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# Classical Elect & Magnetism PHYS 4620

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Notes for Phys 4620 Last revised 051105 These notes are just intended to give an overview of the major equations covered in class 1 Electrodynamics At last we consider timdependent phenomena in electromagnetism In doing so we nd that the electric and magnetic elds are related through timdependence We will arrive at it though the phenomenon of electromagnetic induction and Faraday s law77 but to discuss this we need to learn more about how current ows in a wire Yup7 real currents 11 Electromotive Force The current density within a conductor is empirically proportional to the electric field within the conductor J TE 1 which is the general form of Ohm s law The proportionality factor a is the conductivity of the material its reciprocal p 10 is the resistivity of the material Note7 under conditions where charge is flowing the electric field can be non zero within a imperfect conductor When we discuss the total current I owing from one electrode to another we nd that it is proportional to the potential difference V between them v IR 2 For steady currents the charge density within the conductor is zero excess charge must again lie on the surface A model for the motion of electrons which relates their drift velocity to the electric force which they feel and hence the current density to the electric field in the material gives 2 J A 3 2mvthenmi where n is the number density of molecules7 f is the number of free electrons per molecule7 is the mean free path for the electrons7 q is the electron charge7 m is their mass and uthenml is their thermal speed The power delivered to a resistive material by a current owing through a potential difference V is P Vi 2R 4 Figure 1 Simple Circuit moves through a magnetic eld an emf is set up by the magnetic force on the moving Charges Electromotive force is a name given to the line integral of the force f which drives the current around a circuit 5 fl f d1 5 For electrostatic conditions where a battery supplies the force which drives the current around a circuit7 the potential difference between the terminals of the battery is also equal to the emf of the circuit V7bEdl 6 E can be viewed as the work done per unit charge by the source 12 Motional emf A generator exploits the phenomenon of motional emf which can be understood using the ideas already studied in the course since it only involves steady elds and currents though the circuit is in motion A very simple generator is shown in Fig 1 Here there is a uniform magnetic eld and a rectangular loop of wire with resistance R moves through it with speed 1 as shown The emf set up in the wire comes from the magnetic force on the charges in the wire and is given by jifmagdlvBh and it is related to the current in the wire by E IR The work done on the charges comes from the force which is dragging the loop The emf can expressed in a nice way using the magnetic ux through the loop ltIgtBda 7 Then one can show M E 7 7E 8 The ux rule 8 is more general than our basic example it is true for wires of arbitrary shape moving in arbitrary directions in non uniform magnetic elds The wire can even change its shape as it moves We can have cases of motional emf where we can t use the ux rule The generator shown in example 74 homopolar generator of the text is a good example In general a conductor which moves through a B eld has eddy currents set up in it which dissipate energy V Move Magnet l I B La LB R a b Fl ure 2 Two other thought experiments a Loop is stationary magnet moves b Both are stationary but magnetic eld changes strength 13 Electromagnetic Induction The motional emf phenomenon discussed in the last section did not require any new eld equations other than the ones given so far in the course But two variants of the loop experiment in the last section do require a new equation These are shown in Fig 2 In both of these a current ows in the wire but the since the loop is not moving its charges can t experience a magnetic force If it s a force from an electric eld7 where does E come from Whatever is driving the current7 the same ux rule holds in this case 5 7 What is needed is a new law of electricity for the case of magnetic elds which change in time It is 8B v x E 7 g 9 which does give the ux rule and also reduces to V x E 0 for the case of constant magnetic elds Predicting the direction of the induced current in a loop is made easier with Lenz s law7 