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Unformatted text preview: 27 Field Energy and Field Momentum 27—1 Local conservation It is clear that the energy of matter is not conserved. When an object radiates
light it loses energy. However, the energy lost is possibly describable in some other
form, say in the light. Therefore the theory of the conservation of energy 15
incomplete without a consrderation of the energy which is associated with the light
or, in general, with the electromagnetic ﬁeld. We take up now the law of conserva
tion of energy and, also, of momentum for the ﬁelds. Certainly, we cannot treat
one without the other, because in the relatiVity theory they are different aspects of
the same fourvector. Very early in Volume I, we discussed the conservation of energy; we said
then merely that the total energy in the world is constant. Now we want to extend
the idea of the energy conservation law in an important way—in a way that says
something in detail about how energy is conserved. The new law will say that if
energy goes away from a region, it is because it ﬂows away through the boundaries
of that region. It is a somewhat stronger law than the conservation of energy
without such a restriction. To see what the statement means, let’s look at how the law of the conservation
of charge works. We described the conservation of charge by saying that there is
a current density j and a charge density p, and that when the charge decreases at
some place there must be a ﬂow of charge away from that place. We call that the
conservation of charge. The mathematical form of the conservation law is . 6p
VJ———:9? (27.1)
This law has the consequence that the total charge in the world is always constant—
there 15 never any net gain or loss of charge. However, the total charge in the
world could be constant in another way. Suppose that there is some charge Q1
near some pomt (1) while there is no charge near some point (2) some distance
away (Fig. 27—1). Now suppose that, as time goes on, the charge Q1 were to
gradually fade away and that simultaneously with the decrease of Q1 some charge
Q2 would appear near point (2), and in such a way that at every instant the sum of
Q1 and Q2 was a constant. In other words, at any intermediate state the amount
of charge lost by Q1 would be added to Q2. Then the total amount of charge in
the world would be conserved. That’s a “worldwide” conservation, but not what
we will call a “local” conservation, because in order for the charge to get from
(1) t0 (2). it didn’t have to appear anywhere in the space between pomt (l) and point (2). Locally, the charge was just “lost.” There is a difﬁculty With such a “worldwide” conservation law in the theory
of relativity. The concept of “simultaneous moments” at distant pomts is one which
is not equivalent in different systems. Two events that are simultaneous in one
system are not Simultaneous for another system moving past. For “worldwide"
conservation of the kind described, it is necessary that the charge lost from Q1
should appear simultaneously in Q2. Otherwise there would be some moments
when the charge was not conserved. There seems to be no way to make the
law of charge conservation relativistically invariant without making it a “local”
conservation law. As a matter of fact, the requirement of the Lorentz relativistic
invariance seems to restrict the possible laws of nature in surprismg ways. In
modern quantum ﬁeld theory, for example, people have often wanted to alter the
theory by allowing what we call a “nonlocal” interaction—where something here 27—1 27—1 Local conservation 27—2 Energy conservation and
electromagnetism 27—3 Energy density and energy
ﬂow in the electromagnetic
ﬁeld 27—4 The ambiguity of the ﬁeld
energy 27—5 Examples of energy ﬂow 27—6 Field momentum (I) (2’ A
QI v 02 (b) Fig. 27—]. Two ways to conserve
charge: (a) Q] + 02 is constant; (b)
th/dt = finda = —sz/dt. has a direct effect on something there—but we get in trouble With the relatiVity
principle. “Local” conservation involves another idea. It says that a charge can get
from one place to another only ifthere is something happening in the space between.
To describe the law we need not only the density of charge, p, but also another
kind of quantity, namely j, a vector giving the rate of ﬂow of charge across a
surface. Then the ﬂow is related to the rate of change of the density by Eq. (27.1).
