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Unformatted text preview: 28 Electromagnetic Mass 28—1 The ﬁeld energy of a point charge In bringing together relativity and Maxwell’s equations, we have ﬁnished our
main work on the theory of electromagnetlsm. There are, of course, some details
we have skipped over and one large area that we will be concerned with in the future
——the interaction of electromagnetic ﬁelds with matter. But we want to stop for a
moment to show you that this tremendous ediﬁce, which is such a beautiful
success in explaining so many phenomena, ultimately falls on its face. When
you follow any of our physics too far, you ﬁnd that it always gets into some kind
of trouble. Now we want to discuss a serious trouble~the failure of the classical
electromagnetic theory. You can appreciate that there is a failure of all classrcal
physics because of the quantummechanical effects. Classical mechanics IS a mathe
matically consistent theory; it just doesn’t agree with experience. It is interesting,
though, that the classical theory of electromagnetism is an unsatisfactory theory
all by itself. There are difﬁculties assoc1ated with the ideas of Maxwell’s theory
which are not solved by and not directly associated with quantum mechanics.
You may say, “Perhaps there’s no use worrying about these difﬁculties. Since the
quantum mechanics 15 going to change the laws of electrodynamics, we should
wait to see what difﬁculties there are after the modiﬁcation.” However, when
electromagnetism is joined to quantum mechanics, the difﬁculties remain. So it
will not be a waste of our time now to look at what these difficulties are. Also,
they are of great historical importance. Furthermore, you may get some feeling
of accomplishment from being able to go far enough With the theory to see every
thing—including all of its troubles. The difﬁculty we speak of is associated with the concepts of electromagnetic
momentum and energy, when applied to the electron or any charged particle.
The concepts of simple charged particles and the electromagnetic ﬁeld are in some
way inconsistent. To describe the difficulty, we begin by doing some exercises
with our energy and momentum concepts. First, we compute the energy of a charged particle. Suppose we take a simple
model of an electron in which all of its charge q is uniformly distributed on the
surface of a sphere of radius a, which we may take to be zero for the special case of
a point charge. Now let’s calculate the energy in the electromagnetic ﬁeld. If
the charge is standing still, there is no magnetic ﬁeld, and the energy per unit
volume is proportional to the square of the electric ﬁeld. The magnitude of the
electric ﬁeld is q/41reor2, and the energy density is 2
— £0 2 = i .
u " 2 E 327r260r4 To get the total energy, we must integrate this density over all space. Using the
volume element 4‘Irr2 dr, the total energy, which we will call U,.1..C. is 2
1 q
f87T€0r2 dr. This is readily integrated. The lower limit is a, and the upper limit is 00, so Uelcc ll 2 = q
Uelcc 47.10 1 l
E 5 (28.1) 28—1 28—1 The ﬁeld energy of a point
charge 28—2 The ﬁeld momentum of a
moving charge 28—3 Electromagnetic mass 28—4 The force of an electron on
itself 28—5 Attempts to modify the
Maxwell theory 28—6 The nuclear force ﬁeld ELECTRON SPHERICAL
(1') Fig. 28—l. The fields E and B and the
momentum density 9 for a positive elec
tron. For a negative electron, E and B
are reversed but 9 is not. Fig. 28—2. The volume element
27rr2 sin 6 d0 clr used for calculating the
ﬁeld momentum. If we use the electronic charge qe for q and the symbol e2 for q?/41reo, then 2
g E; . (28.2) Uelec :
It is all ﬁne until we set a equal to zero for a point charge—there’s the great
difficulty. Because the energy of the ﬁeld vanes inversely as the fourth power of
the distance from the center, its volume integral is inﬁnite. There is an inﬁnite
amount of energy in the ﬁeld surrounding a point charge. What’s wrong with an inﬁnite energy? If the energy can’t get out, but must
stay there forever, is there any real difﬁculty With an inﬁnite energy? Of course, a
quantity that comes out inﬁnite may be annoying, but what really matters is only
whether there are any observable physical effects. To answer that question, we
must turn to something else besides the energy. Suppose we ask how the energy
changes when we move the charge. Then, if the changes are Inﬁnite, we will be
1n trouble. 28—2 The ﬁeld momentum of a moving charge Suppose an electron is moving at a uniform velocity through space. assuming
for a moment that the velocity is low compared with the speed of light. Assocrated
With this moving electron there is a momentum—even 1f the electron had no mass
before it was charged—because of the momentum in the electromagnetic ﬁeld.