which states The induced current will ow in such a direction so as to cancel the original change in ux 14 Inductance Now consider two loops of wire7 1 and 2 The current in loop 1 will give a magnetic eld in the vicinity of loop and hence a ux through loop 2 The magnetic eld produced by 1 is iamp lei 131747111 12 ltIgt2B1da2 2 M2111 where M21 amp jg jg 47139 m and the ux through 2 due to 1 is From these one can show that so that M21 is a factor which just depends on geometry We note that M21 M12 so that we can just use the symbol M for the mutual inductance of the two loops The the emf induced in 2 due to changes in current in loop 1 is dltIgt2 dll 7 7M7 dt dt 2 11 It is also true that changes in current in a loop induce an emf in the same loop 1 g 7 i 12 dt Regarding this formula7 we often say that a changing current in a circuit generates a backiemf in the circuit 15 Energy in Magnetic Fields When we increase the current in a circuit we are doing work against the back emf which is induced If we start with zero current in a circuit with self inductance L and over time build it up to a value I7 the work done is W 12 13 One can show that more generally the work done in setting up a volume current distribution J is WAJdr 14 V This expression can be rewritten as an integral over all space involving only the B eld 1 W 7 B2 dT 15 2H0 all space 16 Maxwell s Equations The equations governing the E and B elds before Maxwell were thought to be 1 i V 39 E p 60 ii V B 0 8B 111 V x E 7 75 iv V x B uOJ As it turns out7 these equations are inconsistent They may be xed up by replacing the last of these by 8E V x B J 7 0 060 at The correction gives the important result that a changing electric field will induce a magnetic eld The Maxwell equations are 1 VE 7p 16 60 VB 0 17 8B VxE 7 75 18 8E VXB HOJ l HOEOE 19 Together with the force law7 F qE V x B they give the entire content of classical physics7 at least the part dealing with electrical forces Newton s law of gravity is needed for the gravitational force 17 Magnetic Charge There is a symmetry about the Maxwell equations that is upset by the fact that there is no magnetic charge nor a current of moving magnetic charges If there were7 the Maxwell equations would be i 1 8B 1 V E 605 111 V x E 71101 7 E H i 8E 11 V B IMO7 1V V X B M0Je l 0605 and both kinds of charges would be conserved 8pm 805 J if J 7 V m at V 5 825 But no one has yet found any magnetic charge7 and they ve lookedl It turns out that the existence of magnetic charge would explain why charge is quantized 18 Maxwell s Equations in Matter The Maxwell equations 16 719 also hold in matter but there we want to make a distinction between bound and free charges and currents and in doing so we make use of the new elds D and H We now have to consider time derivatives of the induced charges and currents To deal with this we need to introduce the polarization current JP 8P J 7 20 p at lt gt so that in the Maxwell equations we will substitute for p PPfPbPf V P and for J 8P JJfJbJpJfVgtltME With these substitutions and some algebra we get V 39 D Pf 21 V B 0 22 8B E if 23 V x at 8D H J 7 24 V x f at These equations are every bit as true as 16 719 they only use a division of the charge and current into bound and free parts 19 Boundary Conditions Finally7 we recall the boundary conditions on the elds7 already derived 7 Elli32L af Bf7B O ELEQ 0 Hllngfo 2 Conservation Laws Charge Energy and Momentum 21 Introduction This section deals with the role of energy7 momentum and angular momentum in EM These quantities are conserved but we need to know how to nd the energy and momentum associated with the electric and magnetic elds One conservation law is one we ve already seen is the conservation of charge and it is worth reviewing Within a given volume V the charge contained is go v pltntgtdr lt25 and the total current owing out through the boundary of V is I8Jda 26 Electric charge is conserved locally this implies that Q 7 or d 2 7 J da 27 dt 5 Using the divergence theorem we get 8p idT7 VJdT 28 A87 V lt gt and since this is true for any volume we arrive again at the continuity equation7 80 7 7V J 29 at The continuity equation is not independent of the Maxwell equations it is a consequence of them Though in the treatment of Chapter 77 continuity was assumed and then the displacement current77 term was discovered 22 Poynting s Theorem Consider a volume V and nd the rate at which the electromagnetic elds do work on the charges contained in that volume Only the E eld does work7 and we