18 The Radiation of an Accelerating Charge in the Electromagnetic Field 18.1 MOTION OF A CHARGE IN AN ELECTROMAGNETIC FIELD In previous chapters the Stokes vectors were determined for charges moving in a linear, circular, or elliptical path At first sight the examples chosen appear to have been made on the basis of simplicity However, the examples were chosen because charged particles actually move in these paths in an electromagnetic field; that is, the examples are based on physical reality In this section we show from Lorentz’s force equation that in an electromagnetic field charged particles follow linear and circular paths In the following section we determine the Stokes vectors corresponding to these physical configurations The reason for treating the motion of a charge in this chapter as well as in the previous chapter is that the material is necessary to understand and describe the Lorentz–Zeeman effect Another reason for discussing the motion of charged particles in the electromagnetic field is that it has many important applications Many physical devices of importance to science, technology, and medicine are based on our understanding of the fundamental motion of charged particles In particle physics these include the cyclotron, betatron, and synchrotron, and in microwave physics the magnetron and traveling-wave tubes While these devices, per se, will not be discussed here, the mathematical analysis presented is the basis for describing all of them Our primary interest is to describe the motion of charges as they apply to atomic and molecular systems and to determine the intensity and polarization of the emitted radiation In this chapter we treat the motion of a charged particle in three specific configurations of the electromagnetic field: (1) the acceleration of a charge in an electric field, (2) the acceleration of a charge in a magnetic field, and (3) the acceleration of a charge in perpendicular electric and magnetic fields In particular, the motion of a charged particle in perpendicular electric and magnetic fields is extremely interesting not only from the standpoint of its practical importance but because the paths taken by the charged particle are quite beautiful and remarkable Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved In an electromagnetic field the motion of a charged particle is governed by the Lorentz force equation: F ¼ q½E þ ðv  BÞ ð18-1Þ where q is the magnitude of the charge, E is the applied electric field, B is the applied magnetic field, and v is the velocity of the charge The background to the Lorentz force equation can be found in the texts given in the references The text by G P Harnwell on electricity and magnetism is especially clear and illuminating Quite understandably, because of the importance of the phenomenon of the radiation of accelerating charges in the design and fabrication of instruments and devices, many articles and textbooks are devoted to the subject Several are listed in the references 18.1.1 Motion of an Electron in a Constant Electric Field The first and simplest example of the motion of an electron in an electromagnetic field is for a charge moving in a constant electric field The field is directed along the z axis and is of strength E0 The vector representation for the general electric field E is E ¼ Ex ux þ Ey uy þ Ez uz ð18-2Þ Since the electric field is directed only in the z direction, Ex ¼ Ey ¼ 0, so E ¼ Ez uz ¼ E0 uz ð18-3Þ For simplicity the motion of the electron is restricted to the xz plane and is initially moving with a velocity v0 at an angle from the z axis This is shown in Fig 18-1 Because there is no magnetic field, the Lorentz force equation (18-1) reduces to m€r ¼ ÀeE Figure 18-1 ð18-4Þ Motion of an electron in the xz plane in a constant electric field directed along the z axis Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved where m is the mass of the electron In component form (18-4) is mx€ ¼ ð18-5aÞ my€ ¼ ð18-5bÞ mz€ ¼ ÀeEz ¼ ÀeE0 ð18-5cÞ At the initial time t ¼ the electron is assumed to be at the origin of the coordinate system, so xð0Þ ¼ zð0Þ ¼ ð18-6Þ Similarly, the velocity