We can show that the ﬁeld momentum IS in the direction of the velocity v of the
charge and is, for small velocities, proportional to v. For a point P at the distance
r from the center of the charge and at the angle 6 with respect to the line of motion
(see Fig. 28—1) the electric ﬁeld is radial and, as we have seen, the magnetic ﬁeld
is v X E/cz. The momentum density, Eq. (27.21), is g=€0EXB. It IS directed obliquely toward the line of motion, as shown in the ﬁgure, and has
the magnitude
g = g E2 sin 0. The ﬁelds are symmetric about the line of motion, so when we integrate over
space, the transverse components will sum to zero, giving a resultant momentum
parallel to v. The component ofg In this direction is g sm 0. which we must inte
grate over all space. We take as our volume element a ring with its plane per
pendicular to v, as shown in Fig. 28—2. Its volume is 27172 sin 0 d0 dr. The total
momentum is then
1v c2 E2 sin2 0 27172 sin 0 d0 dr. 1) =
Since E is independent of 6 (for u << c), we can immediately integrate over 0; the
integral is cos3 6 3 [sin3 6d0 = — [(1 — cos2 0) d(cos 0) = —cos 0 + The limits of 0 are 0 and 7r, so the 0integral gives merely a factor of 4/3, and The integral (for 1) << 6) is the one we have just evaluated to ﬁnd the energy; it is
q2/167rzega, and 01'
p = §— v. (28.3) 28—2 The momentum in the ﬁeld—the electromagnetic momentum—is proportional to
v. It is just what we should have for a particle with the mass equal to the coefﬁcient
of v. We can, therefore, call this coeﬁiCient the electromagnetic mass, mph.“ and
write it as 82 Inchw z: 3 525 ‘ (284) N 28—3 Electromagnetic mass Where does the mass come from? In our laws of mechanics we have supposed
that every object “carries” a thing we call the mass—which also means that it
“carries” a momentum proportional to its velocity. Now we discover that it is
understandable that a charged particle carries a momentum proportional to its
velocity. It might, in fact, be that the mass is just the effect of electrodynamics.
The origin of mass has until now been unexplained. We have at last in the theory
of electrodynamics a grand opportunity to understand something that we never
understood before. It comes out of the blue~or rather, from Maxwell and
Poynting—that any charged particle Will have a momentum proportional to its
velocity just from electromagnetic inﬂuences. Let’s be conservative and say, for a moment, that there are two kinds of mass—
that the total momentum of an object could be the sum of a mechanical momentum
and the electromagnetic momentum. The mechanical momentum is the “mechan
ical” mass, mum,” times v. In experiments where we measure the mass of a particle
by seeing how much momentum it has, or how it swmgs around in an orbit, we
are measuring the total mass. We say generally that the momentum is the total
mass (muloch + mom.) times the velocity. So the observed mass can consist of two
pieces (or possibly more if we include other ﬁelds): a mechanical piece plus an
electromagnetic piece. We know that there is deﬁnitely an electromagnetic piece,
and we have a formula for it. And there is the thrilling possibility that the me
chanical piece is met there at all—that the mass is all electromagnetic. Let’s see What size the electron must have if there is to be no mechanical mass.
We can ﬁnd out by setting the electromagnetic mass of Eq. (28.4) equal to the
observed mass m, of an electron. We ﬁnd 2 e2
a —— § — (£2 (28.5)
The quantity
2
e
r0 = mecz (28.6) is called the “classical electron radius”; it has the numerical value 2.82 X 10'13
cm, about one onehundredthousandth of the diameter of an atom. Why is r0 called the electron radius, rather than our a? Because we could
equally well do the same calculation With other assumed distributions of charges—
the charge might be spread uniformly through the volume of a sphere or it might
be smeared out like a fuzzy ball. For any particular assumption the factor 2/ 3
would change to some other fraction. For instance, for a charge uniformly dis
tributed throughout the volume of a sphere, the 2/3 gets replaced by 4/5. Rather
than to argue over which distribution is correct, it was decided to deﬁne r0 as the
“nominal” radius. Then different theories could supply their pet coefﬁcients. Let’s pursue our electromagnetic theory of mass. Our calculation was for
2) << c; what happens if we go to high velocities? Early attempts led to a certain
amount of confusion, but Lorentz realized that the charged sphere would contract
into a ellipsoid at high velocities and that the ﬁelds would change in accordance
with the formulas (26 6) and (26.7) we derived for the relativistic case in Chapter 26.
If you carry through the integrals for p in that case, you ﬁnd that for an arbitrary
velocity v, the momentum is altered by the factor l/\/l — 02/c2: 62 U N 28—3 In other words, the electromagnetic mass rises with velocity inversely as
\/1 — 02/c2—a discovery that was made before the theory of relativity. Early experiments were proposed to measure the changes with velocity in the
observed mass of a particle in order to determine how much of the mass was
mechanical and how much was electrical. It was believed at the time that the elec
trical part would vary with velocity, whereas the mechanical part would not. But
while the experiments were being done, the theorists were also at work. Soon the
theory of relativity was developed, which proposed that no matter what the origin
of the mass, it all should vary as m0/\/1 — vZ/cz. Equation (28.7) was the
beginning of the theory that mass depended on velocity. Let’s now go back to our calculation of the energy in the ﬁeld, which led to
Eq. (28.2). According to the theory of relativity, the energy U will have the mass
U/c2; Eq. (28.2) then says that the ﬁeld of the electron should have the mass Uolcc l 92
mice = 62 = 5 acz’ (28.8) which is not the same as the electromagnetic mass, mm... of Eq. (28.4). In fact. if
we Just combine Eqs. (28.2) and (28.4), we would write 3 2
Uelec = "irnelecC  This formula was discovered before relativity, and when Einstein and others began
to realize that it must always be that U = mc2, there was great confusion. 284 The force of an electron on itself The discrepancy between the two formulas for the electromagnetic mass is
especially annoying, because we have carefully proved that the theory of electro
dynamics is consistent With the principle of relativ1ty. Yet the theory of relatiVity
implies Without question that the momentum must be the same as the energy
times v/c2. So we are in some kind of trouble; we must have made a mistake.