rst nd dW W EJdT but using the Maxwell equations and some vector identities one can express the rhs completely in terms of the elds and then one gets 1W 1 1 1 1 777 7ltEOEZ7B2gt77EXBda 30 dt dt V 2 M0 M0 5 which is Poynting s theorem the workienergy theorem as formulated in EM An important new quantity is the Poynting vector 1 S E 7E x B 31 0 whose meaning is that S da is the energy per time crossing the area element da If we de ne the electromagnetic energy density as 1 1 7 E2 3 32 Uem 2 60 1 do gt and uemch as the energy density of the massive mechanical particles then we can write a local version of the Poynting theorem as 8 altumech l Uem 7V s 23 Momentum and Electromagnetism We can get a theorem similar in spirit at least to the Poynting theorem if we consider the net force acting on the particles contained within a volume V This is the same as the rate of change of the total momentum of these particles We start with FEVgtltBpdTpEJgtltBgtdT V V and then use the Maxwell equations and lots of vector identities to rewrite this in terms of the elds alone The results is very hairy so some de nitions are in order to make the result comprehensible We introduce the Maxwell stress tensor 1 1 1 Ti E 60 7 6HE2gt 1313 7 56MB 34 which is a beastie with two indiltces and will be denoted with a double arrow When we dot a vector with T we get a vector out Thus we write lt gt lt gt 8 a39TjIZ Tu V Tb39 z 873739 lzyz lgtygtz With this de nition the total force on the charges in V can be expressed as dpmech d lt F 7 7 i S d T 1 35 dt 60 dt v T T a Here the rst term on the rhs represents the rate of loss of the momentum contained in elds inside 7 pen HOEO s dT V 7 The second term represents the ow of momentum across the surface of that volume We can get a differential form of Eq 35 with the de nition of the electromagnetic momentum density Pam M0608 so that if h is the density of the momentum of the massive particles we can write IDEC 8 lt gt alt mech Pem V T 37 gt and this gives us an interpretation of T 7T is the momentum in thei direction crossing a surface oriented in the j direction per unit area per unit time whewl We also note that S plays a dual role It represents a ow of em energy but is also related to the linear momentum density 24 Electromagnetic Angular Momentum Last but not least we note that we could carry out a similar discussion for the angular momentum of mechanicalem system One nds that the density of angular momentum contained in the em field is 65m r xPem eor x E x B 38 3 Electromagnetic Waves 31 Review of Waves A wave is a travelling disturbance that maintains a xed shape and travels at a constant velocity Except when it isn t Through absorption the size of a wave can diminish if the medium is dispersive that parts of the wave having different frequencies will travel at different speeds and the wave will distort and nally in the case of electromagnetic waves the waves don t travel through a physical medium at all they just travell Aside from that it s a ne de nition for a wave A wave function of the ideal type to be considered has the form mt gltzi ms 39gt that is the variables only appear as the combination 2 i 1225 A stretched string is perhaps the clearest example of how the wave equation arises lf fz t is the displacement of the bit of the string at z away from the equilibrium position at time if one can show that fz t satis es the equation 82 1 82 EW 40 where T U i u T being the string tension and u being the linear mass density of the string Eq 40 is known as the classical wave equation and it admits all solutions of the form fz t 92 i vt and also linear combinations of solutions of this type A harmonic or sinusoidal wave has the form f27 t Acoskz 7 mt 6 41 A is the amplitude of the wave 6 is the phase constant k is the wave number and is related to the wavelength by i 27139 7 We will usually use complex notation for waves for the simple one dimensional wave we would write m t ReiAeMHHW lt42 and do most of the mathematical operations on the complex wave function 1327 t Aeikziwt6 43 In the end7 the actual wave function will be the real part of Understanding harmonic waves helps to understand all waves because any wave can be built up as a linear combination of sinusoidal waves by m 1 flkeikz tdk 44 where using Fourier theory can be obtained in terms of the initial conditions fz0 and fz0 32 Simple Example of Wave With Boundary Conditions The most basic boundary value problem with waves has a harmonic waves has two different one dimensional media with