EVROPSKÁ UNIE Evropské strukturální a investiční fondy Operační program Výzkum, vývoj a vzdělávání ministerstvo školství MLÁDEŽE A TĚLOVÝCHOVY Název projektu Rozvoj vzdělávání na Slezské univerzitě v Opavě Registrační číslo projektu CZ.02.2.69/0.0./0.0/16_015/0002400 COURSE READER WITH SELECTED PROBLEMS Classical Electrodynamics RNDr. Pavel Bakala, Ph.D. Ing. Alena Bakalová Olga Chadaeva, Ph.D. SLEZSKA UNIVERZITA V OPAVĚ Opava 2019 SLEZSKA UNIVERZITA FILOZOFICKO-PŘÍRODO VĚDECKÁ FAKULTA V OPAVĚ References This Course Reader is based on and compiled from: STOLL, Ivan, Jiří TOLAR a Igor JEX. Klasická teoretická fyzika. Praha: Univerzita Karlova, nakladatelství Karolinum, 2017. ISBN 978-80-246-3545-3. JACKSON, John D. Classical Electrodynamics. 3rd edition, John Wiley & Sons Inc., New York., 1999. ISBN: 978-0-471-30932-1 l TABLE OF CONTENTS 1. ELECTROMAGNETIC FIELD 1 1.1 Maxwell' s equations...................................................................1 1.2 Description of the point charge using Dirac S-function...........................4 1.3 Electromagnetic potentials............................................................6 1.4 Laws of conservation of charge, energy and momentum..........................9 1.5 The equations of electrodynamics in Minkowski space-time.....................12 1.6 Motion of the charged particle in external electric and magnetic field..........14 1.7 Lorentz transformation of potentials and fields, invariants........................16 1.8 Actions for charged particle system and electromagnetic field..................18 1.9 Problems..................................................................................23 2. ELECTROMAGNETIC WAVES 25 2.1 Plane electromagnetic waves..........................................................25 2.2 Monochromatic plane waves..........................................................30 2.3 Monochromatic plane wave on boundary............................................32 2.4 Solution of non-homogeneous wave equations......................................38 2.5 Problems..................................................................................41 2 Chapter 1 Electromagnetic field According to today's knowledge, atoms of all substances are composed of electrically charged particles, i.e. electrons and atomic nuclei. Substances may seem electrically neutral, if the influences of opposite charges interfere with each other. We explain the electric current as motion of electrically charged particles. The contemporary physics is based on these ideas and aims to determine the structure of substances and deduce the laws of physical, or chemical phenomena at the level of atoms and molecules from the laws of the motion of electrically charged particles. The first step in this direction is to clarify the laws of the mutual action (interaction) of electric charges through an electromagnetic field. In this paragraph, we will formulate the basic laws of electrodynamics, i.e. Maxwell and Lorentz-Maxwell equations. These equations play the same role in electrodynamics as Newton's equations in classical mechanics. They were found by generalization of empirically established regularities. Their general validity is, however, based on the fact that all of the consequences of these laws have been experimentally proved and even found a great number of applications in technical practice. The origin of electrodynamics dates back to the nineteenth century (M. Faraday 1831, J. C. Maxwell 1865), and its final formulation was completed only at the end of the nineteenth century (H. A. Lorentz 1892), i.e. considerably later than that of classical mechanics. Therefore, there were efforts to reduce its laws to the laws of mechanics, which seemed to be simpler. However, later development demonstrated that the electrodynamics is independent of classical mechanics, and both these disciplines constitute the two fundamental pillars for the current understanding of the structure and motion of matter. 1.1 Maxwell's equations The fundamental laws of a non-stationary electromagnetic field induced by moving charged particles in vacuum can be summarized in two series of the fundamental Lorentz-Maxwell equations: p dE I. série: divE = —. xotB — eopo—- = poj £q at. OB II. série: votE + — = 0, divB = 0 (1.1) at (Hendrik Antoon Lorentz in 1892). These partial differential equations of the first order determine the spatial distribution and temporal changes of the electromagnetic field induced in vacuum by particles at given density of electric charge p(r, t) and current density j(r, t). Here j = pv, where v(r, t) is the distribution of velocity of charges' motion. The electromagnetic field is determined by the distribution of vector fields of electric field intensity E(r, t) and magnetic induction B(r, t), which are the solution of the set of equations (1.1). Vacuum permittivity eo and permeability po are two constants that have been introduced in the International System of Units (SI) for physical quantities. They have approximate values eq = 8, 854 • 1CT12 Fm_1, p0 = 1,257 ■ 10-6 Hnf1 . (where F denotes the farad unit, H denotes henry), and fulfill the important Weber relation (Wilhelm Eduard Weber, 1804 1891) 1 £° = (1.2) 3 1.1. MAXWELL 'S EQUATIONS where c = 2, 998 • 108 m s-1 is the speed of light in vacuum. Unlike the artificially established constants eo and [to, c is a natural constant whose value has to be determined experimentally. We can, of course, turn from the SI system to another system (e.g. to the so-called 'absolute' system), where constants eo and / iE-df =— IpdV . £o J s0 J OV V in vacuum . . divD = p <=> Bdf = 0 ov expresses the fact that there are no magnetic charges, i.e. magnetic field lines are either closed or are extending from infinity to infinity. 4th law: qj^ TOtE + — 0 -function, we express the one-dimensional density of the point charge Q in point x = 0 as p(x) = Q 5(x) , for the charge in point xo = 0 we have p(x) = Q S(x - x0). For a three-dimensional particle at point ro we express p(r) = Q 53{r - r0), where the three-dimensional (5-function is defined as a product S3(r - r0) = 8(x - x0) S(y - y0) S(z - z0) and meets the relations J 63(r) dV = 1 a <53(r) = 0 pro r ^ 0. Thus, the system TV of point charges ea at points ra where a = l,...,N, will have the charge density p(r) = &3(.r ~ ra) (1.12) Q = l and current density j(r) = va S3(r - ra), (1.13) where the va are the charge velocities. These expressions should be substituted into the right sides of the first series of Lorentz-Maxwell equations 1.1. Mathematical comment 1. Dirac d-{unction Let us introduce the basic rules for working with (5-functions: oo J 5(x) f(x)dx = /(O). —oo X J 6(x - x0) f{x)dx = f(x0), —oo f S3(r-r0) f(r)dV = f(r0), S(-x) = S(x) , x 5{x) = 0 , +oo J S'(x - x0) f{x) dx = -f'(x0) , —oo +oo / 0\x - x0) f{x) dx = {-l)k fW(x0) , fe = l,2,.. 