We did not make an algebraic mistake in our calculations, but we have left some
thing out. In deriving our equations for energy and momentum, we assumed the con
servation laws. We assumed that all forces were taken into account and that any
work done and any momentum carried by other “nonelectrical” machinery was
included. Now if we have a sphere of charge, the electrical forces are all repulsive
and an electron would tend to ﬂy apart. Because the system has unbalanced forces,
we can get all kinds of errors in the laws relating energy and momentum. To get a
consistent picture, we must imagine that something holds the electron together.
The charges must be held to the sphere by some kind of rubber bands—something
that keeps the charges from ﬂying off. It was ﬁrst pointed out by Poincaré that the
rubber bands—or whatever it is that holds the electron together—must be included
in the energy and momentum calculations. For this reason the extra nonelectrical
forces are also known by the more elegant name “the Poincaré stresses.” If the
extra forces are included in the calculations, the masses obtained in two ways are
changed (in a way that depends on the detailed assumptions). And the results are
conSIStent With relativity; i.e., the mass that comes out from the momentum cal
culation is the same as the one that comes from the energy calculation. However,
both of them contain two contributions: an electromagnetic mass and contribution
from the Poincaré stresses. Only when the two are added together do we get a
consistent theory. It is therefore impossible to get all the mass to be electromagnetic in the way
we hoped. It is not a legal theory if we have nothing but electrodynamics. Some
thing else has to be added. Whatever you call them—“rubber bands,” or “Poincaré
stresses,” or something else—there have to be other forces in nature to make a
consistent theory of this kind. 28—4 Clearly, as soon as we have to put forces on the inside of the electron, the
beauty of the whole idea begins to disappear. Things get very complicated. You
would want to ask: How strong are the stresses? How does the electron shake?
Does it oscillate? What are all its internal properties? And so on. It might be
possible that an electron does have some complicated internal properties. If we
made a theory of the electron along these lines, it would predict odd properties,
like modes of oscillation, which haven’t apparently been observed. We say “ap
parently” because we observe a lot of things in nature that still do not make sense.
We may someday ﬁnd out that one of the things we don‘t understand today (for
example, the muon) can, in fact, be explained as an oscillation of the Poincare
stresses. It doesn’t seem likely, but no one can say for sure. There are so many
things about fundamental particles that we still don’t understand. Anyway, the
complex structure implied by this theory is undesirable, and the attempt to explain
all mass in terms of electromagnetism—at least in the way we have described—has
led to a blind alley. We would like to think a little more about why we say we have a mass when
the momentum in the ﬁeld is proportional to the velocity. Easy! The mass is the
coefﬁcient between momentum and veloc1ty. But we can look at the mass in another
way: a particle has mass if you have to exert a force in order to accelerate it. So
it may help our understanding if we look a little more closely at where the forces
come from. How do we know that there has to be a force? Because we have
proved the law of the conservation of momentum for the ﬁelds. If we have a
charged particle and push on it for awhile, there will be some momentum in the
electromagnetic ﬁeld. Momentum must have been poured into the ﬁeld somehow.
Therefore there must have been a force pushing on the electron in order to get it
going—a force in addition to that required by its mechanical inertia, a force due
to its electromagnetic interaction. And there must be a corresponding force back
on the “pusher.” But where does that force come from? (a) (b) Fig. 28—3. The selfforce on an accelerating electron is not zero because of the
retardation. (By dF we mean the force on a surface element do; by d2F we mean the
force on the surface element daa from the charge on the surface element dog.) The picture is something like this. We can think of the electron as a charged
sphere. When it is at rest, each piece of charge repels electrically each other piece,
but the forces all balance in pairs, so that there is no net force. [See Fig. 28—3(a).]
However, when the electron is being accelerated, the forces Will no longer be in
balance because of the fact that the electromagnetic inﬂuences take time to go
from one piece to another. For instance, the force on the piece a in Fig. 28—3(b)
from a piece [3 on the opposite side depends on the position of 6 at an earlier time,
as shown. Both the magnitude and direction of the force depend on the motion
of the charge. If the charge is accelerating, the forces on various parts of the
electron might be as shown in Fig. 28—3(c). When all these forces are added up,
they don’t cancel out. They would cancel for a uniform velocity, even though
it looks at ﬁrst glance as though the retardation would give an unbalanced force
even for a uniform velocity. But it turns out that there is no net force unless the
electron is being accelerated. With acceleration, if we look at the forces between 28—5 the various parts of the electron, action and reaction are not exactly equal, and
the electron exerts a fdrce on itself that tries to hold back the acceleration. It holds
itself back by its own bootstraps. It is possrble, but difﬁcult, to calculate this self—reaction force; however, we
don’t want to go into such an elaborate calculation here. We Will tell you what
the result is for the speCial case of relatively uncomplicated motion in one dimension,
say x. Then, the selfforce can be written in a series. The ﬁrst term in the series
depends on the acceleration )‘c', the next term is proportional to x, and so on.*
The result is 2 9 )
e . 26‘, 671...,
,x—e—Tx+v x+~~ F = a
ac 3 c‘ c4 (28.9) where a and 7 are numerical coefﬁcients of the order of l. The coefﬁcient a of
the jc' term depends on what charge distribution is assumed; if the charge is dis
tributed uniformly on a sphere, then a = 2/3. So there is a term. proportional
to the acceleration, which varies inversely as the radius a of the electron and agrees
exactly with the value we got in Eq. (28.4) for melee. If the charge distribution is
chosen to be different, so that a is changed, the fraction 2/3 in Eq. (28.4) would
be changed in the same way The term in fc‘ is independent of the assumed radius
a, and also of the assumed distribution of the charge; its coefﬁCient is always 2/3.