different wave speeds v1 for z lt 07 region 1 and v2 for z gt 07 region 2 The incident 1 wave enters from the left7 going in the 2 direction with wave number k1 There will be a re ected wave R travelling to the right in region 1 with the same wave number k1 and a transmitted wave T travelling to the right in region 2 with wave number kg The frequencies of oscillation of all parts of the system are the same which is why the re ected wave has the same wave number and that gives us 1 k2 U1 2 7 k1 U2 The wave functions in the two regions are AIeiklz t Agei klz t for z lt 0 27 t k 17mg 45 ATe 2 for z gt 0 The usual statement of the problem is Given the amplitude and frequency of the incoming wave 141 and the wave speeds in the regions7 what are the re ected and transmitted amplitudes We need two equations and these conditions are the continuity of the total wave functions in the two regions and the continuity of the miderivative of the wave function 8 8 MW mm 87 0 87 lt46 0 For the one dimensional two region problem7 the solution is 7 U1 7 U2 1 i 2112 AR 7 ltU2U1gtAl and AT 7 ltU2U1gt A1 47 Hz t Acos0eikz t 2 Asm 0eiltkziwtgt y 48 33 EM Waves in Vacuum The electric and magnetic elds in vacuum also satisfy a wave equation of the form of Eq 40 If we start with the Maxwell equations in vacuum 1 VE0 iii VXE7 ii VB0 iv VXBMOEO and apply the curl operator twice to E or B we can substitute time derivatives of these elds and arrive at separate wave equations for the E and B elds 82E 82B 2 i 2 7 V E H060W7 V B HOEO atz We note that each of these contains three wave equations7 ie one for each component of E and B The wave equations 49 give the speed of these waves it is 7 1 xEOHO so that the speed of light can be deduced from the measured quantities 60 and M0 49 8 300 x 10 50 As the basis for our study of EM waves we consider waves of a de nite frequency which travel in the z direction and have no z or y dependence These are monochromatic plane waves and they have the general form m t Emmi 12 t Emmi 51 where the amplitudes E0 and E0 are complex valued constant vectors When these waves are put into the Maxwell equations we nd some restrictions on the vector amplitudes First7 E0z Boz 0 so that the directions of E and B are always perpendicular to the direction of propagation EM waves are transverse Secondly7 the electric and magnetic elds are related via More generally7 we can consider a plane wave propagating in a general direction by making k into a vector for a wave propagating in the direction ofk7 polarized along direction 7 the monochromatic plane waves are Eu t Egalk39r wt 52 1 A 1A Br t iEoedk39r ka x n 7k x E 53 C C 10 where n I O The actual E and B elds for the monochromatic plane wave are Er7 t E0 cosk rwt 6 54 Br t E0c0sk r 7 wt 30 x n 55 34 Energy and momentum in EM Waves The energy density u Poynting vector S and momentum density for a monochromatic plane wave are all time dependent the only sensible thing to consider is the time average of these quantities since the wave frequencies we consider are so large One uses the fact that the average of a cosine squared factor is and for a wave propagating in the z direction7 one nds A 1 A u 60E3 S 660E3Z EEOEgZ 56 The magnitude of the timeaveraged Poynting vector is the intensity of the plane wave I E lt5 CEOE3 57 I is the average power per unit area carried by the wave Since radiation carries momentum to a surface it exerts a pressure on the surface For a perfect absorber the radiation pressure force per unit area is I EOE 58 35 EM Waves in a Linear Medium In a region with no free charge or no free current the Maxwell equations are i VD0 iii VXE7 ii VB0 iv VXH and if in addition it is a linear homogeneous medium then 1 D 6E H 7B u with e and u constant Then we arrive at wave equations for E and B just as before but now the wave speed is given by 1 c 7 59 v n lt gt where EN 71 E 60 60W is the index of refraction of the material Often7 u is close to uo so that n z EEo All the previous results for vacuum carry over with e and u replacing 60 and M0 11 Figure 3 EM wave with wave Vector k1 incident on interface between linear media 1 and 2 36 Re ection and Transmission at an Interface Linear Media The problem of oblique incidence of an EM wave at an interface of two linear media is rather technical but it can be understood from the wave solutions and the boundary conditions and it is an important problem because the rules of optics come out The basic geometry of the plane waves is shown in Fig 3 The incident wave has wave