1.3. ELECTROMAGNETIC POTENTIALS ö(c x) = — 5(x), c ^ 0 . The last relation is a special case of identity where g is a function differentiable at points x; in which g(x\) = 0 Lastly, let us prove an important relation A- = - 4ir S3(r) . (1.15) r (Compare with the Poisson's equation of electrostatics.) In this case, the LaPlace operator A is reduced to the radial Palt 1 d /, d r r2 dr \_ dr and therefore A- = Ar- = 0 r r at r 4- 0 . On the other side, for any area V containing r = 0, we will receive a non-zero result using the Gauss's theorem J A^ dV - J div grad^ dV = ^gradj • df = - j^•nd/ = - Jdtt = - 4tt , v v ev ev since d£2 is a spatial angle defined by the surface element df, and its vertex r = 0 is assumed to be within the area V. 1.3 Electromagnetic potentials In some special cases, Maxwell's equations can be solved directly as a system of partial differential equations of the first order for six unknown functions Et, By Since the theory of solving partial differential equations of the second order is elaborated better, it is useful to convert Maxwell's equations into an equivalent system of equations of this type. This is done by the method of potentials which uses the fact that the second series of Maxwell's equations does not depend on the sources p,j and gives relations only for E, B. The equation divB = 0 can be easily fulfilled identically by introducing a vector potential A(r, t) by relation B = votA. (1.16) Then the first of the equations of the second series takes the form rot + ~j = 0, (117) which we can easily fulfill identically by introducing the scalar potential

A' = A + gradA, (1.21) then, due to the identity of rot grad A = 0, the new vector potential A' describes the same magnetic field as A. In order not to change the electric field E during the transformation (1.21), it is necessary to compensate for the change A by the change of 9A' A dA E = -grad (p' = (p - —. (1.22) The obtained transformations of potentials (1.21), (1.22) are referred to as gauge (also calibration) because they do not change measurable quantities E and B. We call them gauge invariant quantities. Thus, we see that a whole class of pairs of potentials {(jp,A)} united by gauge transformations corresponds to one of the electromagnetic fields E, B . Equations (1.20) can now be simplified by appropriate selection of potentials from this class. For example, by applying the Lorenz secondary condition3 divA + sn— = 0 (1.23) at we ensure that the equations (1.20) will no longer be bound: a d2

', A'. Furthermore, let us also make sure that the time evolution of the potentials q>, A according to (1.24) does not break the Lorenz condition (1.23), i.e. if (1.23) is fulfilled at time to, it will also apply to all t > to. To do this, it is sufficient to apply the operators eft d/dt and div to the equations (1.24), to sum up the resulting equations and reverse the order of derivations. After the transformation, we will receive the equation / dip\ O2 ( d" = y'--^ = ¥>-^(A + A'), (1.29) A" = A' + grad A' = A + grad (A + A'). (1.30) For example, when p = 0, we may require additional conditions to be met if" = 0. We see that this condition corresponds to such a solution of the wave equation A (1.28), which according to (1.29) fulfills the equation ™ = u/ (1.31) This equation is apparently fulfilled by the function in the form A\r, t) = JQ (p (r, t')dt' + /3(r). According to the wave equation (1.28), fi(r) will then be the solution of Poisson equation A/?(r) = ep.(d p + e~+vc or annihilation of the electron-positron pair e~+e+ —> 2y. In all these processes, the sum of the charges is preserved; therefore, we are speaking of the additive law of conservation. The first experimental proof of this law was given by Michael Faraday in 1843 (See also [22]). However, the charge has other properties, e.g. it is quantized and, unlike mass, it is relativistically invariant, it does not change when the particle moves. The charge is a scalar and can have a positive or negative sign. After the discovery of anti-particles, the charge symmetry of particles and anti-particles turned out to play an important role in the universe and to be related to other spatial and quantum symmetries. However, it took a century to clarify whether electrical phenomena are the manifestation of only one type of charge ('single-fluid theory') or whether there are two oppositely charged electrical 'fluids'. The law of conservation of a charge contained in a given arbitrary spatial region V bounded by the surface dV can be expressed by equality: the rate of loss of charge in the volume V is equal to the amount of charge that passes through the stationary surface dV per unit of time. Mathematically, the equality reads fPdV= ijdf, dt v dV and represents an integral form of the equation of electrical current continuity. According to Gauss's theorem, here we obtain a differential continuity equation in the limit V —> 0 — + divj = 0, (1.33) which expresses the local law of charge conservation. It is evident that the equation of continuity is a consequence of the first series of Maxwell's equations (it is sufficient to use the operators d/dt and div for the equations (1.5) and sum up the resulting equations.) However, the physical meaning of the continuity equation, whose validity was demonstrated outside the framework of Maxwell's equations, illustrates the following point of view in a better way. Maxwell's equations ^ dB rT 3D rot E + — =0, rot //--— = j dt. dt, describe the time development of the electromagnetic field. If we apply div operator for both of them and use the expression (1.33), we will obtain d d — divB = 0, — (divD - p) = 0. We can interpret these eauations in such a wav that due to the validity of the eauation continuity, the eauation div B = 0, div D = p (1-34) is valid at any time t, if at a certain time t = to. Thus, from this standpoint, the expressions (1.34) can be considered as the initial conditions. Energy Conservation Law According to all experience to date, physical phenomena are in accordance with the laws of conservation of energy and momentum. To derive the energy balance in the electromagnetic field, let us consider the system of charged particles and the electromagnetic field. The field energy in the spatial region V bounded by a stationary closed surface dV, which particles do not pass through, can only be consumed by the work W performed by the field while interacting with the particles in the volume V and by the energy flow through the boundary dVof the region V into the outer space. 