t bserve these field changes as they prpagate at the speed f light t ther parts f the same systems f currents, and interact with thse currents. In this paper, we have given the cmplete slutins everywhere in space fr tw such systems, a planar slenid and a spherical f current. Unexpectedly, we find that ur slutins fr these electrmagnetic fields are free f the cncepts f differential calculus, in that ur slutins nly invlve the currents and their time integrals, and d nt invlve the time derivatives f the currents. These prblems are f physical interest because: (1) they shw that current systems f finite extent can radiate even during time intervals when the currents are cnstant; () they explicitly display transit time delays acrss a surce assciated with its finite dimensins; and (3) they allw students t see directly the rigin f the reactin frces fr time-varying systems; (4) they allw fr the direct calculatin f the rati f the energy radiated t the energy stred fr arbitrary time dependence. We present fur prblems belw which bear n this understanding, three f which are apprpriate fr the undergraduate level, and the furth f which is apprpriate fr a graduate level curse in classic electrmagnetism. 6 Sample Prblems 6.1 The planar slenid and Faraday s Law This prblem is apprpriate fr an intermediate level undergraduate curse, that is, the typical junir/senir level curse taken by physics majrs. The prblem prbes the quasi-static apprximatin fr the magnetic field f the planar slenid. In this apprximatin, we ignre prpagatin effects and cnsider the magnetic field inside the slenid t be spatially unifrm and the magnetic field utside the slenid t be zer. Since we have the exact slutins fr the planar slenid, we can cmpare thse exact slutins t the fields in the quasi-static apprximatin and draw cnclusins abut the validity f the apprximatin. In particular, we will cmpare ur exact slutins t the calculatin f the electric field at the surface f the slenid using Faraday s Law applied assuming the quasi-static apprximatin fr the magnetic field. We have tw current sheets in the yz plane as described abve. We assume that in a previus prblem the student has been led thrugh the derivatin f Eq. (9) abve. (a) Assume that t a gd apprximatin the magnetic field as a functin f time and space is given by B( x, t) ( ) zˆ t fr x a and zer therwise. With this assumptin, use Faraday's Law t find the electric field at x a. (b) Find an expressin fr E ( x, t) frm the exact slutin fr the vectr ptential in Eq. (9). Assuming that the time scale T fr significant variatin in () t is much greater than a/c, expand yur exact expressin fr E using Taylr series, keeping nly leading rder
Versin Date July 10, 011 terms in the small quantity a/ct. Shw that if yu keep terms t this rder, then the exact slutin fr the electric field assuming T a/ c reduces t the electric field yu btained in (a) at x a. 6. The planar slenid prblem cmparing radiated and stred energy We have tw current sheets in the yz plane as described abve. We assume that in a previus prblem the student has been led thrugh the derivatin f Eq. (9) and the assciated fields. (a) Assuming that T a/ c, shw that fr x a Eq. (9) can be written t leading rder as x A( xt, ) y ˆ ca t (4) c (b) Assume the functinal frm fr ( t) is similar t the smth turn-n frm given in Eq.(35), and that T a/ c. Using yur slutin fr the vectr ptential in Eq. (4), and its assciated fields, cmpute the rati f the energy radiated away t infinity per unit yz area f the slenid t the energy in the final magnetic field per unit yz area f the slenid fr t >T after the current has stpped increasing in time (Hint: d 5 ). [Answer: a/ ct ]. 1 (c) Nw assume that ( t) is a step functin, that is, ( t) ges instantaneusly frm 0 t at t = 0. Use the fields derived frm yur exact slutin fr the ptential in Eq. (9) t cmpute the same rati as in (b) abve. [Answer: 1] 6.3 Reactin frces fr the unifrmly-charged rtating spherical We have a spherical carrying a current as described in Eq.(10). We assume that in a previus prblem the student has been led thrugh the derivatin f Eq. (19) using the time dmain methd (see the develpment in [6]). (a) In the case that the time scale T fr significant variatin in mt ( ) is much greater than a/c, use Taylr series expansins f the exact slutins in Eq. (19) t shw that the electric field at r a is given by Eq. (33). (b) Assume that current is turned n ver a time T with t 0 fr t < 0 and t fr t > T. Assume als that the turn n and the leveling ff t cnstant current are smth in the sense that the first thrugh third time derivatives f t vanish bth at t = 0 and at t = T. Using Eq. (10) and Eq. (33), calculate the wrk dne (that is, the time and space integral f JE) in the case that the current varies slwly in the sense that the time-scale fr change T is much larger than a/c. Relate yur answer t the energy stred
