Lectue 6 Notes, Electomagnetic Theoy II D. Chistophe S. Baid, faculty.uml.edu/cbaid Univesity of Massachusetts Lowell 1. Radiation Intoduction - We have leaned about the popagation of waves, now let us investigate how they ae geneated. - Physically speaking, wheneve the electic field changes in time, it ceates a magnetic field which changes in time, which ceates an electic field which changes in time, etc. - This ceates a chain eaction that then popagates away fom the system and exists independent of it. - Evey time the fields change non-linealy in time, waves ae adiated. - Any acceleation of an electic chage ceates a non-linealy changing electic field. Theefoe, acceleating chages adiate electomagnetic waves. - The only system that does not adiate is a system whee all the chages ae completely at est with espect to some inetial fame. - In the eal wold, no chages ae eve completely at est. Evey object has some tempeatue, even if vey small, which means its atoms and electons ae constantly expeiencing themal motion. These moving chages adiate waves. - Theefoe, all mateials ae constantly adiating electomagnetic waves because of thei themal motion. - The themal adiation emitted by an object tends to be andom and is often viewed as undesied noise that must be ovecome by a meaningful signal. - Electostatics is an appoximation whee the adiation due to themal motion and othe motion of chages is small enough to be consideed negligible. - Unfotunately, the types of adiation and thei souces ae quite numeous and involved and we only have time in this couse to cove the simplest foms of adiation.. Geneal Solution fo Radiation - Fo the pupose of undestanding adiation, assume we have chages oscillating in fee space. Thee ae no boundaies o mateials pesent. - The Maxwell-Ampee Law states that cuents and changing electic fields give ise to culing magnetic fields: B= 0 J 1 c E t - Define potentials in the usual way, B= A and E= A t, and plug them in: A = 0 J 1 c A t t A A= 0 J 1 c d t 1 c A t
- Because the fields ae defined in tems of the deivatives of the potentials, thee is some feedom in choosing potentials. - Any specific choice of potentials is called a gauge. - The Maxwell equations ae gauge invaiant, meaning that they give the same answe in the end even if we switch fom one gauge to the next. - Let us use the Loenz gauge, defined by: A= 1 c t - This causes two tems in the Maxwell-Ampee Law to cancel, leaving us with: A 1 c A t = μ 0 J Maxwell-Ampee Law in Fee Space in the Loenz Gauge - In the Loenz gauge, the Maxwell-Ampee Law thus educes to a wave equation. - This is an inhomogeneous patial diffeential equation. Its geneal solution is the sum of the solution to the coesponding homogeneous equation and the solution to the inhomogeneous equation: A=A homo A inhomo - The homogeneous equation is just the one whee thee ae no souces, J = 0. We have aleady solved that equation. Its paticula solution is tansvese plane waves popagating in space. - The inhomogeneous equation coesponds to souces ceating fields. - The geneal solution then is just waves that wee aleady popagating along plus new waves and fields that ae ceated by electic chages. - Let us find the Geen function solution to the inhomogeneous Maxwell-Ampee Law. - In the usual way, if thee ae no boundaies pesent, the Geen function solution is: A x,t = 0 4 G x,t,x',t ' J x',t ' d x' d t ' whee the Geen function satisfies: G 1 G = 4 x x ' t t ' c t - Using Fouie Tansfoms and afte much wok, the solution is found to be (the full deivation was done last semeste in Lectue 14). G= 1 x x ' t ' t± 1 c x x ' - This is essentially the static Geen's function times a Diac delta which ensues causality. - Causality means that it takes time fo the effects to popagate. The effect of some souce's
action at a location x' and time t' cannot be felt at the obsevation point x until a late time t = t' + x x' /c. - The plus o minus accounts fo an incoming o an outgoing signal. - Now use this solution fo G to get the final solution fo the potential: A x,t = 0 1 4 x x' t ' t±1 c x x' J x ',t ' d x'd t ' A(x,t)= μ 0 4π 1 x x' J ( x',t 1 c x x' ) d x' - This is essentially the magnetostatics solution but with the time shifted to ensue causality. - Fo the pupose of studying adiation, we only cae about the outgoing signal taveling towads us (the etaded signal): A(x,t)= μ 0 4π 1 x x' J ( x',t 1 c x x' ) d x' 3. Radiation of Hamonically Oscillating Souces - Most adiation systems of inteest involve a localized oscillating souce. We can always buildup a non-localized souce as a sum of localized souces, so the analysis below still has geneal applicability. - A localized souce is a system of electic chages and cuents that all eside within some finite sphee of adius d such that ou obsevation distance fom the souce is extenal to the souce: >> d. - As has been done peviously, the simplest way to handle the time dependence of the adiation is to view the final solution as a supeposition of single-fequency waves. - Theefoe, we only need to solve the poblem fo a single fequency and then in the end we can sum them in the appopiate way to get any abitay solution. - The oscillating souces can be boken down into fequency components: J x,t =J x e i t - Plugging in this cuent density into the geneal solution above, we find: A(x,t)= μ 0 4π 1 x x ' J(x ')e i ω(t x x' / c) d x' A(x,t)= μ 0 4π e i ω t J(x ') ei k x x' x x' d x' - This esult is geneal fo any hamonically-vaying souce. - Note that at this point, J epesents the peak cuent distibution, i.e. the cuent distibution when its oscillation in time hits its peak. We have aleady explicitly extacted the hamonic time-dependence of J and bought it out font.
