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Unformatted text preview: 29 The Motion of Charges in Electrici
and Magnetic Fields 29—1 Motion in a uniform electric or magnetic ﬁeld We want now to describe—mainly in a qualitative way—the motions of
charges in various Circumstances. Most of the interesting phenomena in which
charges are movmg in ﬁelds occur in very complicated situations. With many,
many charges all interacting With each other For instance, when an electromagne
tic wave goes through a block of material or a plasma, billions and billions of
charges are interacting with the wave and with each other. We Will come to such
problems later, but now weJust want to discuss the much Simpler problem of the
motions of a smgle charge in a given ﬁeld. We can then disregard all other charges
—except, of course, those charges and currents which most somewhere to produce
the ﬁelds we Will assume. We should probably ask ﬁrst about the motion of a particle in a uniform elec
tric ﬁeld At low veloc1ties, the motion is not particularly interesting~it is just a
uniform acceleration in the direction of the ﬁeld. However, if the particle picks
up enough energy to become relativistic, then the motion gets more complicated.
But we Will leave the solution for that case for you to play with Next, we conSider the motion in a uniform magnetic ﬁeld With zero electric
ﬁeld. We have already solved this problem—one solution is that the particle goes
in a Circle The magnetic force qv X B is always at right angles to the motion.
so dp/dt is perpendicular top and has the magnitude zip/R, where R is the radius
of the Circle. F=qz)3=% The radius of the Circular orbit is then P R — 213 (29.1) That is only one poss1bility. If the particle has a component of its motion
along the ﬁeld direction, that motion is constant, since there can be no component
ofthe magnetic force in the direction of the ﬁeld. The general motion ofa particle
in a uniform magnetic ﬁeld is a constant velocity parallel to B and a circular motion
at right angles to B—the trajectory is a cylindrical helix (Fig. 29—1). The radius
of the helix is given by Eq (29 1) if we replace p by pi, the component of mo
mentum at right angles to the ﬁeld. 29—2 Momentum analysis A uniform magnetic ﬁeld is often used in making a "momentum analyzer,"
or “momentum spectrometer," for highenergy charged particles. Suppose that
charged particles are shot into a uniform magnetic ﬁeld at the point A in Fig.
29—2(a), the magnetic ﬁeld being perpendicular to the plane of the drawmg. Each
particle will go into an orbit which is a circle whose radius is proportional to its
momentum. If all the particles enter perpendicular to the edge of the ﬁeld, they
will leave the ﬁeld at a distance x (from A) which is proportional to their momentum
p. A counter placed at some pomt such as C Will detect only those particles Whose
momentum is in an interval A]; near the momentum p = qu/2 It is, of course, not necessary that the particles go through 1800 before they
are counted. but the socalled “180° spectrometer” has a special property it is not 29—1 29—1 Motion in a uniform electric
or magnetic ﬁeld 29—2 Momentum analysis 29—3 An electrostatic lens 29—4 A magnetic lens 29—5 The electron microscope
29—6 Accelerator guide ﬁelds 29—7 Alternatinggradient focusing 29—8 Motion in crossed electric
and magnetic ﬁelds Rewew Chapter 30, Vol 1, Dﬂraction (0) (b) Fig. 29—1. Motion of 0 particle in a
uniform magnetic ﬁeld. /’/V / UN I FORM MAGNETIC FIELD 4 ozrecron / POINT SOURCE
(b) Fig. 29—2. A uniformfield, momen
tum spectrometer with 180° focusing:
(a) different momenta; (b) different
angles. (The magnetic ﬁeld is directed
perpendicular to the plane of the ﬁgure.) Fig. 29—3. An axialﬁeld spectrom
eter. Fig. 294. An ellipsoidal coil with
equal currents in each axial interval Ax
produces a uniform magnetic field inside. necessary that all the particles enter at right angles to the ﬁeld edge. Figure 29—2(b)
shows the trajectories of three particles, all with the same momentum but entering
the ﬁeld at different angles. You see that they take different trajectories, but all
leave the ﬁeld very close to the pomt C. We say that there is a “focus.” Such a
focus1ng property has the advantage that larger angles can be accepted at A—
although some limit is usually imposed, as shown in the ﬁgure. A larger angular
acceptance usually means that more particles are counted in a given time, decreasing
the time required for a given measurement. By varying the magnetic ﬁeld, or m0ving the counter along in x, or by usmg
many counters to cover a range of x, the “spectrum” of momenta in the incoming
beam can be measured. [By the “momentum spectrum”f(p), we mean that the number of particles with momenta between p and (p + dp) is f(p) dp.] Such