The next term is proportional to the radius a, and its coefﬁcient 7 depends on the
charge distribution. You will notice that if we let the electron radius a go to zero,
the last term (and all higher terms) will go to zero; the second term remains con
stant, but the ﬁrst term—the electromagnetic mass—goes to infinity. And we can
see that the inﬁnity arises because of the force of one part of the electron on another
—because we have allowed what is perhaps a silly thing, the possibility of the
“point” electron acting on itself. 28—5 Attempts to modify the Maxwell theory We would like now to discuss how it might be possible to modify Maxwell’s
theory of electrodynamics so that the idea of an electron as a simple point charge
could be maintained. Many attempts have been made, and some of the theories
were even able to arrange things so that all the electron mass was electromagnetic.
But all of these theories have died. It is still interesting to discuss some of the
possibilities that have been suggested—to see the struggles of the human mind. We started out our theory of electricity by talking about the interaction of
one charge with another. Then we made up a theory of these interacting charges
and ended up with a ﬁeld theory. We believe it so much that we allow it to tell
us about the force of one part of an electron on another. Perhaps the entire difﬁ
culty is that electrons do not act on themselves; perhaps we are making too great
an extrapolation from the interaction of separate electrons to the idea that an
electron interacts with itself. Therefore some theories have been proposed in which
the possibility that an electron acts on itself is ruled out. Then there is no longer
the inﬁnity due to the selfaction. Also, there is no longer any electromagnetic
mass associated with the particle; all the mass is back to being mechanical, but
there are new difﬁculties in the theory. We must say immediately that such theories require a modiﬁcation of the
idea of the electromagnetic ﬁeld. You remember we said at the start that the force
on a particle at any point was determined by Just two quantities—E and B. If
we abandon the “selfforce” this can no longer be true, because if there is an elec
tron in a certain place, the force isn’t given by the total E and B, but by only those
parts due to other charges. So we have to keep track always of how much of E
and B is due to the charge on which you are calculating the force and how much
is due to the other charges. This makes the theory much more elaborate, but it
gets rid of the difﬁculty of the inﬁnity. * We are usmg the notation: x = dx/dt, X = d2x/dt2, x = dgx/dt3, etc.
28—6 So we can, if we want to, say that there is no such thing as the electron acring
upon itself, and throw away the whole set of forces in Eq. (28.9). However, we
have then thrown away the baby with the bath! Because the second term in Eq.
(28.9), the term in 'x‘, is needed. That force does something very deﬁnite. If you
throw it away, you’re in trouble again. When we accelerate a charge, it radiates
electromagnetic waves, so it loses energy. Therefore, to accelerate a charge, we
must require more force than is required to accelerate a neutral object of the same
mass; otherwise energy wouldn’t be conserved. The rate at which we do work on
an accelerating charge must be equal to the rate of loss of energy per second by
radiation. We have talked about this effect before—it is called the radiation re
sistance. We still have to answer the question: Where does the extra force, against
which we must do this work, come from? When a big antenna is radiating, the
forces come from the inﬂuence of one part of the antenna current on another.
For a single accelerating electron radiating into otherwise empty space, there
would seem to be only one place the force could come from—the action of one
part of the electron on another part. We found back in Chapter 32 of Vol. I that an oscillating charge radiates
energy at the rate dW _ 2 6220:)2
gt— ~3— c3  (28.10)
Let’s see What we get for the rate of doing work on an electron against the boot strap force of Eq. (28.9). The rate of work is the force times the velocity, or F x:
~—=a——xx—~c—xx+~ (28.11) The ﬁrst term is proportional to dxz/dt, and therefore just corresponds to the rate
of change of the kinetic energy % mv2 associated with the electromagnetic mass.