at the initial time is assumed to be x_ ð0Þ ¼ vx ¼ v0 sin ð18-7aÞ z_ð0Þ ¼ vz ¼ v0 cos ð18-7bÞ There is no force in the y direction, so (18-5b) can be ignored We integrate (18-5a) and (18-5c) and find x_ ðtÞ ¼ C1 z_ðtÞ ¼ À ÀeE0 t þ C2 m ð18-8aÞ ð18-8bÞ where C1 and C2 are constants of integration From the initial conditions, C1 and C2 are easily found, and the specific solution of (18-8) is x_ ðtÞ ¼ v0 sin z_ðtÞ ¼ ÀeE0 t þ v0 cos m ð18-9aÞ ð18-9bÞ Integrating (18-9) once more yields xðtÞ ¼ v0 t sin zðtÞ ¼ ÀeE0 t2 þ v0 t cos 2m ð18-10aÞ ð18-10bÞ where the constants of integration are found from (18-6) to be zero We can eliminate t between (18-10a) and (18-10b) to obtain ! eE0 zðtÞ ¼ À ð18-11Þ x2 þ ðcot Þx 2mv20 sin2 which is the equation of a parabola The path is shown in Fig 18-2 Inspecting (18-11) we see that if ¼ then zðtÞ ¼ That is, the electron moves in a straight line starting from the origin along the z axis and ‘‘intercepts’’ the z axis at infinity (1) However, if is not zero, then we can determine the Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-2 Parabolic path of an electron in a constant electric field positions x(t) where the electron intercepts the z axis by setting z(t) ¼ in (18-11) On doing this the intercepts are found to occur at xðtÞ ¼ xðtÞ ¼ mv20 sin 2 eE0 ð18-12aÞ ð18-12bÞ The first value corresponds to our initial condition x(0) ¼ z(0) ¼ Equation (18-12b) shows that the maximum value of x is attained by setting ¼ 45 , so xmax ¼ mv20 eE0 ð18-13Þ This result is not at all surprising, since (18-11) is identical in form to the equation for describing a projectile moving in a constant gravitational field Finally, the maximum value of z is found from (18-11) to be ! mv20 zðtÞ ¼ sin 2 eE ð18-14aÞ or zmax ¼ xmax where we have used (18-12b) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð18-14bÞ 18.1.2 Motion of a Charged Particle in a Constant Magnetic Field We now consider the motion of an electron moving in a constant magnetic field The coordinate configuration is shown in Fig 18-3 In the figure B is the magnetic field directed in the positive z direction The Lorentz force equation (18-1) then reduces to, where the charge on an electron is q ¼ Àe, F ¼ Àeðv  BÞ ð18-15Þ Equation (18-15) can be expressed as a differential equation: m€r ¼ Àeðv  BÞ ð18-16Þ where m and r¨ are the mass and acceleration vector of the charged particle, respectively In component form (18-16) is mx€ ¼ Àeðv  BÞx ð18-17aÞ my€ ¼ Àeðv  BÞy ð18-17bÞ mz€ ¼ Àeðv  BÞz ð18-17cÞ where the subscript on (v  B) refers to the appropriate component to be taken The vector product v  B can be expressed as a determinant ux uy uz y_ z_ ð18-18Þ v  B ¼ x_ Bx By Bz where ux, uy, and uz are the unit vectors pointing in the positive x, y, and z directions, respectively and the velocities have been expressed as x_ , y_ , and z_ The constant magnetic field is directed only along z, so Bz ¼ B and Bx ¼ By ¼ Then, (18-18) and (18-17) reduce to mx€ ¼ Àeðy_ BÞ ð18-19aÞ my€ ¼ ÀeðÀx_ BÞ ð18-19bÞ mz€ ¼ ð18-19cÞ Figure 18-3 Motion of an electron in a constant magnetic field Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Equation (18-19c) is of no interest because the motion along z is not influenced by the magnetic field The equations of motion are then x€ ¼ ÀeB y_ m ð18-20aÞ eB x_ ð18-20bÞ m Equation (18-20a) and (18-20b) can be written as a single equation by introducing the complex variable (t): y€ ¼ ðtÞ ¼ xðtÞ þ iyðtÞ ð18-21Þ Differentiating (18-21) with respect to time, we have _ ¼ x_ þ iy_ ð18-22aÞ € ¼ x€ þ iy€ ð18-22bÞ Multiplying (18-20b) by i and adding this result to (18-20a) and using (18-22a) leads to ieB _ ¼ ð18-23Þ m The solution of (18-23) is readily found by assuming a solution of the form: € À ðtÞ ¼ e!t ð18-24Þ Substituting (18-24) into (18-23) we find that !