- To investigate the behavio of a localized hamonically oscillating souce, we will expand this equation into a seies and use seveal limiting pocedues to only keep a few tems. - We will do this in diffeent egions of space defined by thei poximity to the souce. - These egions only have meaning if the associated wavelength of the oscillation of the chages, λ = πc/ω, is much lage than the dimensions of the souce, λ >> d. This is called the long-wavelength appoximation. If we wee to not make this appoximation, we would have to take into account the fact that the waves fom one side of the souce can tavel ove to the othe side of the souce and scatte off of it. d λ k Nea Field Intemediate Field Fa Field - Define the nea-field egion as the egion of space that is vey fa fom the souce but still much smalle than a wavelength of the light: d << << λ which leads to k << 1. - Define the intemediate-field egion as the egion whee the obsevation distance is on the ode of the wavelength of the light: λ. - Define the fa-field egion as the egion whee the obsevation distance is much geate than the wavelength of the light: >> λ which leads to k >> 1. 4. Nea-Field Radiation - In the nea field, the exponential becomes one and the solution becomes: A(x, t)= μ 0 4 π e i ω t J (x ') ei k x x ' x x ' d x' A(x,t)= μ 0 4π e i ω t 1 J(x ') x x' d x' Nea Field Solution - The spatial pat is the exact same solution as found in magnetostatics. - In the nea field, theefoe, the fields espond instantaneously to the souce. The poblem is essentially the static solution, vaying hamonically in time. - Because the methods fo solving magnetostatics poblems have aleady been coveed, thee is not much left to be said hee. - In the nea-field, electostatic (capacitive) and magnetostatic effects dominate. 5. Intemediate-Field Radiation - In this egion, we can not make any appoximations. - The exponential and invese distance factos ae expanded in vecto spheical hamonics and integated tem by tem.