measurements have been made, for example, to determine the distribution of
energies in the Bdecay of various nuclei. There are many other forms of momentum spectrometers, but we will describe
just one more, which has an especially large solid angle of acceptance. It is based
on the helical orbits in a uniform ﬁeld, like the one shown in Fig. 29—]. Let’s
think of a cylindrical coordinate system—p, 0, z—set up With the zaxis along the
direction of the ﬁeld. If a particle is emitted from the origin at some angle a
With respect to the zaxis, it will move along a spiral whose equation is p = asinkz, 6 = bz, where a, b, and k are parameters you can easily work out in terms ofp, a, and the
magnetic ﬁeld B. If we plot the distance p from the axis as a function of z for a
given momentum, but for several starting angles, we will get curves like the solid
ones drawn in Fig. 29—3. (Remember that this is just a kind of projection of a
helical trajectory.) When the angle between the axis and the starting direction
is larger, the peak value of p is large but the longitudinal velocity is less, so the
trajectories for different angles tend to come to a kind of “focus” near the pomt
A in the ﬁgure. If we put a narrow aperture of A, particles With a range of initial
angles can still get through and pass on to the axis, where they can be counted by
the long detector D. Particles which leave the source at the origin with a higher momentum but
at the same angles, follow the paths shown by the broken lines and do not get
through the aperture at A. So the apparatus selects a small interval of momenta
The advantage over the ﬁrst spectrometer described 15 that the aperture A—and
the aperture A’—can be an annulus, so that particles which leave the source in a
rather large solid angle are accepted. A large fraction of the particles from the
source are used—an important advantage for weak sources or for very precise
measurements. One pays a price for this advantage, however, because a large volume of
uniform magnetic ﬁeld is required, and this is usually only practical for lowenergy
particles One way of making a uniform ﬁeld, you remember, is to wind a cm] on
a sphere, with a surface current density proportional to the sine of the angle
You can also show that the same thing 18 true for an ellips01d of rotation. So such
spectrometers are often made by Winding an elliptical coil on a wooden (or alumi
num) frame. All that IS required is that the current in each interval of axial distance
Ax be the same, as shown in Fig 29—4 29—3 An electrostatic lens Particle focus1ng has many applications. For instance, the electrons that leave
the cathode in a TV picture tube are brought to a focus at the screen—to make a
fine spot. In this case, one wants to take electrons all of the same energy but With
different initial angles and bring them together in a small spot. The problem is
like focus1ng light with a lens, and deVices which do the corresponding job for
particles are also called lenses. 29—2 Fig. 29—5. An electronaiic lens. The field lines shown are “lines of ferce,” that is, of qE. One example of an electron lens is sketched in Fig 29~5. It is an “electro
static” lens whose operation depends on the electric ﬁeld between two adjacent
electrodes. Its operation can be understood by considering what happens to a
parallel beam that enters from the left. When the electrons arrive at the region a,
they feel a force With a sideWise component and get a certain impulse that bends them
toward the axis You might think that they would get an equal and opposite im
pulse in the region I), but that is not so. By the time the electrons reach b they have
gained energy and so spend less time in the region b. The forces are the same, but
the time is shorter, so the impulse is less. In gomg through the regions a and b,
there is a net axial impulse, and the electrons are bent toward a common pomt
In leaving the highvoltage region, the particles get another kick toward the mm
The force is outward in region c and inward in region d, but the particles stay longer
in the latter region, so there is again a net impulse For distances not too far from
the axis, the total impulse through the lens is proportional to the distance from the
ax13 (Can you see why"), and this is just the condition necessary for lenstype
focusmg. You can use the same arguments to show that there is focusing if the
potential of the middle electrode is either pOSitive or negative with respect to the
other two. Electrostatic lenses of this type are commonly used in cathoderay
tubes and in some electron microscopes. 29—4 A magnetic lens Another kind of lens—often found in electron microscopes—is the magnetic
lens sketched schematically in Fig. 29—6. A cylindrically symmetric electromagnet
has very sharp circular pole tips which produce a strong, nonuniform ﬁeld in a