ð! À i!c Þ ¼ ð18-25Þ where !c ¼ eB/m is the frequency of rotation, known as the cyclotron frequency Equation (18-25) is called the auxiliary or characteristic equation of (18-23), and from (18-25) the roots are ! ¼ 0, i!c The general solution of (18-23) can be written immediately as ðtÞ ¼ c1 þ c2 ei!c t ð18-26Þ where c1 and c2 are constants of integration To provide a specific solution for (18-23), we assume that, initially, the charge is at the origin and moving along the x axis with a velocity v0 Thus, we have xð0Þ ¼ x_ ð0Þ ¼ v0 yð0Þ ¼ y_ ð0Þ ¼ ð18-27aÞ ð18-27bÞ which can be expressed in terms of (18-21) and (18-22a) as ð0Þ ¼ xð0Þ þ iyð0Þ ¼ ð18-28aÞ _ð0Þ ¼ x_ ð0Þ þ iy_ ð0Þ ¼ v0 ð18-28bÞ This leads immediately to c1 ¼ Àc2 c2 ¼ iv0 !c Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð18-29aÞ ð18-29bÞ so the specific solution of (18-26) is ðtÞ ¼ À iv0 ð1 À ei!c t Þ !c ð18-30Þ Taking the real and imaginary part of (18-30) then yields v0 sin !c t !c v yðtÞ ¼ À ð1 À cos !c tÞ !c xðtÞ ¼ ð18-31aÞ ð18-31bÞ or v0 sin !c t !c v v y þ ¼ cos !c t !c !c xðtÞ ¼ Squaring and adding (18-32a) and (18-32b) give 2 v v x2 þ y þ ¼ !c !c ð18-32aÞ ð18-32bÞ ð18-33Þ which is an equation of a circle with radius v0/!c and center at x ¼ and y ¼ Àv0/!c Equations (18-32) and (18-33) show that in a constant magnetic field a charged particle does indeed move in a circle Also, (18-32) describes a charged particle moving in a clockwise direction as viewed along the positive axis toward the origin Equation (18-33) is of great historical and scientific interest, because it is the basis of one of the first methods and instruments used to measure the ratio e/m, namely, the mass spectrometer To see how this measurement is made, we note that since the electron moves in a circle, (18-33) can be solved for the condition where it crosses the y axis, which is x ¼ We see from (18-33) that this occurs at y¼0 y¼À ð18-34aÞ 2v0 !c ð18-34bÞ We note that (18-34b) is twice the radius ( ¼ v0/!c) This is to be expected because the charged particle moves in a circle Since !c ¼ eB/m, we can solve (18-34b) for e/m to find that e 2v0 ¼À ð18-35Þ m By The initial velocity 0 is known from equating the kinetic energy of the electron with the voltage applied to the charged particle as it enters the chamber of the mass spectrometer The magnitude of y where the charged particle is intercepted (x ¼ 0) is measured Finally, the strength of the magnetic field B is measured with a magnetic flux meter Consequently, all the quantities on the right side of (18-35) are known, so the ratio e/m can then be found The value of this ratio found in this manner agrees with those of other methods Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 18.1.3 Motion of an Electron in a Crossed Electric and Magnetic Field The final configuration of interest is to determine the path of an electron which moves in a constant magnetic field directed along the z axis and in a constant electric field directed along the y axis, a so-called crossed, or perpendicular, electric and magnetic field This configuration is shown in Fig 18-4 For this case Lorentz’s force equation (18-1) reduces to mx€ ¼ Àeðy_ BÞ ð18-36aÞ my€ ¼ ÀeE þ eðx_ BÞ ð18-36bÞ mz€ ¼ ð18-36cÞ From (18-21) and (18-22), (18-36) can be written as a single equation: € À i!c _ ¼ À ieE m ð18-37Þ where !c ¼ eB/m Equation (18-37) is easily solved by noting that if we multiply by eÀi!c t then (18-37) can be rewritten as d Ài!c t ÀieE Ài!c t _ ðe Þ ¼ ð18-38Þ e dt m Straightforward integration of (18-38) yields eE ic1 i!c t ¼ tÀ e þ c2 m!c !c ð18-39Þ where c1 and c2 are constants of integration We choose the initial conditions to be xð0Þ ¼ yð0Þ ¼ x_ ð0Þ ¼ v0 y_ ð0Þ ¼ ð18-40aÞ ð18-40bÞ The specific solution of (18-39) is ¼ a þ ibð1 À cos Þ þ b sin Figure 18-4 Motion of an electron in a crossed electric and magnetic field Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð18-41aÞ where ¼ !c t ð18-41bÞ a¼ eE m!2c ð18-41cÞ b¼ v0 À eE=m!c !c ð18-41dÞ Equating the real and imaginary parts of (18-41a) and (18-21), we then find that xð Þ ¼ a þ b sin ð18-42aÞ yð Þ ¼ bð1 À cos Þ ð18-42bÞ Equation (18-42) is well known from analytical geometry and describes a general cycloid or trochoid Specifically, the trochoidal path is a prolate cycloid, cycloid, or curtate cycloid, depending on whether a < b, a ¼ b, or a > b, respectively We can easily understand the meaning of this result First, we note that if the applied electric field E were not present