The second term should correspond to the radiated power in Eq. (28.10). But it
is different. The discrepancy comes from the fact that the term in Eq. (28.11) is
generally true, whereas Eq. (28.10) is right only for an osczllating charge. We can
show that the two are equivalent if the motion of the charge is periodic. To do
that, we rewrite the second term of Eq. (28.11) as ————um+géwﬁ which is just an algebraic transformation. If the motion of the electron is periodic,
the quantity xx” returns periodically to the same value, so that if we take the
average of its time derivative, we get zero. The second term, however, is always
pOSitive (it‘s a square), so its average 18 also pOSIIIVC This term gives the net work
done and is Just equal to Eq. (28.10). The term in )‘c' of the bootstrap force is required in order to have energy
conservation in radiating systems, and we can’t throw it away. It was, in fact, one
of the triumphs of Lorentz to show that there is such a force and that it comes from
the action of the electron on itself. We must believe in the idea of the action of the
electron on itself, and we need the term in 'X‘ The problem is how we can get that
term without getting the ﬁrst term in Eq. (28.9), which gives all the trouble. We
don’t know how. You see that the classical electron theory has pushed itself
into a tight corner. There have been several other attempts to modify the laws in order to straighten
the thing out. One way, proposed by Born and Infeld, is to change the Maxwell
equations in a complicated way so that they are no longer linear. Then the electro
magnetic energy and momentum can be made to come out ﬁnite. But the laws
they suggest predict phenomena which have never been observed. Their theory
also suﬂ‘ers from another difﬁculty we will come to later, which is common to all
the attempts to avoid the troubles we have described. The following peculiar possibility was suggested by Dirac. He said: Let’s
admit that an electron acts on itself through the second term in Eq. (28.9) but not
through the ﬁrst. He then had an ingenious idea for getting rid of one but not the 28—7 other. Look, he said, we made a special assumption when we took only the
retarded wave solutions of Maxwell’s equations; if we were to take the advanced
waves instead, we would get something different. The formula for the self—force
would be F=a—.X+~—:—L{"+7€i— x: (28.12) This equation is Just like Eq. (28.9) except for the sign of the second term—and
some higher terms#of the series [Changing from retarded to advanced waves
is just changing the sign of the delay which, it IS not hard to see, is equivalent to
changing the sign oft everywhere. The only effect on Eq. (28.9) is to change the
Sign of all the odd time derivatives] So, Dirac said. let’s make the new rule that
an electron acts on itself by onehalf the difference of the retarded and advanced
ﬁelds which it produces. The difference of Eqs. (28.9) and (28.12). diVided by two.
is then
2 e2 F = — 3 E x + higher terms. In all the higher terms, the radius a appears to some positive power in the numera
tor. Therefore. when we go to the limit of a point Charge. we get only the one
term—just what is needed. In this way, Dirac got the radiation resistance force
and none of the inertial forces. There is no electromagnetic mass, and the claSSical
theory is saved—but at the expense of an arbitrary assumption about the selfforce. The arbitrariness of the extra assumption of Dirac was removed. to some ex
tent at least, by Wheeler and Feynman, who proposed a still stranger theory
They suggest that point charges interact only With other charges, but that the inter
action is half through the advanced and half through the retarded waves. It turns
out, most surprisingly, that in most situations you won’t see any effects of the
advanced waves, but they do have the effect of producing Just the radiation re
action force. The radiation resistance is not due to the electron acting on itself,
but from the following peculiar effect. When an electron is accelerated at the
time t, it shakes all the other charges in the world at a later time 1' 2 t + r/c
(where r is the distance to the other charge), because of the retarded waves. But
then these other charges react back on the original electron through their advanced
waves, which will arrive at the time t”, equal to t’ minus r/c. which is, of course.
Just 1. (They also react back with their retarded waves too. but that just corre—
sponds to the normal “reflected” waves.) The combination of the advanced and
retarded waves means that at the instant it is accelerated an oscillating charge
feels a force from all the charges that are “going to" absorb its radiated waves.
You see what tight knots people have gotten into in trying to get a theory of the
electron! We’ll describe now still another kind of theory, to show the kind of things
that people think of when they are stuck. This is another modification of the laws
of electrodynamics, proposed by Bopp. You realize that once you decide to change
the equations of electromagnetism you can start anywhere you want. You can
change the force law for an electron, or you can change the Maxwell equations
(as we saw in the examples we have described). or you can make a change some
where else. One possibility is to change the formulas that give the potentials in
terms ofthe charges and currents. One of our formulas has been that the potentials
at some pomt are given by the current dens1ty (or charge) at each other point at an
earlier time Using our fourvector notation for the potentials, we write _ 1 ML 1 “ rig/(C)
14,,(1. t) dame? / r12 r~ dVZ (2813) Bopp’s beautifully simple idea is that: Maybe the trouble is in the l/r factor in
the integral. Suppose we were to start out by assuming only that the potential at
one point depends on the charge density at any other pomt as some funmion of
the distance between the points, say as f(r12). The total potential at point (1) 28—8 will then be given by the integral of j, times this function over all space:
Aim = fj.(2)f(r12)dV2. That’s all. No diﬂerential equation, nothing else. Well, one more thing. We also
ask that the result should be relativistically invariant. So by “distance” we should
take the invariant “distance” between two points 1n spacetime. This distance
squared (within a sign which doesn’t matter) is 5'12 = €201 — 12)2 “ riz
= c201 — t2)2 — (X1 ‘ X2)2 " 0/1 “ J’2)2 — (21 — 22)? (2814) 50, for a relativistically invariant theory, we should take some function of the
magnitude of s12, or what is the same thing, some function of sﬁ. So Bopp’s
theory is that A..<1, :1) = fj..(2, r2)F(s122)dV2 dzg. (28.15) (The integral must, ofcourse, be over the fourdimensmnal volume dt2 a’xz dyg 0'22 ) All that remains Is to choose a suitable function for F. We assume only one
thing about Fthat it is very small except when its argument is near zero—so that a
graph of F would be a curve like the one in Fig. 28—4. It is a narrow spike with a
ﬁnite area centered at 32 = 0, and with a width which we can say is roughly 02.