small region. Electrons which travel vertically through this region are focused
You can understand the mechanism by looking at the magniﬁed view of the poletip
region drawn in Fig. 29—7. ConSider two electrons a and b that leave the source
S at some angle With respect to the aXis. As electron (1 reaches the beginning of the
ﬁeld, it is deﬂected awayfrom you by the horizontal component of the ﬁeld But
then it will have a lateral velocity, so that when it passes through the strong vertical
ﬁeld, it will get an impulse toward the aXlS. Its lateral motion is taken out by the
magnetic force as it leaves the ﬁeld, so the net effect is an impulse toward the
axis, plus a “rotation” about the axis. All the forces on particle b are oppos1te,
so it also 18 deﬂected toward the ax1s. In the ﬁgure, the divergent electrons are
brought into parallel paths. The action is like a lens With an object at the focal
point. Another Similar lens upstream can be used to focus the electrons back to a
single pomt, making an image of the source S. 29—5 The electron microscope You know that electron microscopes can “see” objects too small to be seen
by optical microscopes. We discussed in Chapter 30 of Vol. I the basic limitations
of any optical system due to diffraction of the lens opening Ifa lens opening sub 29—3 Fig. 29—6. A magnetic lens. 4—7
/
/ /
/
_/
S
B L Fig. 29—7.
magnetic lens. Electron motion in the LENS
OPENING SOURCE Fig. 29—8. The resolution of a micro
scope is limited by the angle subtended
from the source. BLURRED
IMAGE LENS V OPENING S POINT SOURCE Fig. 29—9. aberration of a lens. Spherical FlELD STRONGER
HERE Fig. 29—l0. Particle motion in a slightly nonuniform ﬁeld. tends the angle 20 from a source (see Fig. 29—8), two neighboring spots at the source
cannot be seen as separate if they are closer than about where >\ is the wavelength of the light. With the best optical microscope, 0 ap
proaches the theoretical limit of 90°, so 6 is about equal to )\, or approximately
5000 angstroms. The same limitation would also apply to an electron microscope, but there
the wavelength is—for 50kilovolt electrons—about 0.05 angstrom. If one could
use a lens opening of near 30°, it would be possible to see objects only § of an
angstrom apart. Since the atoms in molecules are typically 1 or 2 angstroms apart.
we could get photographs of molecules. Biology would be easy; we would have
a photograph of the DNA structure. What a tremendous thing that would be!
Most of presentday research in molecular biology is an attempt to ﬁgure out the
shapes of complex organic molecules. If we could only see them! Unfortunately, the best resolving power that has been achieved in an electron
microscope is more like 20 angstroms. The reason is that no one has yet deSigned
a lens With a large opening. All lenses have “spherical aberration,” Which means
that rays at large angles from the axis have a different pomt of focus than the rays
nearer the axis. as shown in Fig. 29—9 By speCial techniques, optical microscope
lenses can be made With a negligible spherical aberration, but no one has yet
been able to make an electron lens which avoids spherical aberration. In fact, one can show that any electrostatic or magnetic lens of the types we
have described must have an irreduc1ble amount of spherical aberration. This
aberration—together With diffraction—limits the resolvmg power of electron
microscopes to their present value. The limitation we have mentioned does not apply to electric and magnetic
ﬁelds which are not axially symmetric or which are not constant in time. Perhaps
some day someone will think of a new kind of electron lens that Will overcome the
inherent aberration of the simple electron lens. Then we will be able to photograph
atoms directly. Perhaps one day chemical compounds will be analyzed by looking
at the positions of the atoms rather than by looking at the color of some pre
Cipitatel 29—6 Accelerator guide ﬁelds Magnetic ﬁelds are also used to produce special particle trajectories in high
energy particle accelerators. Machines like the cyclotron and synchrotron bring
particles to high energies by passing the particles repeatedly through a strong
electric ﬁeld. The particles are held in their cyclic orbits by a magnetic ﬁeld. We have seen that a particle in a uniform magnetic ﬁeld will go in a Circular
orbit. This, however, is true only for a perfectly uniform ﬁeld. Imagine a ﬁeld
B which is nearly uniform over a large area but which is slightly stronger in one
region than in another. If we put a particle of momentum p in this ﬁeld, it will go
in a nearly Circular orbit With the radius R = p/qB. The radius of curvature will,
however, be slightly smaller in the region where the ﬁeld is stronger. The orbit is
not a closed circle but Will “walk” through the ﬁeld, as shown in Fig. 29—10.