- The methods of vecto spheical hamonic expansion ae complex and involved. Thee is insufficient time in this couse to cove this subject to any satisfaction. - In the intemediate-field, the inductive effects dominate. - In the fa-field, adiation effects dominate. 6. Fa-Field Radiation - The fa field is defined as k >> 1 (o witten equivalently as >> λ). - The long-wavelength appoximation tells us that λ >> d. Since all points ' in the souce ae contained within the sphee of diamete d, this also means that λ >> '. - Combining the long-wavelength appoximation and the fa-field appoximation, we theefee see that >> ' (o equivalently 1 >> '/ ). - Thus any time we have a sum of diffeent powes of '/, we can dop the highe powes of '/ since they will be vanishingly small. - Let us use the notation = x and ' = x'. - Using the law of cosines, we can expand the sepaation distance: x x' = ' ' x x ' - We now ty to get eveything in tems of powes of '/ so that we can dop the highe powes. x x ' = 1 ' ' x x ' - Expand this equation into a binomial seies using 1 u=1 1 u 1 8 u... : ]... x x' = 1 1 [ ' ' x x' ] 8[ 1 ' ' x x ' - Because of '/ << 1, we can dop all tems except the fist two. (If we kept only the fist tem we would end up with dipole adiation. Howeve, ight now we ae tying to get a geneal expession in the fa-field that is valid fo all multipole moments.) x x ' = 1 ' x x ' x x ' = ' x x ' - Using this appoximation, the solution fo the vecto potential in the fa field becomes: A(x, t)= μ 0 4 π e i ω t J (x ') ei k x x ' x x ' d x' A(x,t)= μ 0 4π e i ω t J(x ') ei k( ' x x ') ( ' x x') d x '
i k ' x x ' e A(x,t)= μ 0 4π ei (k ω t) J(x ') ( ' x x ') d x' - Because of '/ << 1, we can dop the second tem in the denominato, leading to: A(x,t)= μ i (k ω t ) 0 e J(x ')e i k ' x x' d x' Fa Field Solution 4π - The facto outside of the integal tells us that we have taveling waves popagating spheically outwads in the fa field. - Note that the integal does not depend on the obsevation distance. At the same time, the integal is the only pat of the equation that does depend on the pola angle and azimuthal angle. Theefoe, the integal specifies the angula adiation patten (o antenna patten ). Once a wave is in the fa-field, its angula patten stays the same as the waves tavels outwads. - Also note that that cuent density J in the integal is just the spatial pat of the cuent density (i.e. the peak cuent density) because its time dependence has aleady been extacted and witten out font. 7. Electic Dipole Radiation in the Fa Field - Let us expand the exponential in the integal accoding to e x =1 x 1 x... A x,t = i k t 0 e J x' e i k ' x x' d x' 4 A x,t = i k t 0 e J x' 1 i k ' x x' 1 4 i k ' x x '... d x ' - Each powe of k' in the expansion coesponds to one of the multipole moments of the souce. - Accoding to the long-wavelength appoximation k' << 1 so that highe powes of k' ae vanishingly small. That means that we only need to keep the fist few multipole moments and we will have a good solution. - The fist tem, o magnetic monopole tem is: A x,t = i k t 0 e J x' d x' 4 - In magnetostatics, the magnetic monopole integal is zeo because the cuents do not divege. - In electodynamics, the cuents can divege, so thee can be a non-zeo monopole tem. - Pefom an integation by pats on the integal to find: A x,t = 0 4 i k t e x' ' J x ' d x ' - In magnetostatics, we had ' J x' =0, which made the magnetic monopole go away. - Howeve, in electodynamics, we can have a diveging cuent accoding to the continuity equation:
x',t ' J x',t = t - Using the continuity equation and applying hamonic dependence on time we have: ' J x' e i t = t x ' e i t ' J x' =i x' - Plugging this into the magnetic monopole equation tansfoms it fom magnetic monopole adiation to electic dipole adiation. Of couse, they ae eally the same thing in electodynamics because the electic and magnetic fields ae coupled. A x,t = i 0 4 i k t e x' x' d x ' A(x,t)= i ωμ 0 4π p ei (k ω t ) whee p= x'(ρ(x ')) d x' Electic Dipole Radiation - The popety p is the electic dipole moment as aleady encounteed in electostatics. - Note that the electic dipole moment p and chage density ρ shown in the equations above ae the peak, instantaneous values since the hamonic time-dependence of both has aleady been explicitly factoed out and witten out sepaately. To make this point moe obvious, we could ewite the solution in the fom: A(x,t)= i ωμ 0 4π p(t) ei k i ωt whee p(t)= x'(ρ(x',t))d x' and ρ(x',t)=ρ(x')e - Calculating the fields fom this vecto potential: B= A B= i 0 4 p ei k t B= i 0 4 p ei k t - We now use the identity shown below, which can be poved by expanding eveything into components and emembeing that by we mean = x : [p f ]= x p f B= i 0 4 x p ei k t