11 1.4. LAWS OF CONSERVATION OF CHARGE, ENERGY AND MOMENTUM If the charges have density distribution p(r, i) and the velocity distribution v(r, i), then the power of the field force dW/dt in the region V will be determined by the Lorentz force density. (1.37) (1.38) f = p(E + vxB) = pE + jxB (1-35) by relation ^ = f f„ &V = J p v • E dV = f j- EdV (1-36) V V V (Note that/dV is a force acting on the charge p dV) Thus, the quantity j ■ E represents the density of the mechanical power of the forces of the field. This quantity induces the Joule heating generated in a unit of volume per second in the conductor. If w denotes the field energy density and S denotes the field energy flux density, we can express the energy balance by equation _ A J w dV = j j ■ E dV + j S ■ df. V V 8V As in the case of the continuity equation, we can arrive from this integral form into a differential form Oiv —— = j ■ E + div S, at J which expresses the local energy conservation law in the system of electromagnetic field and charged particles. To determine how w and S depend on the electromagnetic field, we will transform the expression j ■ E using Maxwell's equations (1.5) and identity div {E x H) = H ■ rot E - E ■ rot H . (1.39) Gradually we will obtain j-E= (rotH-—\E = -div{ExH)+(H-YotE-E-—} = -div (ExH)- (H —+E —\ . \ at) \ at) \ at at J For a soft medium, whose permittivity e and permeability fi do not depend on time, the Poynting theorem in a simple and commonly used form [10, 231 is valid - — - (E ■ D + H ■ B) = j ■ E + div (E x H) . (1.40) For the field energy density, the comparison of the expression (1.40) with (1.38) here gives w = i {E ■ D + H ■ B) = | (eE2 + txH2) and for S (Poynting vector) S = ExH. The energy density of the field in vacuum is w = i (eQE2 + voH2), where H = B/fio. In a material medium, the formula (1.41) includes, in addition to vacuum field energy, the energy consumed by the polarization and magnetization of the medium. The law of momentum conservation We have seen that the electromagnetic field acts on the charge and current carriers by the force with volume density (1.35), which, according to (1.36) is changing their energy. However, we know that according to the law of force, their momentum will also change. The experimental evidence of the existence of the momentum in the electromagnetic field was brought by the discovery of mechanical pressure of radiation (Pyotr Nikolaevich Lebedev (1866-1912) in 1899). Let us write the balance of momentum in the system of electromagnetic field and charged particles: the loss of momentum of this system in the volume V bounded by a motionless surface dV that particles do not pass through, is equal to the momentum of the electromagnetic field, which flows from the volume V through the boundary dV. If we denote the particle momentum 12 Chapter 1. ELECTROMAGNETIC FIELD density as p, g is the field momentum density and (an, a a, er«) is the flux density vector of the field momentum (the flux density of the vector is a tensor), we can express the momentum balance in an integral form (1 di J(in + 9i) dV = j>rrik dfk , («=-1,2,3). V 8V The differential form of these equations dgt dpi dt 1 dt da. ik fj.fi (1.44) (1.45) where dau/dxt is the tensor divergence, is the mathematical expression of the local law of momentum conservation in the system of electromagnetic field and charged particles. In order to explicitly determine the quantities g and (era), we will write the equation of motion for particles in the volume unit, dp -ft = / = RE + jxB (1.46) and we will express the densities p,j using Maxwell's equations (1.5): f = E div D + (rot H If we add terms -tt" x B dt H div B 0 , rot E + dB x D which are zero due to the second series of Maxwell's equations, to the right side, we will obtain a symmetrical expression / = -— (D x B) + [E div D + H div B - D x rot E - B x rot H } (1.47) This equation is identical with the equation (1.45), if we manage to modify the vector in square brackets to the tensor divergence —ci*. Complex algebraic manipulations are made only for the first component: dxi dxo dx3 The last bracket denotes other terms of the same form in the magnetic quantities H, B. For a homogeneous soft medium — = - — (Si Ih) Et ^ = dx\ 2 dxi ' dx2 d.r-2 (Ei D2) - ígL D2, dX2 and thus, after the interruption and regrouping the terms [...]! = DO + ^(Eilh) + ^(EiD3) - l^EiDi + E2D2 + E3D3) + [E^H.D^ B]. Based on this result, we can now write the i-th component of the equation (1.47) as fi ^(D x B)i + 4r {EiDk + HiBk - Uik(E D + H ß)|. dV " dxk | ' " 1 " 2 Comparison with relations (1.45) and (1.46) gives the following relation for the field momentum density g = DxB = e iiExH and for the tensor (era) it gives the expression 1 r Et Dk + Hi Bk - -5ik {ED + HB) (1.48) (1.49) (1.50) 13 1.5. THE EQUATIONS OF ELECTRODYNAMICS IN MINKOWSKI SPACE-TIME In vacuum g = eo /to E x H = -ß S, (1.51) £Q Ei Ek + HoHi Hk - | 5ik (e0 E2 + /*„ H2)J . (1.52) Since in the stationary field dg/dt = 0 and the relation (1.45) transfers into /,. = - ^* , (1-53) i.e., to an equation of the same form as the continuum statics equation (3.277), atk is referred to as Maxwell stress tensor It allows to calculate the force acting on the volume Vof substance as in the continuum mechanics, i.e. as a resultant of 'stresses' acting on the surface of the dV, J fi dV = j(-atk) dfk . (1.54) v av Note that the derived relations (1.49) and (1.50) have been discussed from the very beginning, because it is unclear how the momentum of bound charges in the material medium is included into them in addition to the actual momentum of the field. However, the correct momentum density of the electromagnetic field itself is given by the expression (1.51), which we would derive not from the average macroscopic field, but from the microscopic field. Maxwell stress tensor is also suitable for a field in soft medium (1.50). Detailed relations for different types of media can be found in the recent work4. According to this work, the relation (1.51) is valid not only in vacuum, but in a non-magnetic medium (M = 0) independent of its dielectric properties. The general theory, however, suggests that the electromagnetic field also transmits angular momentum in addition to energy and momentum. 1.5 THE EQUATIONS OF ELECTRODYNAMICS IN MINKOWSKI SPACE-TIME According to Einstein's principle of relativity, the laws of physics have the same form in all inertial systems. Mathematically, the relativistic invariance of physical laws can be expressed by such notations of the relevant equations, where all their terms are clearly of the same transformational properties as in Lorentz transformations. Such equations are referred to as covariant. If they are valid in one inertial system, they are valid in the same form in every other inertial system. In this sense, e.g. vector equations of mechanics are covariant with respect to rotations of the coordinate system. Lorentz-Maxwell equations can also be written in an apparent covariant form, if we determine the correct transformation properties of all the quantities involved. Let us start from the charge we know to be one of the few relativistic invariants. The electric charge density is no longer an invariant, but it is transformed as a zero component of the four-vector, that is, e.g., time, because p p dV de = c -—— = invariant dt dt dV dV Based on this property, a four-vector can be defined r-'s —t. ■ dxv dxv Therefore, we can consider the scalar and vector potential as the time and space part of the four-potential (A")=(^,A\. (1-58) With this definition, equations (1.25) turn into the covariant d'Alembert equation nAf = -pof (1.59) and Lorentz-invariant condition ^— = 0. (1.60) The