We can. if we Wish, consider that the slight “error” in the ﬁeld produces an extra
angular kick which sends the particle off on a new track. lfthe particles are to make
millions of revolutions in an accelerator, some kind of “radial focusing” is needed
which will tend to keep the tl‘ajCCtOl‘lES close to some design orbit. Another difﬁculty With a uniform ﬁeld is that the particles do not remain in a
plane If they start out with the slightest angle—or are given a slight angle by
any small error in the ﬁeld—they Will go in a helical path that Will eventually take
them into the magnet pole or the ceiling or ﬂoor of the vacuum tank Some
arrangement must be made to inhibit such vertical drifts; the ﬁeld must prOVide
“vertical focus1ng” as well as radial focusing. 294 MAGNETIC FIELD MAGNETIC FIELD CIRCULAR” ORBIT ‘ / l l
' l !
CIRCULAR
3 open B :‘\\i
l
l
F__L_*_a..
" r
Fig. 29—l 1. Radial motion of a par Fig. 29—12. Radial motion of a par ticle in a magnetic ﬁeld with a large
positive slope. ticle in a magnetic ﬁeld with a small
negative slope. One would, at ﬁrst, guess that radial focusmg could be provided by making a
magnetic ﬁeld which increases with increasmg distance from the center of the design
path Then ifa particle goes out to a large radius, it will be in a stronger ﬁeld which
will bend it back toward the correct radius. If it goes to too small a radius, the
bending Will be less, and it Wlll be returned toward the ClCSlgn radius. lfa particle
is once started at some angle with respect to the ideal Circle, it Will oscillate about
the ideal Circular orbit, as shown in Fig. 29—1]. The radial focusmg would keep the
particles near the Circular path. Actually there is still some radial focusing even with the opposne ﬁeld slope
This can happen if the radius of curvature of the trajectory does not increase more
rapidly than the increase in the distance of the particle from the center of the ﬁeld.
The particle orbits Will be as drawn in Fig 29—12. If the gradient of the ﬁeld is too
large, however. the orbits Will not return to the design radius but Will spiral inward
or outward. as shown in Fig. 29—13. We usually describe the slope of the ﬁeld in terms of the “relative gradient”
or ﬁeld Index, n: n _ dB/B_
_ dr/r (29.2) A guide ﬁeld gives radial focusing if this relative gradient is greater than — 1. A radial ﬁeld gradient Will also produce vertzcal forces on the particles
Suppose we have a ﬁeld that is stronger nearer to the center of the orbit and weaker
at the outside. A vertical cross section of the magnet at right angles to the orbit
might be as shown in Fig. 29—14. (For protons the orbits would be coming out of
the page ) If the ﬁeld is to be stronger to the left and weaker to the right, the lines
of the magnetic ﬁeld must be curved as shown . We can see that this must be so by using the law that the Circulation ofB is zero in free space. lfwe take coordinates
as shown in the ﬁgure, then 68x 68. “
(VXB)y_ 62 " ax "0’
01'
63, _ 6B,
79? ‘ ‘ax ' (293) Since we assume that aBs/ax is negative, there must be an equal negative 631/62.