then (18-42) would reduce to the equation of a circle of radius b, so the electron moves along a circular path However, an electric field in the y direction forces the electron to move in the same direction continuously as the electron moves in the circular path Consequently, the path is stretched, so the circle becomes a general cycloid or trochoid This ‘‘stretching’’ factor is represented by the term a in (18-42a) We note that (18-40) shows ¼ corresponds to the origin Thus, is measured from the origin and increases in a clockwise motion We can easily find the maximum and minimum values of x( ) and y( ) over a single cycle of The maximum and minimum values of y( ) are simply and 2b and occur at ¼ and , respectively For x( ) the situation is more complicated From (18-42a) the angles where the minimum and maximum values of x( ) occur are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 À1 Æ b À a ¼ tan ð18-43Þ a The negative sign refers to the minimum value of x( ), and the positive sign refers to the maximum value of x( ) The corresponding maximum and minimum values of x( ) are then found to be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À1 Æ b À a ð18-44Þ xða, bÞ ¼ a tan Æ b2 À a2 a In particular, if we set b ¼ in (18-43) and (18-44) we have pffiffiffiffiffiffiffiffiffiffiffiffiffi! Æ À a2 ¼ tanÀ1 a xðaÞ ¼ a Á tan À1 pffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffi Æ À a2 Æ À a2 a Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð18-45Þ ð18-46Þ Equation (18-46) shows that x(a) is imaginary for a > 1; that is, a maximum and a minimum not exist This behavior is confirmed in Fig 18-13 and 18-14 for a ¼ 1.25 and a ¼ 1.5 Equation (18-45) ranges from a ¼ to 1; for a ¼ (no applied electric field) ¼ =2 and 3/2 (or À/2), respectively This is exactly what we would expect for a circular path Following the conventional notation the path of the electron moves counterclockwise, so /2 is the angle at the maximum point and 3/2 (À/2) corresponds to the angle at the minimum point Figure 18-5 shows the change in ðaÞ as the electric field (a) increases The upper curve corresponds to the positive sign of the argument in (18-45), and the lower curve corresponds to the negative sign, respectively We see that at a ¼ the maximum and minimum values converge The point of convergence corresponds to a cycloid This behavior is confirmed by the curve for x(a) in the figure for a ¼ 1, as we shall soon see The maximum and minimum points of the (prolate) cycloid are given by (18-46) We see immediately that for a ¼ we have x(0) ¼ Æ1 This, of course, applies to a circle For < a < we have a prolate cycloid For a cycloid a ¼ 1, and (18-46) gives x(1) ¼ and ; that is, the maximum and minimum points coincide This behavior is also confirmed for the plot of x(a) versus a at the value where a ¼ In Fig 18-6 we have plotted the change in the maximum and minimum values of x(a) as a increases from to The upper curve corresponds to the positive sign in (18-46), and the lower curve corresponds to the negative sign It is of interest to determine the points on the x axis where the electron path intersects or is tangent to the x axis This is found by setting y ¼ in (18-42b) We see that this is satisfied by ¼ or ¼ 2 Setting b ¼ in (18-42a), the points of intersection on the x axis are given by x ¼ and x ¼ 2a; the point x ¼ and y ¼ 0, Figure 18-5 Plot of the angle ðaÞ, Eq (18-45), for the maximum and minimum points as the electric field (a) increases Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-6 Plot of the maximum and minimum values of xð Þ written as x(a), Eq (18-46) as the electric field (a) increases from to we recall, is the position of the electron at the initial time t ¼ Thus, setting b ¼ in (18-42a), the initial and final positions of the electron for a ¼ are at x(i) ¼ and x(f ) ¼ 0, which is the case for a circle For the other extreme, obtained by setting a ¼ 1, the initial and final intersections are and 2, respectively Thus, as the magnitude of the electric field increases, the final point of intersection on the x axis increases In addition, as a increases, the prolate cycloid