gauge transformations (1.21) and (1.22) can also be written in a covariant form dA A* —> A'» = A11 ox i' (1.61) We will now express fields E, B according to (1.19) using the four-potential components 1 E, = Br = d - (-) dA1 dAi dAo d(ct) dx° OA3 OA2 dAo dA? dx2 dx3 dx3 dx2'" ' Thus, they form six independent components of antisymmetric tensor of the second order in four-dimensional space, i.e. electromagnetic field tensor Qj± £)A Ffj,u — dx* dx" (Verify that the tensor FnV is gauge invariant.) Explicitly * r 0, r and after raising both indices Ex Ě ' C ' o, -Bz, By Bz, 0, -Bx -By, Bx, 0 Ex. c ' o, _ĚX c ' -Bz, _Ej_ c By Bz, o, —Bx -By, Bx, 0 The relevant dual tensor is again an antisymmetric tensor of the second order. It has components / U, BX, By, _R n -Bx, — By, \ -Bz, 0, Mi 0, Ex c ' Bz \ Ev o / (1.62) (1.63) (1.64) (1.65) 15 1.6. MOTION OF THE CHARGED PARTICLE IN EXTERNAL ELECTRIC AND MAGNETIC FIELD The Lorentz-Maxwell equations 1.1 can now be written in a compact and relativistically covariant form dF'"' qf*v" 1st series: —- = — Uo j11 , and 2nd series: —-- = 0. (1.66) ox" ox" It is apparent from the previous relations that the transformation of the duality F —> F* corresponds to the current substitution of E/c —> —B, B —> E/c, in which the left sides of the first series of the Lorentz-Maxwell equations will transfer (except for the proportionality constants) to the left side of the second series. Similarly, the dual tensor for the four-rotation (1.62) is the solution of the second series (1.66) with the use of potentials, and the first series is equivalent to the d'Alembert equation (1.59) with the Lorentz condition (1.60). The Lorentz force (1.3) or rather Lorentz force density (1.35) also has a simple covariant form. We will express the first component of the Lorentz four-force K1 = -j£= = e^=*f + en±L=Bz ~ e By using the four-velocity components if and tensor F1™ as K1 = e Fu' uu. Similar relations are applied to all the components of Lorentz four-force eE-v ^e(E + vxB) | ^ 7i where the zero component is proportional to the power of Lorentz force. Similar relations are also valid for covariant Lorentz force density: (1.68) f" = F«»ju, (/") = (^Ej, pE+jxB^j Therefore, relativistic equation of motion (1.4) for a particle of rest mass mo and charge e in a given external electromagnetic field F^v will have a covariant form in Minkowski's four-space dr 1.6. Motion of the charged particle in external electric and magnetic field Let us consider a particle of rest mass mo and positive charge q, which at time t = 0 moves from the origin O in the direction of the x-axis with a relativistic initial velocity vo. A homogeneous electrostatic field E acts in the direction of axis y. We shall determine the form of the particle orbit. Let us write the relativistic equation of motion of a particle in the form dp — = qE (1.70) (the charge of the particle is relativistically invariant). By integrating with the given initial conditions, we will obtain Px - Po - konst. , py = q E t . (1.71) Now we must take into account the fact that the relativistic relation between the momentum and velocity is p c2 v = , kde £ = c Jp2 + c2 . (1.72) To avoid confusion with the electric field, we shall denote the energy of the free particle as G. If we mark the initial value of energy £o = c sjpl + mlc* , (1 73) we can write £ = C yjp2 + q2 £2 t2 + m2 C2 = Jg2 + (c q £ t)2 . (1.74) 16 Chapter 1. ELECTROMAGNETIC FIELD We therefore have v = + (cqE t)» ' (L75) and in the components da* _ c2 po dy _ cr q E t VX ~ * ~ ^£2 + (cqEtf ' Vy ~ dt ~ ^£1 + {cqEtf ' ^ By further integration, we will receive5 c po cqEt 1 / —- £0 (1.77) The equations (1.77) indicate the law of motion and represent the parametric equations of the trajectory in the surface x, y. If we exclude time from them, we will obtain the equation of the particle's orbit y = AXosh^ -lY (1.78) q E \ cpo ) Thus, the motion of the particle follows a curve referred to as catenary. The constants Go and po o are bound by the relation (1.73). Figure 1.1: A relativistic charged particle in a constant electric field At non-relativistic velocities, we have Go = moc2, po = movo and we will expand the hyperbolic cosine according to a small argument . e1 + e~x , x2 cosh a- =--- = ! + _ + .... Then the expression (1.78) transforms into a non-relativistic parabola y = q E 2 m0 t'o x . (1.79) Now let us consider the case where a particle of rest mass mo, charge q and relativistic initial velocity v is moving in a homogeneous magnetic field B. We shall write the equation of motion of a particle in the form dp dt — q v x B . (1.80) If we use the relation between the relativistic velocity and momentum of a particle p = (G/c2) v, we can rewrite (1.80) as dv q c2 „ d^ = -TvxB- 5arg sinh x is an inverse function to the hyperbolic sinus and arg sinh (x/a) = ln|x + Vx2 + a2\ is valid. (1.81) 17 1.7. LORENTZ TRANSFORMATION OF POTENTIALS AND FIELDS, INVARIANTS Since the magnetic field does not perform work on a particle (Lorentz force remains perpendicular to velocity), the energy £ is constant during the motion. Let us remind that in a non-relativistic case, we solve the equation of motion *» = X„xB, (1.82) at mo arrive at the result that the particle performs a helical motion with an axis parallel to the magnetic field (see [22] p. 356). In a plane perpendicular to the magnetic field, this motion appears to be a uniform circular motion with an angular velocity wc = qB/mo and a radius rL = movot/qB. In the direction of the magnetic field, the particle performs a uniform motion at a speed vof, vo< and voi are the components of the initial velocity in the perpendicular and longitudinal direction with respect to the magnetic field, where vo< = vt, vo; = vi, vo = v and they do not change during the motion. The relativistic and non-relativistic equations of motion (1.81) and (1.82) differ only by the constant before the product v x B. Using the same procedure as in the solution of the non-relativistic problem, we will come to the conclusion that even relativistic particles will move along the helix. The cyclotron angular frequency and the Larmor radius will equal q c2 B q B v2 vot vot £ m0 vu 1 t mQ V er ujcr qc* B q B L _ v* If the initial velocity of the particle vo = v increases, the cyclotron frequency cocr decreases and the Larmor radius rLr increases. 