This 18 the more extreme kind of a conservation law. It says that charge is con
served in a special way—conserved “locally.” It turns out that energy conservation is also a local process. There is not only
an energy density in a given region of space but also a vector to represent the rate
of ﬂow of the energy through a surface. For example, when a light source radiates,
we can ﬁnd the light energy movmg out from the source. If we imagine some mathe
matical surface surrounding the light source, the energy lost from inside the surface
is equal to the energy that ﬂows out through the surface. 27—2 Energy conservation and electromagnetism We want now to write quantitatively the conservation of energy for electro
magnetism. To do that, we have to describe how much energy there is in any
volume element of space, and also the rate of energy flow. Suppose we think ﬁrst
only of the electromagnetic ﬁeld energy. We Will let u represent the energy denszty
in the ﬁeld (that is, the amount of energy per unit volume in space) and let the
vector S represent the energy ﬂux of the ﬁeld (that is, the ﬂow of energy per unit
time across a unit area perpendicular to the ﬂow). Then, in perfect analogy with
the conservation of charge, Eq (27 l), we can write the “local” law of energy
conservation in the ﬁeld as 611
71 — —V  S. (27.2) Of course, this law is not true in general; it is not true that the ﬁeld energy is
conserved. Suppose you are in a dark room and then turn on the light sw1tch. All
of a sudden the room is full of light, so there is energy in the ﬁeld, although there
wasn’t any energy there before. Equation (27.2) is not the complete conservation
law, because the ﬁeld energy alone is not conserved, only the total energy in the
world—there is also the energy of matter. The ﬁeld energy Will change if there is
some work being done by matter on the ﬁeld or by the ﬁeld on matter However, if there is matter inSide the volume of interest, we know how much
energy it has: Each particle has the energy muc2/x/l ~— 7‘2/62. The total energy
of the matter isjust the sum of all the particle energies, and the ﬂow of this energy
through a surface is just the sum of the energy carried by each particle that crosses
the surface We want now to talk only about the energy of the electromagnetic
ﬁeld. So we must write an equation which says that the total field energy in a given
volume decreases either because ﬁeld energy ﬂows out of the volume or because
the ﬁeld loses energy to matter (or gains energy, which is just a negative loss).
The ﬁeld energy inside a volume V is fudV,
V and its rate of decrease is minus the time derivative of this integral. The ﬂow of
ﬁeld energy out of the volume V is the integral of the normal component of S over
the surface 2 that encloses V,
/ S ' n da.
2 7% / udV 2 f S  n da + (work done on matter inside V). (273)
_ v x So 27—2 We have seen before that the ﬁeld does work on each unit volume of matter
at the rate E j. [The force on a particle is F = q(E + v x B), and the rate of
doing work is F~ v = qE' v. If there are N particles per unit volume, the rate of
dOing work per unit volume is NqE v, but qu = j] So the quantity E ‘j must
be equal to the loss of energy per unit time and per unit volume by the ﬁeld.
Equation (27.3) then becomes _9_/ udV=/s.nda+/EjdV. (27.4)
at V 2 V This is our conservation law for energy in the ﬁeld. We can convert it into a
differential equation like Eq. (27.2) if we can change the second term to a volume
integral That is easy to do with Gauss’ theorem. The surface integral of the
normal component of S is the integral of its divergence over the volume inSide.
So Eq. (27.3) is equivalent to — @dV=/ vSdV+/E~jdV,
V dt V V where we have put the time derivative of the ﬁrst term inside the integral. Since
this equation is true for any volume, we can take away the integrals and we have
the energy equation for the electromagnetic ﬁelds: —6—u=vS+Ej. (27.5)
6! Now this equation doesn’t do us a bit of good unless we know what u and S
are. Perhaps we should just tell you what they are in terms of E and B, because
all we really want is the result. However. we would rather show you the kind of
argument that was used by Poynting in 1884 to obtain formulas for S and u, so
you can see where they come from. (You won’t, however. need to learn this de rivation for our later work.) 27—3 Energy density and energy ﬂow in the electromagnetic ﬁeld The Idea is to suppose that there is a ﬁeld energy density u and a ﬂux S that
depend only upon the ﬁelds E and B. (For example, we know that in electrostatics,
at least. the energy density can be written %60E  E.) Of course, the u and S might