vector k1 There is a re ected wave with wave vector kg and a transmitted wave with wave vector kT The latter two have directions measured from the normal given by 03 and 0T In the two media the speed of light will be 121 and 122 In each medium the frequency of the wave oscillation will be the same but the wavelengths and wave numbers will differ w kwl va kTvg The incident wave is given by 1 A E1rt Eoel v39r wt 13m t 7k1 x E U1 with similar expressions for the re ected and transmitted waves We don t yet specify the polar ization of the incoming wave The generic structure of the boundary conditions guarantees that the phases of the three waves are the same everywhere and that implies that the three wave vectors lie in the same plane also 01 0R 61 known as the law of re ection and s1n0T E 62 sin 01 n2 7 which is known as Snell s law To get more out of the boundary conditions we need to specify the polarization of the incoming wave Griffiths explicitly considers only the case where the polarization direction of the E eld is in the plane of the page as shown in Fig 4 Then using the boundary condition equations on the amplitudes the phase information has already been used and cancels out one can solve for 12 Figure 4 EM wave incident on interface is polarized in the plane of the page the re ected and transmitted amplitudes in terms of the amplitude of the incoming wave After a lot of work one arrives at F resnel s equations7 0473 2 E E E E 63 OR MB 01 0T 3 01 lt gt where 0 04 cos T and E H1712 cos01 2711 One can also get a pair of Fresnel equation for the case where the incoming wave is polarized perpendicular to the plane of the page Griffiths leaves that as a hard exercise For the initheiplane polarization case7 there is some intermediate angle 03 where there is no re ected wave7 ie it has zero amplitude This occurs where 04 B which gives i 1 7 2 sm2 0B B or tan 03 z E 64 711 7117122 52 where the second approximate condition is for the typical case where m z W 37 EM Waves in Conductors In a conductor there is a free current Jf that will enter into the Maxwell equations if the conductor follows Ohm s law then we can use Jf 0E There is also a free charge pf but one can show that in a good conductor the free charge density quickly dissipates and we can take it to be zero Combining the equations similar to the vacuum case7 one gets modi ed wave equations E 82B 8B 2 i 2 7 VEilueWaqzaE VBilueatZ 4107 65 which now contain a term with a single time derivative The equations still admit plane wave type solutions 82 8E 1327 t ma a47m 7 327 t Boei gziwt 66 but now the wave number I is complex we will use I k m Here7 k determines the wavelength of wave but amp gives the length over which the wave is attenuated 13 We can solve for k and a and we get kw 1gt1l12 rew 1gt71l12 67 the distance that it takes the wave to attenuate in amplitude by 1e is the skin depth at7 1 d E In a conductor the E and B elds are no longer in phase The B eld lags the E field by an amount 7 63 7 6E where tan 1rltk 38 Frequency Dependence of Permittivity The discussions so far has dealt with monochromatic plane waves where are idealizations a real wave is a wave packet of nite length Such a wave can be treated as a sum of harmonic waves with different wavelengths by Fourier analysis But now we need to realize that the speed light in a medium can depend on the frequency of the wave If there is such a dependence then we say we are dealing with a dispersive medium In such a medium a travelling wave will not retain its shape In this situation one must be very careful about what we mean by wave speed For a harmonic wave with frequency w and wavenumber k the wave velocity is given by a number which in some cases can be bigger than cl For a wave packet one can show that the envelope travels at a speed dw E which gives the speed of the information and energy ow7 and generally U9 is smaller than 0 U9 One can make a very simple model for the way that the interaction of an EM wave with matter can give rise to a dispersion relation The model has an electron oscillating in one dimension on a spring with natural frequency we and a damping force proportional to the velocity with damping constant 39y The frequency of the EM wave is w After arriving at an expression for the oscillating dipole for the system7 we build up a macro scopic system by assuming that for each molecule in the substance there are fj electrons for which the natural frequency is W and the damping constant is 39ng the number density of molecules is N When