B= i 0 4 x p ei k t i k ei k t - In the fa-field, only the last tem contibutes since is so lage, leading to: B= 0 c k 4 x p ei k t - It is impotant to note that the souce is at the oigin and the waves popagate adially outwad, so that the adial vecto and the popagation vecto point in the same diection: x= k. B= μ 0 ck p ( 4π k (k ωt ) ei p) Magnetic Field Due to Electic Dipole Radiation - The unit vecto pointing in the popagation diection used hee must not be confused with the unit vecto in the z diection, although they may be labeled the same. - Again note that the electic dipole moment p shown in the equation above is the peak instantaneous dipole moment, since its hamonic time dependence has aleady been witten out sepaately. - This equation has thee main pieces. - The fist facto (including all vaiables up to the fist paenthesis) gives the oveall stength of the magnetic field adiated by an electic dipole oscillating hamonically. As we would expect, inceasing the stength of the dipole p inceases the magnetic field stength. Pehaps unexpected though is that inceasing the wavenumbe of the oscillation inceases the field stength. - The second facto in the equation (the tem in paentheses) completely detemines the diection of the magnetic field. It is pependicula to the diection of popagation, as expected of a tansvese taveling wave. It is also pependicula to the electic dipole moment vecto. This is expected because the magnetic field should pependicula to the electic field, which should be geneally aligned with the electic dipole moment. - The second facto also modulates the field accoding to k p =sin wee γ is the angle between the two vectos. Fo instance, the popagation diection that is along the axis of the dipole has a 0 angle and theefoe thee ae no waves popagating in this diection. - The last facto in the equation modulates the oveall stength such that it is an oscillating taveling wave that tavels adially outwad and dies in stength as it tavels and speads out. - Let us now find the coesponding electic field. - The Maxwell-Ampee Law in fee space states: B= 0 J 1 c E t - In the fa field we ae extenal to the localized cuent distibution so that J = 0, leading to: B= 1 c E t - Apply hamonic time dependence and solve fo the electic field:
E=i c k B - If we take this geneal esult and apply it to the magnetic field of the electic dipole adiation, we get: E=i c k 0 c k p 4 k p ei k t E=i 0 c k p 4 k p ei k t E=i 0 c k p k 4 k p ei k t - Evaluating the deivative and dopping the highe ode tem, we find: E= k p e k ( k i (k ω t ) p) 4π ϵ 0 Electic Field Due to Electic Dipole Radiation - Again, the unit vecto pointing in the popagation diection must not be confused with the unit vecto in the z diection, although they ae labeled the same. - As expected, the electic field points pependicula to the magnetic field, pependicula to the wavevecto, and is in the same plane as the electic dipole moment vecto. We theefoe have standad tansvese taveling waves. - This equation may be moe undestandable if we align the dipole with the z-axis, use spheical coodinates, and look at only the magnitude of the electic field vecto: E= μ 0 c k p 4π sin θ ei (k ωt ) - This equation tells as that waves ae not adiated along the axis of electic dipoles and ae most stongly adiated pependicula to the dipole. - We can sketch these esults fo bette undestanding. Fo ease of illustation, imagine the dipole oiented along the z axis and imagine we can take snapshots at some fixed adius at some time and then at a late time. The electic field and magnetic field vectos lie tangential eveywhee to the spheical wavefont. B E p E B p
APPENDIX Let us look at the mathematical deivation of simple taveling waves in diffeent coodinate systems. The wave equation in fee space when no souces ae pesent is: E 1 c E t =0 A wavefont is a suface in space along which the electic field has the same phase at a given moment in time. In othe wods, a wavefont is a suface that connects all the adjacent points in space that ae eaching thei peak electic field stength at the same time. Rathe than estict ouselves to the specific cases of waves that have constant phase and amplitude acoss thei wavefonts, we will look at the moe geneal cases of waves that have only constant phase acoss thei wavefonts. With this in mind, we now use the phase plane wave to indicate a wave with plana wavefonts and not necessaily a wave with constant amplitude acoss its wavefont. In addition to the wave equation, the complete set of Maxwell's equations in thei standad fom esticts what types of field pattens can have wavefonts with a simple shape. Howeve, we will not investigate hee all the estictions placed on field pattens by Maxwell's equation. Instead, we seek hee to only get a ough pictue of what electomagnetic waves looks like with wavefonts of vaious shapes. Plane Waves Let us fist look at electomagnetic waves with plana wavefonts. Expand the wave equation into ectangula coodinates and focus on the i th vecto component of the electic field: E i E i E i E i x + y + z 1 c t =0 As should be obvious, thee is complete symmety between the vaious ectangula coodinates. Theefoe, if we align the plana wavefonts with the x-y plane, ou esults ae still geneal. Thus, fo definiteness, we choose the wavefonts to be paallel to the x-y plane and taveling in the positive o negative z diection. To keep the discussion