1.7 Lorentz transformation of potentials and fields, invariants We derive transformation formulas for potentials -/?. The transformation of fields E, B should be based on the transformation law for the tensor of the electromagnetic field F^v F,flu{x') = a?p al,a Fpa{x). (1.87) By substituting into rfV we get transformational relations for its six independent components F<01 = F01 j Frt)2 = 7 (F02 _ ^12^ F/03 = 7 (F03 _ ^13^ F'i2 _ 7 (pi2 _ 0j?o9\ f'13 = 7 (F13 - (3f03), f'23 = F23. If we substitute into components of the fields according to the relation (1.64), we obtain e'x = ex , e'y = 7 (Fy - (3 c bz) , e'z = 7 (ez + p c by) , 18 Chapter 1. ELECTROMAGNETIC FIELD b'x = bx, b'y = 7 (bv + £ e\ , b'z = 7 (bz - | ey) . (1.88) In the case of general direction of velocity V, we obtain different transformation relations for components parallel to V and perpendicular to V: E\ = E\\ , E'x = 7 {Ex + v x b) , fijj = B,, , S'x = 7 (b± - ^ x E^j . We obtain an inverse transformation by substituting V —> — V. If we have a pure electric field (B = 0) in the system S, then in the system 5" in addition to the electric field E', there is also the magnetic field B': £(, - En , EL = 7 E± , Bji = 0, B'x = - 7 3 x E . (1.89) c2 If V 1 -E is selected, then the generated field B' is perpendicular to E' and V, B' = -X;*E' . d-90) Analogously, in the case of a purely magnetic field, E = 0, V 1 B, an electric field is generated £' = V X B' . (191) In each transformation of quantities, we are concerned with the combinations of those quantities which do not change during the transformation and remain relativistically invariant. Thus, in the orthogonal transformation of coordinates in the Euclidean space, the distances of two points, the length are invariant, while in the Lorentz transformation in space-time, the interval is invariant. Let us try to find out what relativistic invariants can be created from the components of the electromagnetic field Ex, Ey, Ez, Bx, By, Bz. These components form the electromagnetic field tensor F^, or the dual tensor F*1™, whose transformation properties are known, i.e. they are antisymmetrical contravariant four-tensors of the second order. We will obtain the simplest invariants that can be constructed by contraction. Since the diagonal elements of the antisymmetric tensor F^ are equal to zero, Then Jo = = 9fiU = 0. TP f„„ wy = - E B. (1.92) (1.93) We could thus construct other invariants by contraction the products of three, four and more F1™ tensors, or the dual ones. However, these invariants would be either equal to zero or a combination of invariants I\ and h. This fact is generally valid. The fact that the Lorentz group is locally isomorphic to the group SL(2, C) of complex matrices 2x2 with determinant equal to one implies that the invariants of the tensor F1™ are invariants of a complex matrix / Fz , Fx -{ Fx + i F„ , ) ' kde F = — + i B, with respect to complex linear transformations of the group SL(2, C). According to the algebraic theory of invariants, matrix invariants are the coefficients of the characteristic equation Ft Fx + lFy, A, Fx Fz = A2 and the only (complex) coefficient of this characteristic equation - B2 + — E • B = F = 0 -h + i/2 19 1.8. ACTIONS FOR CHARGED PARTICLE SYSTEM AND ELECTROMAGNETIC FIELD determines the two independent invariants I; and h. The knowledge of these invariants allows to perform a relativistically invariant classification of electromagnetic fields. These invariants can take on zero, positive or negative values, whose sign does not change in the Lorentz transformations. Therefore, one can distinguish nine cases (canonical forms) of the mutual position of vectors E and B. If h= h = 0, then E = cB, E • B = 0 in all Inertial systems. Both vectors are perpendicular to each other and correspond to the case of plane electromagnetic wave propagating in vacuum at a velocity c. The electromagnetic plane wave thus remains an electromagnetic plane wave in all inertial systems. 1.8 Actions for charged particle system and electromagnetic field The actual solution to the problem of motion of charged particles that mutually interact through the electromagnetic field is very complex. However, the problem itself can easily be formulated in a compact form of Hamilton's variational principle. The apparatus of the Lagrangian formalism then allows to introduce (through the Noether theorem) basic preserving quantities. We will construct the respective action based on an analysis of two marginal cases: A. A system of charged particles in a given external electromagnetic field. B. An electromagnetic field generated by a system of charged particles moving in the prescribed manner. In the case A, we know the Lagrange function for one particle I tf2 L = - m0 c2W 1 - — - e(

* pA»{x) At R3 Ui and using the definition of the four-current (1.55), we will transform it to the form t2 dV , (1.100) Smf = - i If Ap dV* = - j j{(*p-pv- A)dV dt, (1.101) ti R3 which will then be used in situation B including the case of continuous distribution of charges and currents. Let us remind that action for the relativistic field is S = (1/c) / L dV* , where dV* = c dt dV. Thus, the interaction part of the action corresponds to the density of the Lagrange function Cmf = -fA? (1.102) referred to as an interaction Lagrangian. Let us demonstrate that in the case of a single particle, variation of the action Sa with respect to x11 yields the equation of motion (1.69). Let us write the action Sa in the form _ SA = I I -"»0 c \/ —--37- - e Ay. (x) H L dt dt and then we will proceed to the integral through the proper time of the particle T T2 in whose integrand there is the Lagrange function L = L(*rM,/rr) = T — U* — — mo c ^/u+*u^^ — e (x) t/M , with U* velocity-dependent potential. The action principle (1.95) is equivalent to the Euler-Lagrange equations _d_ — = o tj — (—\ - -— + — dr \ du„ J dxv dr \ 3uv / dxu dr dU* duv The transformation of both sides mo du" dr" ' dA» .1,1" c- U/i + e - = — e dxv dr \ dxv actually leads to the covariant motion equation (1.69) with the Lorentz four-force. How the action in the case B should be constructed? We will look for an action principle which would serve as a source for Lorentz-Maxwell's equations for the electromagnetic field, whose source would be the specified four-current f. In mechanics, the Lagrange function is the scalar function of general coordinates and velocities determining the state of the system v at any time t. In electrodynamics, the state of the field in vacuum at time t is determined either by the values of the fields E, B, or by potentials AM and their first derivatives Aftv = dA/dxv at any point in space. In order to make the Lagrange equations linear, the sought density of the Lagrange function Lb can be at most quadratic with respect to the field variables. An important limitation is the covariance of the resulting equations: to do this, the scalar LB must be an invariant. As we already know, only two independent invariants can be constructed from the electromagnetic field tensor F1™: F^F*" = 2 (b2 - ^\ , F*VF^ - -c E b. (1.103) 21 1.8. ACTIONS FOR CHARGED PARTICLE SYSTEM AND ELECTROMAGNETIC FIELD Both of them are quadratic, but the second is a pseudoscalar, and therefore it is not applicable. Quadratic invariant A^A1 also cannot be used because it violates the gauge invariance. As a candidate for the field part of the density £/ of the Lagrange function Cb = Cf + Cmf (1104) there remains Cf = a F,VF^. (U05) The constant a is determined so that the Lagrange equations for LB relative to the general coordinates A1, dAp ' dx» \dA^J led precisely to the equation of the first series (1.66). Since dA, ~ 3 ' ~ we obtain a = -l/(4/" — ___—(a fup\ = Mo "p " Mo dxPK " ' ~ dxp we create a gauge-invariant symmetrical tensor of energy-momentum according to (6.152) Tpv = f- gpa Fw Fvo + ± tv Fp* Fpo^j ■ d-U°) 22 Chapter 1. ELECTROMAGNETIC FIELD By substituting for F*™ we find that the tensor T" has a known structure (6.159) V ir \ (1.111) where w, S and (era) are the quantities derived in paragraph 7.4. for the electromagnetic field in vacuum. The tensor T" has zero four-divergence in a free field (6.157), but with the sources present we can derive the equation (1.112) where/'' is covariant Lorentz force density (1.68). This important covariant equation summarizes, in a nutshell, the local laws of energy and momentum conservation for a system of charged particles and the electromagnetic field in vacuum, which we derived in paragraph 7.4. 