we do this7 we get complexivalued polarization for the substance 2 nq Pi E 68 m gwgiwziiww which gives a complex dielectric constant The planekwave type solutions are actually attenuated waves of the form 13239 t Eoeimeikziwt 69 Figure 5 Waveguide extends along the z axis 0r7 a plane wave where the wave number k is complex k km An approximate expression for k is w N612 fj k z 7 1 70 c 2m60w7w27iww The simple model explains 71 generally rises with frequency except in the vicinity of a natural frequency w where it can drop sharply Away from resonances7 where damping can be ignored the model gives a simple formula for n 277160 Luiw N2 n1 q 7sz 2 71 7 7 39 Guided Waves Lastly we consider waves which are confined7 in particular by conducting pipe whose cross section has an arbitrary shape7 as shown in Fig 5 We ll assume the pipe extends along the z axis The relevant boundary conditions for the inner wall are EH 0 BL 0 Again we consider planewave type solutions but now they have the form 1339072472775 1330907 mew 1957 247 27 t Bow y5ilz t where the coefficient of the z t dependent wiggly part has a dependence on x and y the cross section coordinates We put these into the Maxwell equations with the boundary conditions and see how this restricts the functions E0m7 y and B0z7 With E0meEyyE12 B0BmxByyBzi b Y Figure 6 Rectangular waveguide which extends along the z axis The sides are 1 along x and I along y with a gt b we get equations for Em Ey Bm and By Ex we 7 k2 88 Ey we 7 k2 88 B keel 528822 By Ti 52 so that it suffices to specify the longitudinal components Ez and B1 since these equations will then give the other components The longitudinal components satisfy the equations 82 82 2 2 TTg leC klEz 0 82 82 2 2 78278y2wc ilez 7 0 We can solve for the case where E1 0 and such a solution is a transverse electric TE wave Likewise the case where B1 0 gives a solution which is a transverse magnetic TM wave The case where both E1 and B1 are zero is called a TEM wave but this cannot occur for a hollow waveguide For a rectangular waveguide whose cross section has sides a along x and I along y with a 2 b as shown in Fig 6 we solve for the TE modes a solution using separation of variables does the trick If we try Bzm y XYy applying the boundary conditions gives Bzz y B0 c0sm7rza c0smryb where k We 7r2lma2 nWl and n and m are indices for the particular mode solution also called the TEmn mode For mode mm the frequency must be greater than the cutoff frequency wmn cwxma2 nb2 16 Interestingly enough the wave velocities for the various modes are greater than 0 LA 7 C k 1 7 wmnw2 but the energy is transported at the group velocity which is 119 Ex1 wmnw2 U which is lt c 4 Potentials and Fields In this section we will finally find the elds due to time dependent charge densities and currents without the fudges quasi static approximation used earlier To find out what happens in the general case we will have to make serious use of the scalar and vector potentials V and A Note in introducing timekdependence and obtaining the full set of Maxwell equations we actually ignored what happened to the potentials so that is our first order of business 41 Scalar and Vector Potentials Back in electrostatics and magnetostatics we had E 7VV B V x A but in electrodynamics the first of these is no longer true we only wrote it down because E had zero curl but that is no longer true It is true that B has zero divergence so the second of these still holds Even with time dependence we still have a vector potential A such that B V x A Faraday s law tells us that it is the quantity E 8A8t which has zero curl not E so that it is this quantity that we must set equal to the negative gradient of some scalar function V This gives V a new meaning and the new relation between E an the potentials is E 7VV 7 7 72 One can write down a couple equations connecting V and A which follow from the Maxwell equations the simpler of which is 8 1 2 V 7 A if V 8tV E00 but they can be simplified with a particular choice for V and A The freedom to make such a choice is now discussed 4 2 Gauge Transformations The potentials V and A are not uniquely determined by the sources There is always of choice of these functions of space and time which will give the same fields E and B If we have potentials V and A it is always possible to find new potentials V and A which will also work by the formulae 8A A AV V U7E 73 for any function of space and time This is called a gauge transformation There are two important conditions one can impose