advances so that for a ¼ (a circle) the midpoint of the path is at x ¼ and for a ¼ the midpoint is at x ¼ We now plot the evolution of the trochoid as the electric field E(a) increases The equations used are, from (18-42) with b ¼ 1, xð Þ ¼ a þ sin ð18-47aÞ yð Þ ¼ À cos ð18-47bÞ It is of interest to plot (18-47a) from ¼ to 2 for a ¼ 0, 0.25, 0.50, 0.75, and 1.0 Figure 18-7 is a plot of the evolution of x( ) from a pure sinusoid for a ¼ to a cycloid for a ¼ The most significant feature of Fig 18-7 is that the maxima shift to the right as a increases This behavior continues until a ¼ 1, whereupon the maximum point virtually disappears Similarly, the minima shift to the left, so that at a ¼ the minimum point virtually disappears This behavior is later confirmed for a ¼ 1, a cycloid The paths of the electrons are specifically shown in Figs 18-8 to 18-15 The curves are plotted over a single cycle of (0 to 2) For these values (18-45) shows Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-7 Plot of xð Þ, Eq (18-47a), for a ¼ to Figure 18-8 The trochoidal path of an electron, a ¼ (a circle) that the path intersects the x axis at and 2a, respectively We select a to be 0, 0.25, 0.5, , 1.5 The corresponding intersections of the path on the x axis are then (0, 0), (0, /2), (0, ), , (0, 3) With these values of a, Figs 18-8 to 18-15 show the evolutionary change in the path Figure 18-15 shows the path of the electron as it moves over four cycles Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-9 The trochoidal path of an electron a ¼ 0.25 Figure 18-10 The trochoidal path of an electron, a ¼ 0.5 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-11 The trochoidal path of an electron, a ¼ 0.75 Figure 18-12 The trochoidal path of an electron, a ¼ 1.0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-13 The trochoidal path of an electron, a ¼ 1.25 Figure 18-14 The trochoidal path of an electron, a ¼ 1.5 18.2 STOKES VECTORS FOR RADIATION EMITTED BY ACCELERATING CHARGES We now determine the Stokes vectors for the radiation emitted by the accelerating charges undergoing the motions described in the previous section, namely, (1) the Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-15 The trochoidal path of an electron over four cycles, a ¼ 0.25 motion of an electron in a constant electric field, (2) the motion of an electron in a constant magnetic field, and (3) the motion of the electron in a crossed electric and magnetic field The components of the radiation field in spherical coordinates were shown in Chapter 16 to be E ¼ e ½x€ cos À z€ sin 4"0 c2 R ð16-8Þ E ¼ e ½y€ 4"0 c2 R ð16-9Þ These equations refer to the observation being made in the xz plane; that is, at ¼ The angle is the polar angle in the observer’s reference frame Recall that the Stokes parameters of the radiation field are defined by S0 ¼ E E à þ E Eà ð16-10aÞ S1 ¼ E E à À E Eà ð16-10bÞ S2 ¼ E Eà þ E E à ð16-10cÞ S3 ¼ iðE Eà À E E Ã Þ ð16-10dÞ In the following problems we represent the emitted radiation and its polarization in the form of Stokes vectors Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 18.2.1 Stokes Vector for a Charge Moving in an Electric Field The path of the charge moving in a constant electric field in the xz plane was found to be xðtÞ ¼ v0 t sin zðtÞ ¼ ÀeE0 t2 þ v0 t cos 2m ð18-10aÞ ð18-10bÞ We see that the accelerations of the charge in the x and z directions are then x€ ðtÞ ¼ z€ðtÞ ¼ À ð18-48aÞ eE0 m ð18-48bÞ Substituting (18-48) into (16-8) and (16-9) yields E ¼ e2 E0 sin m4"0 c2 R E ¼ ð18-49aÞ ð18-49bÞ and we immediately find from (18-49) that the Stokes vector is 1 !2 B À1 C e2 E C sin2 B S¼ @0 A m4"0 c R ð18-50Þ Equation (18-50) shows that the emitted radiation is linearly vertically polarized It also shows the accelerating electron emits the familiar dipole radiation pattern described by sin2 , so the intensity observed along the z axis is zero ( ¼ 0) and is a maximum when viewed along the x axis ( ¼ /2) Before we finish the discussion of (18-50) there is another point of interest that should be noted We observe that in (18-50) there is a factor of e2 =4"0 mc2 We now ask the