23 1.9. PROBLEMS 1.9 Problems 1 Motion of a charged non-relativistic particle in electric and magnetic field. Solve equations of motion for a non-relativistic charged particle with mass m and charge e moving a) in a constant homogeneous electric field E = (E, 0, 0) (initial conditions r = 0, ř = Vo for t = 0); b) in a constant homogeneous magnetic field B = (0, 0, B) (initial conditions r = 0, ř = (vo, 0, 0)). [ a) a- = f£ í2 + v0xt, y = v0yt, z = v0zt; b) x = ^ sinat, y - -^-(1 - cosať), z = 0;a = ^\ 2 Zacek's magnetron. Demonstrate that the trajectory of a particle of mass m and charge e in crossed constant homogeneous fields E = (0, E, 0), B = (0, 0, B) is a cycloid, if initially r (0) = (0,0, zo), ř (0) = 0. This cycloid is generated by rolling a circle with radius ro = mEleB2 on surface z = Zo along the x-axis with the angular frequency wc = eB /m (the cyclotron frequency). [:r = Z^E^t - sinwcř), y=1£E{l- cosu;cť), z = z0 } 3 Normal Zeeman effect. Electron of mass m 9,1.10—31 kg and charge e = — 1, 6.10-19 C bound to the origin by force —kr is harmoniously oscillating with angular frequency a>o = ^jk/m (isotropic harmonic oscillator). Determine how the angular frequency of oscillations changes if this spatial oscillator is placed in a constant homogeneous magnetic field B = (0,0, B). Instructions: Write the equations of motion of the electron. Look for solutions in the x\x2 plane in the form x; = A; exp(icot), i = 1, 2. To determine w, use the condition \eBI2m\ « ojq. Consider to what extent this condition is met for an electron emitting visible light and placed in a magnetic field ~ IT. [ In the plane XiX2: ui = ui0 ± ve směru osy x3: u> — lj0 ] 4 Constant homogeneous magnetic field. Show that the vector potential A = B x r I 2 determines a constant homogeneous magnetic field B. When selecting a coordinate system such that B = (0,0, B), where B = const, specify: a) Cartesian components Ax, Ay, Az, b) components AR, A,r, Az in cylindrical coordinates R, Idl = 0 13 The magnetic dipole moment of current distribution j(r, t) in the finite volume Vis defined by the relation: m{t) = ±fvrxj(r,t)dV. a) What is the magnetic dipole moment of the system TV of point charges ea located at points ra(t) and moving at velocities va(t)7 b ) What is the magnetic dipole moment of the closed plane curve Y with linear current / flowing though it? Instructions:] dV= I dl. N [ a) m(t) = \ Yl eQrQ(£) x vQ(t), b)m = ^rxdi = ISn, a=l where S is the surface bounded by a loop and n is its normal. ] 14 Pulling a dielectric medium between capacitor plates Calculate how the energy of the electrostatic field will change, if we fill the capacitor space with a homogeneous soft dielectric medium. Instructions: compare the solution of Maxwell's electrostatic equations in vacuum and in a dielectric medium. 15 Force in the capacitor Plate capacitor consists of two parallel conductive plates with area S carrying charges +Q and —Q. Plates are placed in a dielectric medium with permittivity e. If the plates are large and their distance is small, then the electric field is concentrated practically between the plates and is homogeneous. Calculate the forces with which the plates interact. Instructions: Calculate the capacitor field energy and its change when the plates spacing changes: a) if the plates are isolated and their charge is constant, b) if the plates have a constant potential difference S of the four-potential A*1 = (

S of the electromagnetic field tensor components F,flv(xA). B = y/jB*) (fi0 = 4tt • 1(T7 N A"2). [510 V/m, 1,7 • 10-6 T] 47TS0y/{x-Vt)3 + {l-(P){y*+Z*) ' A = (V/c2)p E = q(l- (32)(r -Vt) x E 4ks0[(x - Vt)2 + (1 - p2){y2 + z2)]3/2 26 Chapter 2 Electromagnetic waves 20 Plane electromagnetic waves In order to understand the physical meaning of Maxwell's equations, we need to know their solutions under different physical conditions. Whenever we solve them, we will learn something new about their character. It is usually recommended to visualise the solution e.g. using lines of force. In this way we gradually get to a real physical understanding of the equations. If you analyze equations mathematically, you should not think you already understand physics. The real physical situations of the real world are so complex that equations need to be understood much deeper. Physical conditions also sometimes allow to replace the solution to the complex problem by an approximation, and our task is then to assess the conditions for the applicability of such approximate solutions. In the stationary case, Maxwell's equations do not give anything new that could not be derived from the Coulomb's law for charges and the Ampere's force law. In the basic course of physics, the main types of electric and magnetic fields are derived, which correspond to the stationary configurations that most often occur in practice. This chapter is therefore devoted solely to to the time-varying fields, which bring new consequences typical for non-stationary Maxwell's equations: the existence of electromagnetic waves and their radiation with accelerated charged particles. Let us begin by solving Maxwell's equations in an 'empty space', that is, in an area where there are no free sources p = 0, j = 0. Maxwell's equations (in a homogeneous isotropic soft medium with material constants e, fi) obviously have a trivial constant solution. However, we will demonstrate that there are non-zero time-dependent solutions. Using the method of potentials, we can substitute Maxwell's equations with a simple equivalent system in the Coulomb gauge, where j = 0,

