18The Radiation of an AcceleratingCharge in the Electromagnetic Field

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 accelerati

The Radiation of an Accelerating

Charge in the 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 parti-cles 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 dis-cussed 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 accel-eration of a charge in perpendicular electric and magnetic fields In particular, the motion of a charged particle in perpendicular electric and magnetic fields is extre-mely interesting not only from the standpoint of its practical importance but because the paths taken by the charged particle are quite beautiful and remarkable

In an electromagnetic field the motion of a charged particle is governed by the Lorentz force equation:

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

Since the electric field is directed only in the z direction, Ex ¼Ey¼0, so

For simplicity the motion of the electron is restricted to the xz plane and is initially moving with a velocity v0

Because there is no magnetic field, the Lorentz force equation (18-1) reduces to

Figure 18-1 Motion of an electron in the xz plane in a constant electric field directed along the z axis

where m is the mass of the electron In component form (18-4) is

At the initial time t ¼ 0 the electron is assumed to be at the origin of the coordinate system, so

Similarly, the velocity at the initial time is assumed to be

_

x

There is no force in the y direction, so (18-5b) can be ignored We integrate (18-5a) and (18-5c) and find

_

x

_zzðtÞ ¼ eE0t

where C1and C2are constants of integration From the initial conditions, C1and C2 are easily found, and the specific solution of (18-8) is

_

x

_zzðtÞ ¼eE0t

Integrating (18-9) once more yields

zðtÞ ¼eE0t

2

where the constants of integration are found from (18-6) to be zero We can elim-inate t between (18-10a) and (18-10b) to obtain

zðtÞ ¼  eE0

2mv2

0sin2

!

which is the equation of a parabola The path is shown inFig 18-2

moves in a straight line starting from the origin 0 along the z axis and ‘‘intercepts’’

positions x(t) where the electron intercepts the z axis by setting z(t) ¼ 0 in (18-11).

On doing this the intercepts are found to occur at

xðtÞ ¼mv

2

The first value corresponds to our initial condition x(0) ¼ z(0) ¼ 0 Equation

, so

xmax¼mv20

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

zðtÞ ¼1 2

mv20 eE

!

ð18-14aÞ

or

zmax¼1

where we have used (18-12b)

Figure 18-2 Parabolic path of an electron in a constant electric field

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 inFig 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 e,

Equation (18-15) can be expressed as a differential equation:

where m and r¨ are the mass and acceleration vector of the charged particle, respec-tively In component form (18-16) is

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

v  B ¼

ux uy uz _

x y_ _zz

Bx By Bz

















ð18-18Þ

where ux, uy, and uzare the unit vectors pointing in the positive x, y, and z directions, respectively and the velocities have been expressed as _x, _y, and _zz The constant magnetic field is directed only along z, so Bz¼B and Bx¼By¼0 Then, (18-18) and (18-17) reduce to

Figure 18-3 Motion of an electron in a constant magnetic field

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

€ y

y ¼eB

Equation (18-20a) and (18-20b) can be written as a single equation by introducing the complex variable (t):

Differentiating (18-21) with respect to time, we have

_

€

Multiplying (18-20b) by i and adding this result to (18-20a) and using (18-22a) leads to

€

 ieB

The solution of (18-23) is readily found by assuming a solution of the form:

Substituting (18-24) into (18-23) we find that

where !c¼eB/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 c1þc2ei!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

_ x

which can be expressed in terms of (18-21) and (18-22a) as

_

This leads immediately to

c2¼iv0

so the specific solution of (18-26) is

ðtÞ ¼ iv0

!cð1  e

Taking the real and imaginary part of (18-30) then yields

xðtÞ ¼v0

yðtÞ ¼ v0

or

xðtÞ ¼v0

y þv0

!c¼

v0

Squaring and adding (18-32a) and (18-32b) give

x2þ y þv0

!c

¼ v0

!c

2

ð18-33Þ

which is an equation of a circle with radius v0/!cand center at x ¼ 0 and y ¼ v0/!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 0 We see from (18-33) that this occurs at

y ¼ 2v0

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 !c¼eB/m, we can solve (18-34b) for e/m

to find that

e

2v0 By

ð18-35Þ

The initial velocity 0is 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

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 inFig 18-4

For this case Lorentz’s force equation (18-1) reduces to

From (18-21) and (18-22), (18-36) can be written as a single equation:

€

  i!c ¼ _ ieE

where !c¼eB/m Equation (18-37) is easily solved by noting that if we multiply by

ei!c t then (18-37) can be rewritten as

d

dtðe

i!ctÞ ¼_ ieE

m

ei!c t

ð18-38Þ Straightforward integration of (18-38) yields

m!c

t  ic1

!c



where c1and c2are constants of integration We choose the initial conditions to be

_ x

The specific solution of (18-39) is

Figure 18-4 Motion of an electron in a crossed electric and magnetic field

a ¼ eE

b ¼v0eE=m!c

Equating the real and imaginary parts of (18-41a) and (18-21), we then find that

Equation (18-42) is well known from analytical geometry and describes a gen-eral 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 0 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 0 and 2b and occur at  ¼ 0 and , respectively For x() the situation is more complicated From (18-42a) the angles where the minimum and maximum values of x() occur are

 ¼tan1 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2a2

p a

!

ð18-43Þ

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

xða, bÞ ¼ a tan1 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2a2

p a

!

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2a2

p

ð18-44Þ

In particular, if we set b ¼ 1 in (18-43) and (18-44) we have

 ¼tan1 

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1  a2

p a

!

ð18-45Þ

xðaÞ ¼ a tan1 

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1  a2

p a

!



ffiffiffiffiffiffiffiffiffiffiffiffiffi

1  a2

p

ð18-46Þ

Equation (18-46) shows that x(a) is imaginary for a > 1; that is, a maximum and a minimum do not exist This behavior is confirmed in Fig 18-13 and 18-14 for

a ¼1.25 and a ¼ 1.5

Equation (18-45) ranges from a ¼ 0 to 1; for a ¼ 0 (no applied electric field)

 ¼ =2 and 3/2 (or /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 (/2) corre-sponds to the angle at the minimum point.Figure 18-5shows 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, respec-tively We see that at a ¼ 1 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 0 we have x(0) ¼ 1 This, of course, applies to a circle For 0 < a < 1 we have a prolate cycloid For a cycloid a ¼ 1, and (18-46) gives x(1) ¼ 0 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 1

as a increases from 0 to 1 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 0 in (18-42b) We see that this is satisfied by  ¼ 0 or  ¼ 2 Setting b ¼ 1 in (18-42a), the points of intersection on the x axis are given by x ¼ 0 and x ¼ 2a; the point x ¼ 0 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

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 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 used are, from (18-42) with b ¼ 1,

It is of interest to plot (18-47a) from  ¼ 0 to 2 for a ¼ 0, 0.25, 0.50, 0.75, and 1.0

cycloid for a ¼ 1

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 1 the minimum point virtually disappears This behavior is later confirmed for a ¼ 1, a cycloid

The paths of the electrons are specifically shown inFigs 18-8 to 18-15 The curves are plotted over a single cycle of  (0 to 2) For these values (18-45) shows Figure 18-6 Plot of the maximum and minimum values of xðÞ written as x(a), Eq (18-46)

as the electric field (a) increases from 0 to 1