depend on the potentials or something else. but let’s see what we can work out
We can try to rewrite the quantity E j in such a way that it becomes the sum of
two terms. one that is the time derivative of one quantity and another that is the
divergence ofa second quantity. The ﬁrst quantity would then be u and the second
would be S (With suitable signs). Both quantities must be written in terms of the
ﬁelds only; that is, we want to write our equality as Ej=—_—v.s. (27.6) The lefthand side must ﬁrst be expressed in terms of the ﬁelds only. How
can we do that" By using Maxwell’s equations. of course. From Maxwell’s
equation for the curl of B, . _ 2 _ £3 — soc V X B 60 6t Substituting this in (27 6) we Will have only E’s and B’s: Ej = eoczE(V x B) — 605%? (27.7)
We are already partly ﬁnished. The last term is a time derivative—it is
(a/at)(§eoE  E). So ﬁeOE  E is at least one part of u. It’s the same thing we
found in electrostatics. Now, all we have to do is to make the other term into the
divergence of something. 27—3 Notice that the ﬁrst term on the righthand side of (27.7) is the same as
(V X B) ' E. (27.8) And, as you know from vector algebra, (a X b) c is the same as a (b X c);
so our term is also the same as v . (B x E) (27.9) and we have the divergence of “something,” just as we wanted. Only that‘s
wrong' We warned you before that V is “like” a vector, but not “exactly" the
same. The reason it is not is because there is an additional convention from cal
culus: when a derivative operator is in front of a product, it works on everything
to the right. In Eq (27.7). the V operates only on B, not on E But in the form
(27 9), the normal convention would say that V operates on both B and E So
it’s not the same thing In fact, if we work out the components of V  (B X E)
we can see that it is equal to E~ (V X B) plus some other terms. It’s like what
happens when we take a derivative ofa product in algebra For instance, d __ dj‘ dg
g;(fg)—Eg+/g; Rather than working out all the components of V  (B X E), we would like
to show you a trick that is very useful for this kind of problem. It is a trick that
allows you to use all the rules of vector algebra on expressions With the V operator.
without getting into trouble The trick is to throw out—for a while at leastthe
rule of the calculus notation about what the derivative operator works on You
see. ordinarily, the order of terms is used for two separate purposes. One is for
calculus: f(d/dx)g is not the same as g(a’/dx)f; and the other is for vectors:
a X b is difTerent from b X a. We can, ifwe want.chooseto abandon momentarily
the calculus rule. Instead of saying that a derivative operates on everything to the
right, we make a new rule that doesn’t depend on the order in which terms are writ
ten down Then we can Juggle terms around Without worrying Here is our new convention. we show. by a subscript. what a diﬂ‘erential op—
erator works on; the order has no meaning. Suppose we let the operator D stand
for 0/6X. Then Df means that only the derivative of the variable quantity 1‘ IS
taken. Then (if
M = a; fog = (3%) g. But notice now that according to our new rule,fD,g means the same thing We
can write the same thing any which way. Dxfg = gD/f = fog = fg D! You see, the Df can even come after everything. (It‘s surprismg that such a handy
notation is never taught in books on mathematics or physics.) You may wonder: What if I want to write the derivative of ﬁg? I want the
derivative of both terms. That’s easy, you Just say so; you write D/(fg) + Dg(fg).
That is Just g(6f/6X) —l— f(6g/6x), which is what you mean in the old notation by
a(fg)/6x. You Will see that it is now going to be very easy to work out a new expreSSion
for V  (B X E). We start by changing to the new notation; we write But if we have D/ g. it means V(BXE)=V,;(BXE)+VE~(B><E). (27.10) The moment we do that we don’t have to keep the order straight any more We
always know that VE operates on E only. and V]; operates on B only In these
circumstances, we can use V as though it were an ordinary vector. (Of course, 27—4 when we are ﬁnished, we will want to return to the “standard” notation that
everybody usually uses ) So now we can do the various things like interchanging
dots and crosses and making other kinds of rearrangements of the terms. For
instance, the middle term of Eq. (27.10) can be rewritten as [5 VB X B. (You
remember that a  b X c = b  c X a.) And the last term is the same as B  E X
VE. It looks freakish, but it is all right. Now if we try to go back to the ordinary
convention, we have to arrange that the V operates only on its “own” variable.
