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 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-PŘÍRODOVĚDECKÁ FAKULTA V OPAVĚ ELECTROMAGNETIC FIELD - 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, f = vo for t = 0); b) in a constant homogeneous magnetic field B = (0, 0, B) (initial conditions r = 0,r = (v0, 0, 0)). [ a) x = fj| t2 + vDxt, y = Vo„t, z = v0zt; b) x = ^ ainof, y = -^(1 - cosat), z = 0; a = ] 2 Zdcek'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, z0), f (0) = 0. This cycloid is generated by rolling a circle with radius r0 = mEleB2 on surface z = z0 along the x-axis with the angular frequency coc = eB/m (the cyclotron frequency). [ x = sfffK* - sin^t), y = ^C1 - CQ&uct), 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 co0 = yjk/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\xi plane in the form x, = A, exp(/co^), / = 1, 2. To determine m, use the condition \eB/2m\ « co0. 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: w = ua ± f^, ve smSm osy xz: u = cjq ] 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, A2, b) components AR, A9,A2in cylindrical coordinates R, , dr/dz. [ a) A = {-By/2, Bx/2,0), b) AR = AZ = 0, A,p = BR/2 ] 5 Constant homogeneous magnetic field. Characterize the class of vector potentials A, which determine the constant homogeneous magnetic fieldB = (0,0, B), B = const. Do they include^1 = (—By, 0,0)? [ A' = {-By/2,Bx/2, 0) + gradA(z, y, z), A = Bxy/2 ] 6 Preserving quantities in a constant magnetic field. The vector potential A = B x r I 2 determines a constant homogeneous magnetic field B. Write a Hamiltonian H (r, p) of a particle a mass of mass m with charge q in this magnetic field. Prove that vector quantity p + qA represents three integrals of motion of this system. Instructions: Select the axis z in the direction of the magnetic field. 7 Show that when a relativistic particle with charge q and rest mass m0 is moving in an external magnetic field determined by vector potentials = (0,0, A(x, y)), the quantity Pa = J^U + QA(x, y). is preserved. Instructions: Determine the canonical momentum corresponding to the cyclic coordinate z. 8 Magnetic flux through the surface. Use the Stokes theorem to specify the physical meaning of the line ^A- 61, Al, where A is the vector potential of a magnetic field and T is a closed curve bounding a two-dimensional surface / What physical quantity does this integral determine? Is this physical quantity gauge-invariant? [§AAl = JB-Af\ r / 9 Singular magnetic field. Let the vector potential in the area of space R3, where x 4- 0, y 4- 0, have components a * y a — ® x a _n a) Determine the magnetic field in this area, b) Calculate the line integral ^ A ■ &l, of this vector potential along the circle x2 + y1 =r2 40. Explain which singular magnetic field this vector potential corresponds to. Instructions: Integrate in polar coordinates. [ a) B = O, b) $ ] 10 Coulomb potential in anisotropic medium. Determine the electrostatic potential field induced by a point charge e located at the origin in a homogeneous anisotropic dielectric medium with relative permittivity tensor e,t, D, = eoYjc e,kEt Instructions: Derive the Poisson equation in the principal axes of the tensor e,k. By substituting xt = x^^/el transform into the usual form of the Poisson equation. A') tp(r) 1 V ei =a =a and in the general Cartesian coordinates with the matrix e = (e,t) we can write tp(r) ^saJ\£\J2{£~1}ikXtxk 11 The electric dipole moment. Electric dipole moment of the charges distributed in the bounded volume V with the density p(r, t) is p(t) = \vrp (r, f)dV. a) Determine the electric dipole moment of the point charge system at points ra(t), a = 1, ... , n. b) Prove that the electric dipole moment of a neutral charge system does not depend on the origin selection. Instructions: Translate the coordinate system by the formula r' = r + a. c) Prove that the electric dipole moment of the point symmetric charge distribution p (—r) = p(r) is equal to zero. [a) p(t) = Y,eara(t) ] a 12 Stationary current. Show that in the case of stationary currents diyy = 0 in bounded volume V, I jdV = 0 is valid. Instructions: Divide the current into closed current loops for which I jdV = cf> Idl = 0 13 The magnetic dipole moment of current distribution j(r, t) in the finite volume J7 is defined by the relation: m(t) = ±fvrxj(r,t)dV. a) What is the magnetic dipole moment of the system N of point charges ea located at points ra(t) and moving at velocities va(f)7 b ) What is the magnetic dipole moment of the closed plane curve r with linear current / flowing though it? Instructions: j dV = I dl. n [ a) m{i) = | E eara(t) x va[t), l))m=^rxdl= ISn, m = 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 , Fa/S = -Fb/S = eE2/2 | 16 Maxwell stress tensor Analyze the meaning of Maxwell stress tensor of electrostatic field in a homogeneous soft dielectric medium. Using the relation (1.54), show that in the direction of field E, the dielectric medium is subjected to tensile stress eE212, while in the direction perpendicular to E, it is subjected to pressure stress of the same magnitude. (Analogous results also apply to a constant magnetic field.) F = j[eE{E ■ n) - ^£E2n]df; sv in cases E || df and E ± df, where df= ndf we obtain 7 Electromagnetic radiation of the Sun near the Earth has an intensity I — 1365 W in-2 called solar irradiance (formerly a solar constant). Based on this, calculate the mean values of the magnitude of the electric field and the magnetic field in the electromagnetic field of solar radiation near the Earth. These are the quantities E = y/{E2} I B = vTB3}" (fta = 4x ■ 10"7 N A"2). [510 V/m, 1,7-10"6 T] 18 Potentials of a moving charge. Calculate the potentials of the electromagnetic field induced in vacuum by a point particle with charge q, which moves uniformly in a straight line at the velocity V= (V, 0,0) in the inertial system S of an observer. What is the form of the equipotential surfaces of the scalar potential? Instructions: Use the rest system 5" of a particle located at the origin of the system 5" and the special Lorentz transformation 5" —> S of the four-potentials = {

S of the electromagnetic field tensor components Fvv (xA). f = 4xs0 y/(x-Vt)* + (l-P*)(tf+z*)' A = (F/C> E = q(l-ß2)(r-Vt) 47t£D[(x - Vi}* + (1 - ß*)(y* + 22)]3/2 ELECTROMAGNETIC WAVES - Problems 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, xi = —a and e2 = —1, x2 = a, formulate the equation of the line of force that originates from the charge ex at an angle a with an x-axis. At what angle /? does this line of force enter the charge e27 Specify the minimum and maximum value, and such that the lines of force still end up in the charge e2. 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 e, in points x, of the x-axis is ft (x, y) = y. I y-ejcos^j = C, kde cosijj = , x~x\ 1 2 Refraction offorce lines of electric intensity. Determine the law of refraction of force lines of electric intensity E and the surface density nP of a charge bound on the boundary of two soft dielectric mediums e, ft. Instructions: Eu — E2I = 0, eiEi„ - e2E2„ = 0 3 Linear polarization. What is a complex notation of a monochromatic plane wave propagating in the soft medium e, ft in the direction of the positive axis z? The wave is linearly polarized in the direction of the x-axis. [ E = (E0eiikx-'1,t\ 0, 0), B = (0, ^n.e»(**-<"*), 0)] 4 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)],Eoexp[-iiu(^x +1) - f ]), B = (0, -n^ exp[-^{=2- +1) - fBJtp+1)])] 5 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. £ fa) A= Eoe^r-^-^-b) A = sinffe - wi), ±^ cosffcz - wt),0) 1 „