We can say, crudely, that when we calculate the potential at point (1), only those
points (2) produce any appreciable effect if £2 = c2(t2 — 102 — rfg is within
tar" of zero. We can ind1cate this by saying that F is important only for {1’2 = 02(t1 — 22V — ﬁg z =ta2. (28.16)
You can make it more mathematical if you want to, but that’s the idea. Now suppose that a is very small in companson With the size of ordinary
objects like motors, generators, and the like so that for normal problems r12 >> a.
Then Eq. (28.16) says that charges contribute to the integral of Eq. (28.15) only
when t1 — t2 is in the small range a2
C(tl — t2) 2 vrﬁz :1: 02 N r12\/l =1: 72" 12 Since a2/rfg << 1, the square root can be a roximated by l =t are/2&2, so
pp 2 2
t __ t = r12 1 :1: a = r12 :1: a .
1 2 2 2 2
C r12 C 7120 What is the signiﬁcance? This result says that the only times t2 that are im
portant in the integral of A,“ are those which differ from the time t1, at which we
want the potential, by the delay r12/c—with a negligible correction so long as
r12 >> a. In other words, th1s theory of Bopp approaches the Maxwell theory—so
long as we are far away from any particular charge—in the sense that it gives the
retarded wave eﬁ‘ects. We can, in fact, see approximately what the integral of Eq. (28.15) is going
to give. If we integrate ﬁrst over t2 from — ac to +oo—keeping r12 ﬁxed~then
£2 is also going to go from — 00 to + so. The integral will all come from t2’s in
a small interval of width Atz = 2 X a2/2r12c, centered at [1 — rig/C. Say
that the function F(sz) has the value K at 52 = 0; then the integral over t2 gives
approximately KjﬂAtg, or 2 .
Ka at.
C r12 We should, of course, take the value of j” at t2 = t1 — r12/c, so that Eq. (28.15)
becomes 2  __
ML“) = Eta/MW, C r 1 2
289 f(52) Fig. 28—4. The function H52) used in
the nonlocol theory of Bopp. If we pick K = q2c/47reoa2, we are right back to the retarded potential solution
of Maxwell’s equations—including automatically the l/r dependence! And it
all came out of the simple proposition that the potential at one pornt in space
time depends on the current denSity at all other points in space—time, but With
a weighting factor that is some narrow function of the fourdimensional distance
between the two pomts. This theory again predicts a ﬁnite electromagnetic mass
for the electron, and the energy and mass have the right relation for the relativrty
theory They must, because the theory is relativistically invariant from the start,
and everything seems to be all right There is, however, one fundamental objection to this theory and to all the
other theories we have described. All particles we know obey the laws of quantum
mechanics, so a quantum—mechanical modiﬁcation of electrodynamics has to be
made Light behaves like photons It isn’t 100 percent like the Maxwell theory.
So the electrodynamic theory has to be changed. We have already mentioned that
it might be a waste of time to work so hard to straighten out the classical theory,
because it could turn out that in quantum electrodynainics the difﬁculties Will
disappear or may be resolved in some other fashion. But the difficulties do not
disappear in quantum electrodynamics. That is one of the reasons that people
have spent so much effort trying to straighten out the class1cal difﬁculties, hoping
that if they could straighten out the claSSical difﬁculty and then make the quantum
modiﬁcations, everything would be straightened out The Maxwell theory 511]]
has the difﬁculties after the quantum mechanics modiﬁcations are made. The quantum effects do make some changes—the formula for the mass 18
modiﬁed, and Planck’s constant it appears—but the answer still comes out inﬁnite
unless you cut oﬁ“ an integration somehow—just as we had to stop the classical
integrals at r = (1. And the answers depend on how you stop the integrals. We
cannot, unfortunately, demonstrate for you here that the difﬁculties are really
baSically the same, because we have developed so little of the theory of quantum
mechanics and even less of quantum electrodynamics. So you must just take our
word that the quantized theory of Maxwell‘s electrodynamics gives an inﬁnite
mass for a point electron. It turns out, however,that nobody has ever succeeded in making a S'c’lf(‘lﬂﬂnlellf
quantum theory out of any of the modiﬁed theories. Born and lnfeld's ideas have
never been satisfactorily made into a quantum theory The theories With the ad
vanced and retarded waves of Dirac, or of Wheeler and Feynman, have never
been made into a satisfactory quantum theory The theo1y of Bopp has never been
made into a satisfactory quantum theory. So today, there is no known solution
to this problem. We do not know how to make a conSistent theory—including
the quantum mechanics—which does not produce an inﬁnity for the selfenergy of
an electron, or any pOint Charge. And at the same time, there is no satisfactory
theory that describes a nonpoint charge. It‘s an unsolved problem In case you are dec1ding to rush off to make a theory in which the action ofan
electron on itself is completely removed, so that electromagnetic mass is no longer
meaningful, and then to make a quantum theory of it, you should be warned that
you are certain to be in trouble. There is deﬁnite experimental eVidence of the
exrstence of electromagnetic inertiaﬁthere is eVidence that some of the mass of
charged particles is electromagnetic in origin It used to be said in the older books that Since Nature Will obViously not pre
sent us With two particles~one neutral and the other charged, but othci‘Wise the
same—we Will never be able to tell how much of the mass is electromagnetic and
how much is mechanical But it turns out that Nature has been kind enough to
present us with just such objects, so that by comparing the observed mass ol‘the
charged one With the observed mass of the neutral one, we can tell whether there
is any electromagnetic mass. For example, there are the neutrons and protons.