question, what, if any, is the meaning of this quantity? The answer can be obtained by recalling that the electric field E ‘‘outside’’ of an electron is given by e ur ð18-51Þ E¼ 4"0 r2 where r is the distance from the center of the electron and ur is the unit radius vector We now imagine the electron has a radius a and compute the work that must be done to move another (positive) charge of the same magnitude from the surface of this electron to infinity The total work, or energy, required to this is Z1 W ¼ Àe E Á dr ð18-52Þ a where dr is drur Substituting (18-51) into (18-52) gives Z1 e2 dr e2 ¼ W¼ 4"0 a r2 4"0 a Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð18-53Þ We now equate (18-53) to the rest mass of the electron mc2 and find that a¼ e2 4"0 mc2 ð18-54Þ Thus, the factor e2/4"0mc2 is the classical radius of the electron The value of a is readily calculated from the values e ¼ 1.60  10À19 C, m ¼ 9.11  10À31 kg, and c ¼ 2.997  108 m/sec, which yields a ¼ 2:82  10À15 m ð18-55Þ We see that the radius of the electron is extremely small The factor e2/4"0mc2 appears repeatedly in radiation problems Later, it will appear again when we consider the problem where radiation is incident on an electron and is then re-emitted, that is, the scattering of radiation by an electron 18.2.2 Stokes Vector for a Charge Accelerating in a Constant Magnetic Field In the previous section we saw that the path described by an electron moving in a constant magnetic field is given by the equations: xðtÞ ¼ v0 sin !c t !c yðtÞ ¼ À v0 ð1 À cos !c tÞ !c ð18-31aÞ ð18-31bÞ where 0 is the initial velocity and !c ¼ eB/m is the cyclotron frequency Using the exponential representation: Refei!c t g ¼ cos !c t ð18-56aÞ RefÀiei!c t g ¼ sin !c t ð18-56bÞ we can then write x ¼ c ðÀiei!c t Þ ð18-57aÞ y þ c ¼ c ðei!c t Þ ð18-57bÞ where c ¼ v0 !c ð18-57cÞ The accelerations x€ ðtÞ and y€ ðtÞ are then x€ ðtÞ ¼ i c !2c ei!c t ð18-58aÞ y€ ðtÞ ¼ À c !2c ei!c t ð18-58bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and the radiation field components become E ¼ Àie c !2c cos ei!c t 4"0 c2 R ð18-59aÞ E ¼ e c !2c i!c t e 4"0 c2 R ð18-59bÞ From the definition of the Stokes parameters in (16-10) the Stokes vector is 1 þ cos2 2 B À cos2 C e c C ð18-60Þ S¼ !4c B @ A 4"0 c R cos which is the Stokes vector for elliptically polarized light radiating at the same frequency as the cyclotron frequency !c Thus, the Stokes vector found earlier for a charge moving in a circle is based on physical reality We see that (18-60) reduces to right circularly polarized light, linearly horizontally polarized light, and left circularly polarized light for ¼ 0, /2, and , respectively 18.2.3 Stokes Vector for a Charge Moving in a Crossed Electric and Magnetic Field The path of the electron was seen to be a trochoid described by xð Þ ¼ a þ b sin ð18-42aÞ yð Þ ¼ bð1 À cos Þ ð18-42bÞ where ¼ !c t eE a¼ m!2c v À eE=m!c b¼ !c ð18-41bÞ ð18-41cÞ ð18-41dÞ Differentiating (18-42a) and (18-42b) twice with respect to time and using (18-56) then gives x€ ðtÞ ¼ ib!2c ei!c t ð18-61aÞ b!2c ei!c t ð18-61bÞ y€ðtÞ ¼ and we immediately find that the Stokes vector is 1 þ cos2 B À cos2 C C S ¼ b2 !4c B @ A cos which, again, is the Stokes vector for elliptically polarized light Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð18-62Þ With this material behind us we now turn our attention to the Lorentz–Zeeman effect and see how the role of polarized light led to the acceptance of Maxwell’s electrodynamical theory in optics REFERENCES Books Jackson, J D., Classical Electrodynamics, John Wiley, New York, 1962 Sommerfeld, A., Lectures on Theoretical Physics, Vols I–V, Academic Press, New York, 1952 Harnwell, G P., Principles of Electricity and Electromagnetism, McGraw-Hill, New York, 1949 Humphries, S., Jr., Charged Particle Beams, John Wiley, New York, 1990 Hutter, R C E and Harrison, S W., Beam and Wave Electronics in Microwave Tubes, D Van Nostrand Princeton, 1960 Panofsky, W K H and Phillips, M., Classical Electricity and Magnetism, AddisonWesley, Reading, MA, 1955 Goldstein, H., Classical Mechanics, Addison-Wesley Reading, MA, 1950 Corben, H C and Stehle, P., Classical