geneal, we elabel the fist two tems in the above wave equation as the tansvese Laplacian of the electic field: t E i + E i z 1 E i c t =0 whee t = x + y The wod tansvese used hee means that that Laplacian only acts on the dimensions along the extent of a wavefont, i.e. the dimensions that ae tansvese to the diection the waves ae taveling. To ensue constant phase acoss each wavefont, the pat of the electic field solution that depends on x and y (which we label E i,t ) must be eal-valued. Since the wave is taveling in the z diection, we expect the electic field's dependence on z and t to be allowed to be hamonic. Putting these concepts togethe, we get ou tial solution: ±i k z i ωt E i =E i,t (x, y)e whee E i,t is a eal-valued function depending only on tansvese dimensions Inseting this tial solution into the wave equation, we find:
t E i, t =(k ω /c )E i,t Combine the two paametes on the ight-hand side of the above equation into one paamete (-κ ), leading to: t E i, t +κ E i, t =0 This equation just specifies the tansvese behavio of the electic field and this is as fa as we want to go. In summay, the paticula solution to the wave equation fo plana wavefonts is: E i =E i,t (x, y)e ±i k z i ωt whee k= ω /c κ and E i,t is eal-valued and obeys t E i, t = κ E i,t We call this a plane wave because fo a given z and t, the electic field has the same phase acoss the entie flat plane in the x and y diections. Conceptually, a plane wave could by ceated by a sheet chage distibution acoss an infinite plane that is oscillating. Cylindical Waves Let us next look at electomagnetic waves with cylindical wavefonts that ae taveling towads o away fom the z axis. We expand the wave equation into cylinde coodinates and elabel the tansvese tems in tems of the tansvese Laplacian: 1 ρ ρ( ρ E i ρ ) + t E i 1 E i =0 whee c t t = 1 ρ ϕ + z Since we ae taking the wavefonts to be cylindical, the pat of the electic field that depends on the tansvese dimensions (the azimuthal angle and z) must be eal-valued. Since the wave is taveling in the cylindical adial diection, we expect the electic field's dependence on t to be allowed to be hamonic, and the its dependence on ρ to be complex-valued. These concepts suggest a tial solution of the fom: i ωt E i =R (ρ)e i,t (ϕ,z)e whee E i,t is eal-valued and R is complex-valued Plugging this fom into the wave equation and using sepaation of vaiables, we get: 1 1 R(ρ) ρ ρ( ρ R(ρ) ρ ) m ρ k z+ ω c ( =0 whee m ρ +k z ) E = i,t t E i, t The equation on the ight just specifies the tansvese behavio of the electic field and it is as fa as we want to go with the tansvese behavio. Fo the equation on the left, elabel the last two tems as the squaed adial wavevecto k accoding to k = k z +ω /c so that the equation becomes: 1 1 R(ρ) ρ ρ( ρ R(ρ) ρ ) 1 m ρ +k =0 Make the substitution u=k ρ and eaange to find:
u R(u) +u R(u) u u +(u m ) R(u)=0 This equation is Bessel's diffeential equation and the solutions ae Bessel functions: R(u)=a J m (u)+b N m (u) which leads to: R(ρ)=a J m (k ρ)+b N m (k ρ) In ode to get taveling waves in the adial diection, R must be complex-valued. Howeve, since the Bessel functions ae eal-valued, the foms above fo R ae not complex-valued. We can tansfom them to Hankel functions which ae complex valued. We can do this because of the sum of any solutions to a linea diffeential equation is also a solution of the diffeential equation. The solution now becomes: R(ρ)=a H m (1) (k ρ)+b H m () (k ρ) whee H m (1) = J m +i Y m, H m () = J m i Y m Fo lage ρ (fa away fom the z axis), the Hankel functions asymptotically appoach the foms: H m (1) ( x)=c 1 x e i x and H m ( ) ( x)=d 1 x ei x Theefoe, fo lage ρ, the adial pat of the electic field solution becomes: R(ρ)=a 1 ρ e±i k ρ In summay, the paticula solution to the wave equation fo cylindical wavefonts fa fom the z axis is: k ρ i ωt e±i E i =E i,t (ϕ, z) ρ whee k= ω /c k z and E i,t is eal and obeys t E i, t = ( m ρ +k z ) E i, t These equations tell us that the paticula solution to the wave equation with cylindical wavefonts fa away fom the z axis is sinusoidal taveling waves that ae taveling outwads o inwads in the adial diection. We call these waves cylindical waves since on a given cylindical suface centeed on the z axis, the electic field has constant phase acoss the suface. Conceptually, cylindical waves ae ceated by an infinitely-long, staight, oscillating line chage. The facto of 1/ ρ is needed in the above equation to ensue consevation of enegy. Since the total enegy caied along by an expanding wavefont must be constant, and since the total enegy is the enegy density times the aea, the enegy density must decease at the ate that the cylinde's aea is inceasing. Since a cylinde's aea is πρh, and thus inceases linealy with adius, the enegy density must be popotional to 1/ρ. Because the electic field is popotional to the squae oot of the enegy density, the electic field stength must be popotional to 1/ ρ. Thus we see that enegy consevation is the explanation of the appeaance of this facto in the solution above.