29—5 / MAGNETIC
’ FIELD CIRCULAR l
ORBIT [
Bl—
l \\‘
a).a. r
Fig. 29—13. Radial motion of a par ticle in a magnetic ﬁeld with a large
negative slope. TO CENTER
OF ORBIT CENTRAL
ORBIT Fig. 29—l4. A vertical guide ﬁeld as
seen in a cross section perpendicular to
the orbits. Fig. 29—15. A horizontal focusing
quadrupole lens. If the “nominal” plane of the orbit is a plane of symmetry where B, 2 0, then the
radial component BI Will be negative above the plane and positive below The lines
must be curved as shown. Such a ﬁeld Will have vertical focusing properties. Imagine a proton that is
travelling more or less parallel to the central orbit but above it. The horizontal
component ofB Will exert a downward force on it. lfthe proton is below the central
orbit, the force is reversed. So there IS an effective “restoring force” toward the
central orbit. From our arguments there Will be vertical focusing, prowded that
the vertical ﬁeld decreases with increasing radius; but ifthe ﬁeld gradient is positive,
there Will be “vertical defocusmg.” So for vertical focusmg, the ﬁeld index n must
be less than zero. We found above that for radial focusing n had to be greater
than — 1. The two conditions together give the condition that —l<n<0 if the particles are to be kept in stable orbits. ln cyclotrons, values very near zero
are used; in betatrons and synchrotrons. the value n = —0.6 is typically used. 29—7 Alternatinggradient focusing Such small values ofn give rather “weak” focusing. It is clear that much mOre
effective radial focusmg would be given by a large positive gradient (n >> 1), but
then the vertical forces would be strongly defocusmg Similarly, large negative
slopes (11 << —1) would give stronger vertical forces but would cause radial de—
focusing. It was realized about 10 years ago, however, that a force that alternates
between strong focusmg and strong defocusing can still have a net focusmg force To explain how alternatinggradient focusing works, we Will ﬁrst describe the
operation ofa quadrupole lens, which is based on the same prinCiple. Imagine that
a uniform negative magnetic ﬁeld is added to the ﬁeld of Fig 29—14, With the
strength adjusted to make zero ﬁeld at the orbit. The resulting ﬁeld—«for small
displacements from the neutral pomt—Would be like the ﬁeld shown in Fig 29—15
Such a fourpole magnet is called a “quadrupole lens.” A pos1tive particle that
enters (from the reader) to the right or left of the center is pushed back toward
the center. If the particle enters above or below, it is pushed away from the center.
This 18 a horizontal focusing lens If the horizontal gradient is reversedas can
be done by reversmg all the polarities——the signs of all the forces are reversed
and we have a vertical focusmg lens, as in Fig. 29—16 For such lenses, the ﬁeld
strength—and therefore the focusmg forces—increase linearly With the distance
of the lens from the was. Fig. 29—16. A vertical focusing quad
rupole lens. 29~6 VERTICAL DISPLACEMENT
FROM AX IS __—’ 0 DISTAN CE HOIZ TAL ’
DISTANCE HORIZONTAL VERTICAL VERTICAL
FOCUS'NG DEFOCUSING DEFOCUSING FOCUSING
F'ELD FIELD FIELD FIELD (0) (b) Fig. 29—17. Horizontal and vertical focusing with a pair of quadrupole lenses. Now imagine that two such lenses are placed in series. If a particle enters With
some horizontal displacement from the axis, as shown in Fig. 2917(a), it will be
deﬂected toward the aXis in the ﬁrst lens. When it arrives at the second lens it is
closer to the axis, so the force outward is less and the outward deﬂection is less
There IS a net bending toward the axis; the average eﬂect is horizontally focusmg
On the other hand, if we look at a particle which enters off the ax1s in the vertical
direction, the path will be as shown in Fig. 29—17(b). The particle is ﬁrst deﬂected
away from the axis, but then it arrives at the second lens With a larger displacement,
feels a stronger force, and so is bent toward the axis. Again the net effect is focusing.
Thus a pair of quadrupole lenses acts independently for horiZOntal and vertical
motion—very much like an optical lens. Quadrupole lenses are used to form and
control beams of particles in much the same way that optical lenses are used for
light beams. We should point out that an alternatinggradient system does not always
produce focusing. If the gradients are too large (in relation to the particle momen
tum or to the spacing between the lenses), the net effect can be a defocusing one.