on V and A so that the equation involving the potentials are simplified The rst is a condition called the Coulomb gauge also transverse or radiation gauge Here the condition we pick is V A 0 7 4 In this gauge the scalar potential is given by Vrt 1 0r7tdT 75 7 4713960 m which looks odd because it has V being determined by the present value of the charge density with no time for propagation includedl However before we nd the elds we have to include A and the A field will take care of the propagation time The second condition is called the Lorentz gauge Here the condition is 8V A 7 7 76 V uoeo at With this choice the messy equations for V and A have symmetrical look 82A 82V 1 2 7 7 7 2 7 7 77 V A 1060 atz qu V V 1060 atz 600 77 If we introduce the d Alembertian operator 82 1 82 2 7 7 2 7 77 E E2 V 060 8252 V c 822 then we get simpler equations for the potentials 1 DZV 76p DZA 7120 78 From now on in the text we will assume that the Lorentz gauge condition holds unless otherwise stated 43 The Retarded Potentials We recall that in the case of static charges and currents we had solutions for the potentials given by Vr 1 prdT Ami MUT 79 7 4713960 m 7 47139 m but what do for the time dependent case where we have to gure in the time for news to travel from the source If we consider a source point at r the time it takes news to travel from that point to the observation point r is 20 so the news left the point r at the retarded time trEtii 80 A reasonable guess as to generalizing 79 for the time dependent case would be to include tr in the argument of the source function inside the integral That would give Vr 1 Lr trdT Ar i L t M 81 7 47139 60 m 7 47139 z where we mean that pr 25 is the charge density at the point r at the time 25 The potentials in 81 which are the correct ones are called the retarded potentials Some comments In electrostatics we also had expressions for the elds E and B as integrals over the sources Coulomb s law and the Biotisavart law But taking those integrals and adding 25 to the argument would not be correct because those elds would not satisfy the Maxwell equations Secondly one can show that the potentials in 81 do satisfy the Maxwell equations and also the Lorentz condition But the proof takes some care because of the complicated nature of the integrals The argument tr contains r and r so taking the space derivatives is tricky Griffiths shows that the Vr t given in 81 does satisfy the condition on V in 77 The proof for A would be similar and he leaves it for you to show that they satisfy the Lorentz condition 44 The Right Equations for E and B Starting from the retarded potentials we can get the fields but the result is not so simple It is Ert 1 pr t iPr t i773r trll 14 82 4713960 12 OL 62 i0 mat Jltrztgt A Brt E l 12 67 de39r 83 two equations which are noteworthy for not having been written down explicitly in the literature until the 60 s 45 The Li nard7Wiechert Potentials We close the chapter by using the formulae for the retard potentials to get the potentials from a moving point charge One might think that a moving point charge should be very simple and indeed we will arrive at a de nite formula for the potentials but the derivation of these formulae is tricky because of the singular nature of the charge distribution The rst thing to specify is the timeedependent location of the charge q It is given by the function wt At a time t we are getting the news of the charges location at some earlier time 25 We are assured that there is only one such time though one needs to think about that At that time the charge was at wtr so that the time of travel of the news was lr 7 wtrlc But the time of travel is the same as t7 tr so that we have tit lr7wtrlc 84 which generally is some equation for 25 which can be solved Note that this is little bit different from the previous usage of 25 where we were given a point r and then we calculated tr t 7 10 from it We then put things into the formula which is not as easy as it looks A careless evaluation of the integral just gives V q47TEOL the static Coulomb potential but evaluated at the retarded position but that isn t correct there are two ways to see why not The rst way is to set up the charge density correctly with a delta function and do the integral carefully The steps will be given in a separate handout on the web page an additional factor arises from the complicated argument of the delta function The second way to see it is to consider the point charge q as the limiting case of a continuous charge distribution