Versin Date July 10, 011 3 in the magnetic field after t = T (see Eq. (36)) and t the energy radiated away in magnetic diple radiatin between 0 and T. [Hints: the rate at which energy is radiated 3 int all slid angles in magnetic diple radiatin is given by m /6 c, and the magnetic diple mment fr this prblem is given in Eq.(11)] (c) Nw assume that ( t) ges instantaneusly frm 0 t at t = 0. Cnsider nly the radiatin terms in the electric field assciated with the ptential in Eq.(19), that is the terms ging as 1/r, and fr these terms assume that the crrespnding B field radiatin terms are in magnitude just the E field radiatin terms divided by c. Using yur exact slutins, cmpute the energy radiated away t infinity fr this prcess f instantaneusly spinning up the sphere, and shw that it is equal t the energy stred in the magnetic field lng after the current has stpped increasing in time. 6.4 The spherical graduate level prblem Cnsider a current carrying in which the current in the -directin depends n the plar angle in the fllwing manner. m where P cs where l,, ˆ ( ) 1 cs J r t t r a P l (43) is an assciated Legendre plynmial. (a) Shw that the slutin fr the vectr ptential A fr this prblem is given by ( n) m l l km m ( km1) 3c 1 lk, lm, c ( 1) m ( t ) A (,) r t P 3 l cs k m l1 ( km1) (44) 8 ar k0 m0( r) ( r) ( 1) m ( t) t is defined in Eq. (0) and Eq. (1), and l,m (l m)! m!(l m)! (45) (b) Hw wuld yu slve the general prblem fr any azimuthally symmetric current distributin n a spherical by superpsitin f the slutins given in Eq.(44)? (c) Nw assume that ( ) t ges instantaneusly frm 0 t at t = 0. Cnsider nly the radiatin terms in Eq.(44). Using yur exact slutins, shw that the energy radiated away t infinity fr this prcess f instantaneusly spinning up the sphere is equal t the energy stred in the static magnetic field a lng time after the current has stpped increasing in time. That energy in the static magnetic field fr general l is given by
Versin Date July 10, 011 4 U mag l l l l1 9 m 1 8 a 0 3 (46) (d) Argue frm energy cnsideratins that fr any azimuthally symmetric current distributin n a spherical, the energy required t establish the currents in a time very shrt cmpared t a/c must be apprximately equal t twice the energy stred in the magnetic field after they the currents are established. (e) Can yu think f a way t shw (d) directly frm yur slutin in Eq. (44), rather than relying n an energy argument? 7 Acknwledgments RHP acknwledges supprt frm NSF grant PHY0554367. 8 References [1] Jacksn, J. D., Classical Electrdynamics, nd Editin (Jhn Wiley and Sns, 1975) see Chapter 16. [] H. Sphn, Dynamics f charged particles and their radiatin field (Cambridge University Press 004); W. Appel and M. Kiessling, "Scattering and Radiatin Damping in Gyrscpic Lrentz Electrdynamics," Lett. Math. Phys. 60, 31-46 (00) and "Mass and Spin Renrmalizatin in Lrentz Electrdynamics," Annals Phys. 89, 4-83 (001); M. Kunze, "On the absence f radiatinless mtin fr a rtating classical charge" Adv. Math., 3, 163-1665 (010). [3] R. P. Feynman, R. B. Leightn, and M. Sands, The Feynman Lectures n Physics, Vl. II, Sec. 18-4 (Addisn-Wesley, Reading, MA, 1964). See als: J.-M. Chung, Revisiting the radiatin frm a suddenly mving sheet f charge, Am. J. Phys. 76(), 133-136 (008); Barry R. Hlstein, Radiatin frm a suddenly mving sheet f charge, Am. J. Phys. 63(3), 17 1 (1995); P. C. Peters, Electrmagnetic radiatin frm a kicked sheet f charge, Am. J. Phys. 54(3), 39 45 (1986), see in particular Eq. (6) fr the electric field f a single sheet f mving charge and the accmpanying discussin in Sec. II f this paper; T. A. Abbtt and D. J. Griffiths, Acceleratin withut radiatin, Am. J. Phys. 53(1), 103-111 (1985), see in particular, Sectin III f this paper. [4] An x cmpnent f the electric field wuld arise frm the scalar ptential and frm the relatinship EA / t. We ignre and hence mit the x-cmpnent f E. Since we treat the charge density as a cnstant in time the scalar ptential and E x wuld be cnstant, and hence irrelevant t ur cnsideratins. Nte that we culd have chsen t have the surface current cnsist f ppsite charge densities driven int mtin in ppsite directins, prducing the same ttal current density as in Eq. (), but withut any charge density, and hence withut any E. These same cnsideratins, with x