They interact With tremendous forces—the nuclear forces~whose origin is un
known However, as we have already described, the nuclear forces have one re
markable property. So far as they are concerned, the neutron and proton are
exactly the same The nuclear forces between neutron and neutron, neutron and
proton, and proton and proton are all identical as far as we can tell Only the little 28—10 electromagnetic forces are different; electrically the proton and neutron are as
different as night and day. This is just what we wanted There are two particles,
identical from the pomt of View of the strong interactions, but different electrically.
And they have a small difference in mass. The mass difference between the proton
and the neutron—expressed as the difference in the restenergy me2 in units of
Mev—is about 1.3 Mev, which is about 2.6 times the electron mass. The classical
theory would then predict a radius of about % to % the classical electron radius,
or about 10‘” cm. Of course, one should really use the quantum theory, but by
some strange accident, all the constants—271’s and h's, etc—come out so that the
quantum theory gives roughly the same radius as the classical theory. The only
trouble is that the Sign is wrong! The neutron is heavier than the proton. Table 28—1
Particle Masses P t 1 Charge Mass Am*
ar m e (electronic) (Mev) (Mev) n (neutron) 0 939.5
p (proton) +1 938.2 —1.3 7r (1rmeson) 0 135.0
dz] 139.6 +4.6 K (Kmeson) 0 497.8
:1 493.9 —3.9 2 (sigma) 0 1191.5
+1 1189.4 —2.1
—1 1196.0 +4.5 * Am = (mass of charged) — (mass of neutral). Nature has also given us several other pairs—or triplets—of particles which
appear to be exactly the same except for their electrical charge. They interact With
protons and neutrons, through the socalled “strong” interactions of the nuclear
forces. In such interactions, the particles of a given kind—say the 7rmesons—
behave in every way like one object except for their electrical charge. In Table
28—1 we give a list of such particles, together with their measured masses. The
charged wmesons—positive or negative—have a mass of 139.6 Mev, but the
neutral 7rmeson is 4 6 Mev lighter. We believe that this mass difference is electro
magnetic; it would correspond to a particle radius of3 to 4 X 10—14 cm. You Will
see from the table that the mass differences of the other particles are usually of the
same general Size. Now the Size of these particles can be determined by other methods, for in
stance by the diameters they appear to have in highenergy collisions. So the
electromagnetic mass seems to be in general agreement with electromagnetic
theory, if we stop our integrals of the field energy at the same radius obtained by
these other methods. That’s why we believe that the differences do represent
elecnomagnetic mass. You are no doubt worried about the different signs of the mass differences in
the table. It is easy to see why the charged ones should be heaVier than the neutral
ones. But what about those pairs like the proton and the neutron, where the mea
sured mass comes out the other way? Well, it turns out that these particles are
complicated, and the computation of the electromagnetic mass must be more
elaborate for them. For instance, although the neutron has no net charge, it does
have a charge distribution inside it—it is only the net charge that is zero In fact,
we believe that the neutron looks—at least sometimes—like a proton with a nega
tive 7rmeson in a “cloud" around it, as shown in Fig. 28—5. Although the neutron
is “neutral,” because its total charge is zero, there are still electromagnetic energies 28—11 WNegahve
' 7T meson . , ' ‘PROTON Fig. 28—5. A neutron may exist, at
times, as a proton surrounded by a
negative 7rmeson. (for example, it has a magnetic moment), so it’s not easy to tell the Sign of the
electromagnetic mass difference Without a detailed theory of the internal structure. We only Wish to emphaSize here the followmg pomts‘ (l) the electromagnetic
theory predicts the existence of an electromagnetic mass, but it also falls on Its
face in dOing so, because it does not produce a conSistent theory—and the same is
true With the quantum modiﬁcations; (2) there is experimental ev1dence for the
existence of electromagnetic mass; and (3) all these masses are roughly the same
as the mass of an electron. So we come back again to the original idea of Lorentz—
maybe all the mass of an electron is purely electromagnetic, maybe the whole
0 511 Mev is due to electrodynamies Is it or isn’t it? We haven’t got a theory. so
we cannot say. We must mention one more piece ofinformation, which is the most annoying
There is another particle in the world called a muon—or umeson—which, so far
as we can tell, diﬁers in no way whatsoever from an electron except for its mass. It
acts in every way like an electron: it interacts w1th neutrinos and With the electro
magnetic ﬁeld, and it has no nuclear forces. It does nothing different from what
an electron does~at least, nothing which cannot be understood as merely a con—
sequence of its higher mass (206 77 times the electron mass). Therefore, whenever
someone ﬁnally gets the explanation of the mass of an electron. he will then have
the puzzle of where a muon gets its mass. Why? Because whatever the electron