Spheical Waves Let us next look at electomagnetic waves with spheical wavefonts. If we expand the wave equation into spheical coodinates and elabel the pats that opeate on the tansvese dimensions as the tansvese Laplacian, we get: 1 ( E i )+ t E i 1 E i c t =0 whee t = 1 sin θ θ( θ) sin θ + 1 sin θ Since we ae taking the wavefonts to be spheical, the pat of the electic field that depends on the tansvese dimensions (the azimuthal angle and pola angle) must be eal-valued. Since the wave is taveling in the spheical adial diection, we expect the electic field's dependence on t to be allowed to be hamonic, and its dependence on to be complex-valued. These concepts suggest a tial solution of the fom: ϕ E i =R () E i, t (ϕ,θ)e i ω t whee E i,t is eal-valued and R is complex-valued Plugging this fom into the wave equation and using sepaation of vaiables, we get: 1 R() ω ( R()) l(l+1)+ c =0 whee t E i,t = l(l+1) E i,t The equation on the ight just specifies the tansvese behavio of the electic field and it is as fa as we want to go with the tansvese behavio. Fo the equation on the left, elabel u=k = ω c to find: 1 R(u) u u (u R (u)) l (l+1)+u =0 Now make the substitution R(u)=U (u)/ u to find afte much wok: u U U +u u u +(u (l+1/) )U =0 This is Bessel's Diffeential Equation and its complex-valued solutions ae again Hankel functions: (1) U (u)=a H l +1/ () (u)+b H l+1 / (u) which when substituting back u=k and R( u)=u (u)/ u leads to: R()=a H (1) l +1/ (k ) +b H () l+1/(k ) Again, fo lage (fa away fom the oigin), the Hankel functions asymptotically appoach the foms: H m (1) ( x)=c 1 x e i x and H m ( ) ( x)=d 1 x ei x
Theefoe, fo lage, the adial pat of the electic field solution becomes: R()=a e±i k In summay, the paticula solution to the wave equation fo spheical wavefonts fa fom the oigin is: E i =E i,t (ϕ,θ) e±i k i ω t whee k= ω c and E i,t is eal and obeys t E i,t = l(l+1) E i,t These equations tell us that the paticula solution to the wave equation with spheical wavefonts fa away fom the oigin is sinusoidal taveling waves that ae taveling outwads o inwads in the adial diection. We call these waves spheical waves since on a given spheical suface centeed on the oigin, the electic field has constant phase acoss the suface. Conceptually, spheical waves ae ceated by localized oscillating chage distibutions. The facto of 1/ is needed in the above equation to ensue consevation of enegy. Since the total enegy caied along by an expanding wavefont must be constant, and since the total enegy is the enegy density times the aea, the enegy density must decease at the ate that the sphee's aea is inceasing. Since a sphee's aea is 4π, and thus inceases quadatically with adius, the enegy density must be popotional to 1/. Because the electic field is popotional to the squae oot of the enegy density, the electic field stength must be popotional to. 1/. Thus we see that enegy consevation is the explanation of the appeaance of this facto in the solution above.