(2.29) where ay(0\ at}x) are the components of Maxwell stress tensor of the incident and reflected wave, n is the unit vector of the normal to the wall (in the direction of the body). In the system fixedly connected to the wall according to equation (2.28) (2.30) where unit vectors s'<0), determine the propagation directions of the incident and reflected waves, therefore % = w^s'^s'^nj + w^s'^sfnj. (2.31) Using the reflectivity R = w(l)/w(0) T = u/°>[s/(0V(0) -n)+fts'(1)(s'(1) •»»)], (2.32) where, according to the law of reflection s'<0) ■ n = • n = cos 9. Radiation pressure is the normal component of this force p = Tn= wm[{s'{0) ■ n)2 + n(s'w ■ n)2} = (1 + ft)u><°> cos20. (2.33) 31 2.2. MONOCHROMATIC PLANE WAVES The tangent component of area force is % = {1- 1l)w(0) sintfcastf. (2.34) In 1899, a leading Russian physicist Pyotr Nikolaevich Lebedev (1866-1913) experimentally investigated the pressure effect of the beam of light that caused the deflection of lycopodium particles falling in an evacuated container (the drug lycopodium is spruce spores). Radiation pressure is also an important factor in astrophysics, since it affects the shape of the comet tails, and, being an important thermodynamic quantity, plays a role in the internal dynamics of stars. The formula for the pressure P of the equilibrium radiation enclosed in a cavity is derived based on the assumption that radiation from all directions is incident on a given surface with equal probability. Radiation pressure P can be equivalently calculated by averaging the normal force components T for a fixed plane wave through all possible spatial orientations of the surface, i.e. through all directions of its normal n, P = (T - n)iiV = (TijiniTij)^^. (2.35) For i ^ j je {niiij)s^= 0, since n, attains both positive and negative values with equal probability at a given i. Mean values (n2t) ^ are all the same and their sum is equal to one. Therefore (ninj)s,tf = —Sij ^2.36) and the pressure of P equilibrium radiation is equal to a third of the tensor trace ay P=\aii5ii = ^, (2.37) 3 3 3 3 where u indicates the energy density of equilibrium radiation. However, the relation (2.36) can be equally obtained by laborious calculation of mean values over angles 9,

*,v = —--• //sin tfdi9d<£ o o Using the result (2.37), L. Boltzmann theoretically derived the law u = aT4 (T is the thermodynamic temperature) experimentally found by J. Stefan. 2.2 Monochromatic plane waves The described d'Alembert solution of the wave equation is sufficiently general. Due to the linearity of the wave equation, the principle of superposition applies, i.e. the solution is again the the linear combination of two solutions of the wave equation. Therefore, any solution can be decomposed into a superposition of suitable simple wave types. The sophisticated mathematical theory of Fourier analysis offers monochromatic (harmonic) plane waves as the simplest components of linear decompositions. The vector potential A (r, t) of a monochromatic plane wave has a harmonic dependence on the phase f = s.r - vt, A{£) = A0 cos(A:£+ ^ \v = v avelength 1 = 2n/k, frequency v = w 12% = l/T and wavenumber k!2% = 1/1 indi It is often useful to consider a vector function (2.38) as a real part of a complex vector function A(r, t) = Re A(r,t), where A(r,t)= A0eI 0. In the xy plane, the following equations apply ~bj + ~bj-1- (2-49) 33 2.3. MONOCHROMATIC PLANE WAVE ONA BOUNDARY This result geometrically means that at each point of space, the vector £ of a monochromatic plane wave rotates in the plane perpendicular to k with its end following an elliptic trajectory: a monochromatic plane wave is in general elliptically polarized. The special case b\ = bi is referred to as circular polarization, the case b\ =0 or bi = 0 is linear polarization. According to the vector E rotation direction (upper or lower sign of Ey in (2.48)), we distinguish between the right-handed or left-handed circular polarization. If the energy quantity in a monochromatic wave quickly oscillate with a period T, we are concerned with their time mean values, which for periodic functions /(t + T) =f(t) are defined by the integration over a time interval T, to+t (/{%• = i J f(t)dt. (2.50) to Using a standard formula l (cos2 {ut + S))T = - then we derive for a linearly polarized wave E(r,t) = E0cos(k • r — ijjt + a) (2.51) time mean values (w)T = e{E*)T = \eEl (S)T = {w)Tvs = ^E2s, ,„v 0, the integral through/ disappears, limh^oph = a, and there remains J'(Di. - D2) ■ ndf = Jadf, (2.58) / / where n is the normal to the boundary pointing into the medium 1,/is the cross-section of the cylinder with the boundary, D\ and D2 are the limits of the vector D from the medium 1 and 2, and a is the surface density of free charge on the boundary. Since the surface/was arbitrary, DivD = n • (Di - D2) = o. (2.59) The formal operation Div is referred to as surface divergence. Note that if the boundary is a plane x = 0, the surface charge on this plane is described by the singular volume density p(x,y,z)=6(x) B ■ df = J(Bi- Ba) ■ ndf = 0 (2.62) h av and therefore DivB = n ■ (br - B2) = 0. (2 63) The behavior of the magnetic intensity H on the boundary is derived from the Maxwell's equation rot// — (dD/dt) = j written in the integral form, dD a/ f f jH dl-j-^df = J jdf. (2.64) According to Fig. 8.6, the surface/is a rectangle of length I along the boundary and height h perpendicular to the boundary, with a normal TV lying in the plane of the boundary. Its circumference df = T\ +T2 + T3 + Ta is oriented in accordance with normal and consists of four sides, with Ti and T2 lying in media 1 and 2, Y3, Y4. connecting the two media. If we perform the limit 35 2.3. MONOCHROMATIC PLANE WAVE ONA BOUNDARY t n* O x r r 1 1 ■Ä j 2v Figure 2.6: the intensity of the magnetic field on the boundary limit h —> 0 in the equation (2.64), the term with the time derivation disappears (provided that the integrand is limited everywhere). The integrals through Ti and T2 will be on the left side, and the possible surface current through the surface/with a surface density i will be on the right side: J (Hj. - H2) tdl = ft- Ndl. (2.65) r r Here we used lim^o7'.Nh = i ■ N and introduced a unit vector t = N x n, therefore (Hi — H2) t = nx (Hi - H2) ■ N. (2.66) Since the curve Y (the intersection of the surface/with the boundary) was arbitrary, the relations (2.65) and (2.66) yield nx(H1-H2)-N =i-N. (267) Now the vectors n x (Hi - H2), i and N lie in the plane of the boundary and because N has an arbitrary direction in this plane, it is valid that RotH = n x (Hi - H2) = i. (2.68) A formal operation Rot is referred to as a surface rotation. Note that when choosing the axes x, y, z in directions n, —t, —N, the surface current on the boundary is described by the singular volume density j(x,y,0) =6{x)i(y,Q). (2.69) Then the current through the surface fis J j ■ Ndxdy = j i ■ Ndy. f r (2.70) The electric intensity E behavior on the boundary follows the same limit procedure from the Maxwell's equation rotE (dB/dt) = 0 written in the integral form, lim and therefore or in the vector form Of f r (£^1 — E2) ■ t — En — E2t — 0, RotJS = nx{El- E2) = 0. (Ei - E2) ■ tál = 0. (2.71) (2.72) (2.73) 36 Chapter 2. ELECTROMAGNETIC WAVES Fresnel formulae Let us consider the plane boundary of two media 1, 2, with material constants £i, £2, P-2, on which a monochromatic plane wave Eo(r, t) is incident with frequency coo and wave vector ko. According to figure 8.7, we denote the quantities related to waves as incident, reflected, and transmitted with the indices i = 0, 1,2, respectively. The respective angles 9t are measured from the normal to the boundary characterized by unit vector n pointing to the medium 1. We will