You can see how that could happen if you imagine that the spacing between the
two lenses of Fig. 29—17 were increased, say, by a factor of three or four. Let’s return now to the synchrotron guide magnet. We can consider that it
consrsts of an alternating sequence of “positive” and “negative” lenses with a
superimposed uniform ﬁeld. The uniform ﬁeld serves to bend the particles, on the
average, in a horizontal circle (with no effect on the vertical motion), and the
alternating lenses act on any particles that might tend to go astray—pushing them
always t0ward the central orbit (on the average). There is a nice mechanical analog which demonstrates that a force which
alternates between a “focusing” force and a “defocusing” force can have a net
“focusing” effect.
rod with a weight on the end, suspended from a pivot which is arranged to be moved
rapidly up and down by a motor driven crank. Such a pendulum has two equrli
brium positions. Besrdes the normal, downwardhanging position, the pendulum
is also in equilibrium “hanging upward”—with its “bob” above the pivot! Such a
pendulum is drawn in Fig. 29—18. By the f0110wing argument you can see that the vertical pivot motion is
equivalent to an alternating focusmg force. When the pivot is accelerated down
ward, the “bob” tends to move inward, as indicated in Fig. 29—19. When the
pivot is accelerated upward, the effect is reversed. The force restoring the “bob”
toward the axis alternates, but the average effect is a force toward the axis. So the
pendulum Will swmg back and forth about a neutral pOSition which isjust opposite
the normal one. There is, of course, a much easier way of keeping a pendulum upside down,
and that is by balancing it on your ﬁnger' But try to balance two independent
sticks on the same ﬁnger! Or one stick with your eyes closed! Balancing involves
making a correction for what is going wrong. And this is not possible, in general.
if there are several things going wrong at once. In a synchrotron there are billions
of particles going around together, each one of which may start out with a different
“error.” The kind of focusmg we have been describing works on them all. 29—7 Imagine a mechanical “pendulum” which consists of a solid Fig. 29—18. A pendulum with an
oscillating pivot can have a stable posi
tion with the bob above the pivot. / \
i l
a
\\ \ \ Xvi \L‘ Fig. 29—19. A downward accelera
tion of the pivot causes the pendulum to
move toward the vertical. Vd
vo —’
TE
0
B
Fig. 29—20. Path of a particle In crossed electric and magnetic ﬁelds. 29—8 Motion in crossed electric and magnetic ﬁelds So far we have talked about particles in electric ﬁelds only or in magnetic
ﬁelds only. There are some interesting effects when there are both kinds of ﬁelds
at the same time. Suppose we have a uniform magnetic ﬁeld B and an electric
ﬁeld E at right angles. Particles that start out perpendicular to B will move in a
curve like the one in Fig 29—20 (The ﬁgure is a plane curve, not a helix!) We can
understand this motion qualitatively. When the particle (assumed posmve) moves
in the direction of E, it picks up speed. and so it is bent less by the magnetic ﬁeld.
When it is going against the Eﬁeld. it loses speed and is continually bent more by
the magnetic ﬁeld. The net eﬁect is that it has an average “drift” in the direction
ofE X B. We can. in fact, show that the motion IS a uniform Circular motion super
imposed on a uniform Sidewise motion at the speed I'd = E/B—the trajectory in
Fig. 29—20 is a cyclmd. Imagine an observer who is mov1ng to the right at a con—
stant speed. In his frame our magnetic ﬁeld gets transformed to a new magnetic
ﬁeld plus an electric ﬁeld in the downward direction. If he has Just the right speed.
his total electric ﬁeld Will be zero. and he Will see the electron gomg in a Circle. So
the motion we see is a Circular motion. plus a translation at the drift speed
I»; = E/B The motion of electrons in crossed electric and magnetic ﬁelds is the
baSis of the magnetron tubes, 1 e.. osc1lators used for generating microwave energy. There are many other interesting examples of particle motions in electric and
magnetic ﬁelds~such as the orbits of the electrons and protons trapped in the
Van Allen belts—but we do not, unfortunately, have the time to deal With them here 29—8 ...
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This note was uploaded on 06/18/2009 for the course PHYSICS none taught by Professor Leekinohara during the Spring '09 term at Uni. Nottingham  Malaysia.
 Spring '09
 LeeKinohara
 Physics, Charge

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