Mechanics, John Wiley, New York, 1957 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... sin2 , so the intensity observed along the z axis is zero ( ¼ 0) and is a maximum when viewed along the x axis ( ¼ /2) Before we finish the discussion of (18-50) there is another point of interest that should be noted We observe that in (18-50) there is a factor of e2 =4"0 mc2 We now ask the question, what, if any, is the meaning of this quantity? The answer can be obtained by recalling that the. .. final intersections are 0 and 2, respectively Thus, as the magnitude of the electric field increases, the final point of intersection on the x axis increases In addition, as a increases, the prolate cycloid advances so that for a ¼ 0 (a circle) the midpoint of the path is at x ¼ 0 and for a ¼ 1 the midpoint is at x ¼ We now plot the evolution of the trochoid as the electric field E(a) increases The equations... of the maximum and minimum values of xð Þ written as x(a), Eq (18-46) as the electric field (a) increases from 0 to 1 we recall, is the position of the electron at the initial time t ¼ 0 Thus, setting b ¼ 1 in (18-42a), the initial and final positions of the electron for a ¼ 0 are at x(i) ¼ 0 and x(f ) ¼ 0, which is the case for a circle For the other extreme, obtained by setting a ¼ 1, the initial and... path of an electron over four cycles, a ¼ 0.25 motion of an electron in a constant electric field, (2) the motion of an electron in a constant magnetic field, and (3) the motion of the electron in a crossed electric and magnetic field The components of the radiation field in spherical coordinates were shown in Chapter 16 to be E ¼ e ½x€ cos À z€ sin 4"0 c2 R ð16-8Þ E ¼ e ½y€ 4"0 c2 R ð16-9Þ These... polarization in the form of Stokes vectors Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 18.2.1 Stokes Vector for a Charge Moving in an Electric Field The path of the charge moving in a constant electric field in the xz plane was found to be xðtÞ ¼ v0 t sin zðtÞ ¼ ÀeE0 t2 þ v0 t cos 2m ð18-10aÞ ð18-10bÞ We see that the accelerations of the charge in the x and z directions are then x€ ðtÞ... that the radius of the electron is extremely small The factor e2/4"0mc2 appears repeatedly in radiation problems Later, it will appear again when we consider the problem where radiation is incident on an electron and is then re-emitted, that is, the scattering of radiation by an electron 18.2.2 Stokes Vector for a Charge Accelerating in a Constant Magnetic Field In the previous section we saw that the. .. Figure 18-8 The trochoidal path of an electron, a ¼ 0 (a circle) that the path intersects the x axis at 0 and 2a, respectively We select a to be 0, 0.25, 0.5, , 1.5 The corresponding intersections of the path on the x axis are then (0, 0), (0, /2), (0, ), , (0, 3) With these values of a, Figs 18-8 to 18-15 show the evolutionary change in the path Figure 18-15 shows the path of the electron... recalling that the electric field E ‘‘outside’’ of an electron is given by e ur ð18-51Þ E¼ 4"0 r2 where r is the distance from the center of the electron and ur is the unit radius vector We now imagine the electron has a radius a and compute the work that must be done to move another (positive) charge of the same magnitude from the surface of this electron to in nity The total work, or energy, required to... refer to the observation being made in the xz plane; that is, at ¼ 0 The angle is the polar angle in the observer’s reference frame Recall that the Stokes parameters of the radiation field are defined by S0 ¼ E E à þ E Eà ð16-10aÞ S1 ¼ E E à À E Eà ð16-10bÞ S2 ¼ E Eà þ E E à ð16-10cÞ S3 ¼ iðE Eà À E E Ã Þ ð16-10dÞ In the following problems we represent the emitted radiation and its polarization... trochoidal path of an electron, a ¼ 1.25 Figure 18-14 The trochoidal path of an electron, a ¼ 1.5 18.2 STOKES VECTORS FOR RADIATION EMITTED BY ACCELERATING CHARGES We now determine the Stokes vectors for the radiation emitted by the accelerating charges undergoing the motions described in the previous section, namely, (1) the Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 18-15 The trochoidal