and then observing that we get an additional factor from the geometry of points over which the integral is done Of course the additional factor is the same in both cases The upshot is that the potential we get is 1 qt V t if 85 F7 47reomcimv We also nd using Jr tr pr trvtr M0 qCV V A t 7 7V t 86 n WM 62 n lt gt Equations 85 and 86 are called the LienardiWiechert potentials for a moving point charge It is worth pointing out that although there is a correction to the naive delayed potentials the correction factor comes from geometry and not from relativity and in any case the only element put into the general formulae for the delayed potentials was the idea of a delay time for the news One can say this is really one manifestation of relativity but that s a matter of taste 46 Fields of a Moving Point Charge We re not done yet We would like to get the E and B elds from the moving charge considered in the last section First we want to nd E from the V and A in 85 and 86 This involves taking a gradient of V and a time derivative of A but from the complicated dependence of the variables on the retarded time these operation require a lot of care and Griffiths exercises this care over 3 full pages of the book In the end the result is Er t m 3 82 7 v2u m x u x a 87 7 i 7 T 4713960 m u where the vector u is de ned as u E 027V and r V and a are the position velocity and acceleration of the charge at the retarded time The two terms in 87 To get the B eld carefully take the curl of A The result is Br t it x Er t 88 To finish things off we put these into the Lorentz force law to get the total force on moving charge charge Q location r velocity V from another moving charge charge q location r velocity V acceleration a where these are all evaluated at the retarded time It is drum roll 41 OW027v2upx u x a ix 02 7v2umx u x a 89 This equation fin principlei contains all of classical EM But it would not have been a good idea to start with this equation Coulomb s law is easier to handle It s so messy that it s only good for showing off my ability to put equations into ETEX And it looks good doesn t it 20 47 An Important Example In Example 104 Griffiths solves the problem of the elds from a point charge q moving at constant velocity V At t 0 it is at the origin so that its path is given by wt vt Putting a 0 into 87 gives qlteewn 7 4713960 m u3 If one makes the de nition R E r 7 vt which is the vector pointing from the present location of the charge to the observation point7 one can write E as 17 11262 R EUquot7 t 90 3950 17 v2 sin 002 The B eld from the moving charge is given by 1 A 1 BEmegvxE 91 For the case of U lt c the elds reduce to 7 1 1 A 7 0 q A Er7 t 7 47m R2 Brt 7 4 R2 V x R 92 The rst of these is no news since it s just the eld from Coulomb s law But the second of these gives the low velocity approximation for the B eld from a point charge Naively one might have thought it was the exact answer from replacing the Jdr in the Biot Savart law by qv But that is not the correct answer 5 Radiation We now make the connection between electromagnetic wave and the time dependent sources that give rise to them When charges accelerate or when currents change with time a radiation eld is generated charges moving at constant velocity and steady currents do not generate such a eld We will always talk about localized sources near the origin We know that we have electro magnetic radiation when we have an average ow of energy outward at very large distances Since the total power passing out through a spherical surface of radius r is Prj1Sdai70jziExBda and the area of the sphere is 47rrz7 the Poynting vector must decrease in magnitude no faster than 1r2 to get energy transported out to in nity If we only have static elds7 that can t happen from Coulomb s law the E eld decreases at least as fast as 1r2 and from the Biot Savart law a static magnetic eld for localized sources also goes as 1r2 or faster7 and that would mean that S falls off at least as fast as 1r4 Thus7 static elds cannot radiate The sources and the resulting elds have to be timeidependent lndeed7 Eqs 82 and 83 indicate that timekvarying sources with non zero 4 and give rise to E and B elds which go as 1r and thus give rise to an S which goes as 1r2 We begin the discussion with a calculation for two simple but important radiating ystems7 the oscillating electric dipole and the oscillating magnetic dipole This will help to re ne some ideas Then we do a more general treatment

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