does, the muon does the same—so the mass ought to come out the same There
are those who believe faithfully in the idea that the muon and the electron are the
same particle and that, in the ﬁnal theory of the mass, the formula for the mass
will be a quadratic equation with two roots—one for each particle There are also
those who propose it Will be a transcendental equation With an inﬁnite number of
roots, and who are engaged in guessmg what the masses of the other particles in
the series must be, and why these particles haven‘t been discovered yet 28—6 The nuclear force ﬁeld We would like to make some further remarks about the part of the mass of
nuclear particles that is not electromagnetic Where does this other large fraction
come from 9 There are other forces besides electrodynamics—like nuclear forces—
that have their own ﬁeld theories, although no one knows Whether the current
theories are right These theories also predict a ﬁeld energy which gives the
nuclear particles a mass term analogous to electromagnetic mass; we could call
it the “7rmesicﬁeld—mass." It is presumably very large, because the forces are
great, and it is the poss1ble origin of the mass of the heavy particles. But the
meson ﬁeld theories are still in a most rudimentary state Even With the well
developed theory of electromagnetism, we found it imposs1ble to get beyond ﬁrst
base in explaining the electron mass. With the theory of the mesons, we strike out We may take a moment to outline the theory of the mesons, because of its
interesting connection With electrodynamics. ln electrodynamics, the field can be
described in terms of a fourpotential that satisﬁes the equation [12A, : sources. Now we have seen that pieces of the ﬁeld can be radiated away so that they exist
separated from the sources. These are the photons of light, and they are described
by a differential equation without sources: [32/1, 2 0. People have argued that the ﬁeld of nuclear forces ought also to have its own
“photons”—they would presumably be the 7rniesons—and that they should be
described by an analogous differential equation. (Because of the weakness of the
human brain, we can’t think of something really new, so we argue by analogy
With what we know.) So the meson equation might be D2¢ = 0.
28—12 where (6 could be a different fourvector or perhaps a scalar. It turns out that the
pion has no polarization, so ¢ should be a scalar. With the Simple equation
[12¢ = 0, the meson ﬁeld would vary with distance from a source as l/r2, just
as the electric ﬁeld does. But we know that nuclear forces have much shorter dis
tances of action, so the simple equation won’t work. There is one way we can
change things without disrupting the relativistic invariance: we can add or subtract
from the D’Alembertian a constant, times (1). So Yukawa suggested that the free
quanta of the nuclear force ﬁeld might obey the equation [1% — Ma = 0, (2817) where ,u2 is a constant—that is, an invariant scalar. (Since 3 2 is a scalar differ—
ential operator in four dimensions, its invariance is unchanged if we add another
scalar to it.) Let’s see what Eq. (28.17) gives for the nuclear force when things are not
changing with time. We want a spherically symmetric solution of V2¢> —M2¢ = 0 around some point source at, say, the origin. If ¢ depends only on r, We know that 1 62
V24) = 7 3’3 (rdu).
So we have the equation
1 a2 2
; W ("13) — IL 4’) = 0
or
(92 (Ht) = M2(r¢) m Thinking of (rqb) as our dependent variable, this is an equation we have seen many
times. It’s solution is r¢> = K93”. Clearly, ¢ cannot become inﬁnite for large r, so the + Sign in the exponent is
ruled out. The solution IS (28.18) This function is called the Yukawa potential. For an attractive force, Kis a negative
number whose magnitude must be adjusted to ﬁt the experimentally observed
strength of the forces. The Yukawa potential of the nuclear forces dies off more rapidly than l/r
by the exponential factor. The potential—and therefore the force—falls to'zero
much more rapidly than l/r for distances beyond 1/,u, as shown in Fig. 28 6
The “range” of nuclear forces 15 much less than the “range” of electrostatic forces
It is found experimentally that the nuclear forces do not extend beyond about
10"”cm, so ,u :2 1015 m”1 Finally, let’s look at the freewave solution of Eq (28 17) If we substitute ¢‘ : ¢0€1<wt—k2) into Eq. (28 17), we get that 2
CU E—k2—u2=0. Relating frequency to energy and wave number to momentum, as we dld at the
end of Chapter 36 of Vol. I, we get that which says that the Yukawa “photon” has a mass equal to Mfr/c. If we use for M
28—13 J t 4’
O 1/}; 2/}; 3/}; r Fig. 28—6. The Yukawa potential
e—W/r, compared with the Coulomb
potential l/r. the estimate 1015 m‘l, which gives the observed range of the nuclear forces, the
mass comes out to 3 X 10'25 gm, or 170 Mev, which is roughly the observed mass
of the 7rmeson. So, by an analogy with electrodynamics, we would say that the
7rmes0n is the “photon” of the nuclear force ﬁeld But now we have pushed the
ideas of electrodynamics into regions where they may not really be valid—we have
gone beyond electrodynamics to the problem of the nuclear forces. 28—14 ...
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 Spring '10
 Chaikin
 Physics, Mass

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