further consider only linearly polarized waves, for which the vectors Et(r, t) are real, so they can be written in the form Ei(r, t) = Ei cos(fci • r - wkxr~w*t) (2-75) and must apply for any time t and at all points r of the boundary. The linear independence of exponential functions yields U>o = tjJi — UJ2 (2.76) and Hence, using dispersion relations where i 1 v2 = we obtain important 'kinematic' relations for reflection and refraction: the law of reflection (2.77) (2.78) (2.79) (2.80) 37 2.3. MONOCHROMATIC PLANE WAVE ONA BOUNDARY • Snett's law of refraction sin #2 ki v2 i»i V £ini sin??! k2 Vi n2 \e2p2 01, We will further use only angles 9\ and #2. Maxwell's theory allows further determination of 'dynamic' relations—the intensity ratios of individual waves depending on the angle of incidence and polarization. Since all the exponentials in the relation (2.75) are equal on the boundary, we can express the conditions on the boundary from relations (2.59), (2.73), (2.63), (2.68) using the amplitudes Et: n-[e1(E0 + E1)-e2E2}=0, n x {E0 + Ei -E2) = 0. n ■ [v/šI72 (2.: 38 Chapter 2. ELECTROMAGNETIC WAVES and the permeability coefficient for the amplitude They also can be written using the characteristic impedance Z = ^ti/d. In the case that fii = fi2, which is usually fulfilled at optical frequencies, the Fresnel formulae (Augustin Jean Fresnel, 1788-1827) are valid ± _ sin(i92 - ďt) ± _ 2cos^ sin i>2 n qm B " sin(tf2 + 0x)" £ ~ sin(t^ + *9) ' 1 ; 1,0 - «,>/72 aB=33°40' St=41°46' i i i i 0 10 30 50 70 90 Figure 2.9: reflectivity for the transition from glass to air Case Eo \\p. Here we will mention only the results: (£iHi 2fi2 sin 2i?i and for /íi = /í2 II _ /ti sin 2ďt -fi2 sin 2ů2 T\\ E Hi sin2$i + H2 sin2i?2' E p|| _ tg(tfx-ď2) || _ íťp — :———:——-, -t £ — —, e2\i2 Mi sin 2i?i + sin 2#2 2 cos i?! sin i?2 tg(^i+tfa) At perpendicular incidence (9\ = fh = 0) both cases yield ^2 — Zl and at /íi = /í2 there are known formulae Re Re = sin^i + ^Jcos^!-^)' TE = 1 + RE In practice, the amplitudes £; are not measured, but the intensities TE = 27Í1 r»i 4- n2 Then the reflectivity R defined by the relation (2.91) (2.92) (2.93) (2.94) (2.95) 39 2.4. SOLUTION OF NON-HOMOGENEOUS WAVE EQUATIONS is equal to the square of the reflection coefficient for the amplitude, and for fi\ = [12 The graphs show the dependence of the reflectivity R on the angle of incidence 9o = 9\ for the transition from air to glass (Fig.8.8) and from glass to air (Fig. 8.9); the values n\ = 1 and ni = 1.5 are selected. The angle 9 is Brewster's angle (R1 = 0 pfi 9\ + 92 = 7i/2, therefore tg9B = n^n\); 9r'\& the limit angle, from which total reflection only occurs when n\ < m (when 9\ = 9r, 9i = ti/2, therefore sin 9r = njni). 2.4 Solution of non-homogeneous wave equations We have so far only discussed the propagation of electromagnetic waves, not their formation. We shall see that the necessary condition for moving charged particles to induce electromagnetic radiation is nonzero acceleration. Now we will derive the potentials of the electromagnetic field induced by any specified nonstationary distribution of charges p(r, t) and currents j ((r, t). We will again assume a homogeneous isotropic medium at rest (with material constants e, p), for which Maxwell's equations apply in their usual form. These equations were equivalently replaced by a system of four inhomogeneous equations 1 #V p (2.97) 1 cPA AA~^^ = (298) for electromagnetic potentials = 0 for r ^ 0; 2. spherical symmetry

0 for r —> oo; 40 Chapter 2. ELECTROMAGNETIC WAVES 4. The Gauss theorem oo, the potential transits into the instantaneous Coulomb potential ^(r,f) = w; (2106) 4. the condition of radiation, which will be further specified. The wave equation for spherically symmetrical potential after substitution and multiplying r ^ 0 we will transform into the form This is however a wave equation for the function r —t. However, the boundary conditions of the problem under discussion do not possess this invariance: the source induces the electromagnetic field which propagates as a diverging wave and, as we will see, irretrievably carries the energy away. The advanced solution contradicts the relativistic principle of causality, since it inverses the causal relation between the cause (source P') and the consequence (an electromagnetic signal, received by an observer P at time t > ť). 42 2.5. PROBLEMS 2.5 Problems 1 Lines of force of electric intensity. Start from the differential equation of the force lines E x dr = 0 and derive an implicit equation for a single-parameter system of force lines of the field generated by a system of point charges lying on the x-axis. In the special case when the system consists only of two charges e\ = 2, x\ = —a and ei = — 1, xi = a, formulate the equation of the line of force that originates from the charge e\ at an angle a with an x-axis. At what angle /? does this line of force enter the charge e{l Specify the minimum and maximum value, and such that the lines of force still end up in the charge e%. What is the angle a of the infinite line of force originating from the charge e\ and directed perpendicularly to the x-axis? Formulate the equation of its asymptote. Instruction: Integrating factor of the differential equation of the lines of force Eydx — Exdy = 0 for charges et in points xt of the x-axis is [i (x, y) = y. [ y\eiCQsi?i = C, kde costf* = . x~x> -1 Refraction of force lines of electric intensity. Determine the law of refraction of force lines of electric intensity £ and the surface density rjp of a charge bound on the boundary of two soft dielectric mediums e, [i. Instructions: E\, — Ei, = 0, £\E\n - £2E2n = 0 [ jj&fll _ Elt/Eln _ «1 . „__TP Z7 _ £3 — €i t? 1 l #7 - e^w^ - 1]p - lln ~t2n - "V*^ Linear polarization. What is a complex notation of a monochromatic plane wave propagating in the soft medium e, p in the direction of the positive axis z? The wave is linearly polarized in the direction of the x-axis. [ E = (E0^k*-Ut),0,0), B = (0, ike«(**-<*t), o)] Circular polarization. Write the complex expression for a monochromatic plane wave propagating in a dispersed medium with the refractive index n in the direction of the negative x-axis. The wave is circularly polarized in the right-hand direction. [ E = (0,Eoexp[-iu(*x + t)],E0 exp[-iw(^x +*) - f]), B = (0, -n^- exphM?* + *) - f]."^1 exp[-zw(?a- + ()])] Vector potential of an electromagnetic wave A monochromatic plane wave propagates in a dispersed medium in the direction of a positive axis z. Calculate its vector potential in Coulomb gauge if the wave is a) linearly polarized, b) circularly polarized. [ a) A = Eoei(k-r-„t-%) h\ A = (Eosinlkz - wt), ±E^cos(kz - wt),0) 1 1 ' <•> \u \ n a, v /' ' J zos(k ■ r — cot). Show that magnetic field B is automatically transverse, while transverse E requires that b = wk ■ a/k2. Show that under this condition the vector B will be perpendicular to E. Determine the calibration transformation that transforms the potentials from the example 8.6 (b = cok ■ a/k2) to the form A' = a' cos(& ■ r - cot),