The ﬁrst one is already that way, so we canjust leave off the subscript. The second
one needs some rearranging to put the V in front of the E, which we can do by
reversing the cross product and changing sign: B(EX VE) = —B(VE x E). Now it is in a conventional order, so we can return to the usual notation. Equation
(27.10) is equivalent to V(BXE)=E(VXB)—B(VXE). (27.11) (A quicker way would have been to use components in this special case, but it
was worth taking the time to show you the mathematical trick. You probably
won’t see it anywhere else, and it is very good for unlocking vector algebra from
the rules about the order of terms with derivatives.) We now return to our energy conservation discussion and use our new result,
Eq. (27.11), to transform the V X B term of Eq. (27.7). That energy equatlon
becomes Ej = Eoc2V  (B x E) + 60623 (V X E) — .331 @6055) (27.12) Now you see we’re almost ﬁnished. We have one term which is a nice derivative
with respect to t to use for u and another that is a beautiful divergence to represent
S. Unfortunately, there is the center term left over, which is neither a divergence
nor a derivative with respect to t. So we almost made it, but not quite. After
some thought, we look back at the differential equations of Maxwell and discover
that V X E is, fortunately, equal to —6B/6t, which means that we can turn the
extra term into something that is a pure time derivative: B‘(VXE)=B(—%>= —%<!%5>. Now we have exactly what we want. Our energy equation reads . 2 6 60C2 60
EJ=V.(€OCBXE)—a_tTB.B+7EIE, (2713) which is exactly like Eq. (27.6), if we make the deﬁnitions 2 u=%EE+E°TCB.B (27.14)
and
= eoc2E x B. (27.15) (Reversing the cross product makes the signs come out right.) Our program was successful. We have an expression for the energy densrty
that IS the sum of an “electric” energy density and a “magnetic” energy density,
whose forms are just like the ones we found in statics when we worked out [he
energy in terms of the ﬁelds. Also, we have found a formula for the energy ﬂow
vector of the electromagnetic ﬁeld. This new vector, S = eoczE X B, is called
“Poynting’s vector,” after 1ts discoverer. It tells us the rate at which the ﬁeld
energy moves around in space. The energy which ﬂows through a small area do
per second is S ' n da, where n is the unit vector perpendicular to do. (Now that
we have our formulas for u and S, you can forget the derivations if you want.) 27—5 S ‘4vv———A4—e»
V MRECHON OF WAVE
PROPAGKHON Fig. 27—2. The vectors E, B, and S
for a light wave. 27—4 The ambiguity of the ﬁeld energy Before we take up some applications of the Poynting formulas [Eqs. (27.14)
and (27.15)], we would like to say that we have not really “proved” them. All
we did was to ﬁnd a possible “it” and a posszble “S.” How do we know that by
juggling the terms around some more we couldn’t ﬁnd another formula for “u”
and another formula for “S”? The new S and the new u would be different, but
they would still satisfy Eq. (27.6). It’s possrble. It can be done, but the forms that
have been found always involve various derivatives of the ﬁeld (and always with
secondorder terms like a second derivative or the square of a ﬁrst derivative).
There are, in fact, an inﬁnite number of different possibilities for u and S, and
so far no one has thought of an experimental way to tell which one is right! People
have guessed that the simplest one is probably the correct one, but we must say
that we do not know for certain what is the actual location in space of the electro
magnetic ﬁeld energy. So we too will take the easy way out and say that the ﬁeld
energy is given by Eq. (27.14). Then the ﬂow vector S must be given by Eq. (27.15). It is interesting that there seems to be no unique way to resolve the indeﬁnite
ness in the location of the ﬁeld energy. It is sometimes claimed that this problem
can be resolved by using the theory of gravitation in the following argument.
In the theory of gravity, all energy is the source of gravitational attraction. There
fore the energy density of electricity must be located properly if we are to know in
which direction the gravity force acts. As yet, however, no one has done such a
delicate experiment that the precise location of the gravitational inﬂuence of
electromagnetic ﬁelds could be determined. That electromagnetic ﬁelds alone can
be the source of gravitational force is an idea it is hard to do without. It has, in
fact, been observed that light is deﬂected as it passes near the sun—we could
say that the sun pulls the light down toward it. Do you not want to allow that the
light pulls equally on the sun? Anyway, everyone always accepts the Simple
expreSSions we have found for the location of electromagnetic energy and its ﬂow.
And although sometimes the results obtained from using them seem strange,
noboby has ever found anything wrong With them—that is, no disagreement with
experiment. So we will follow the rest of the world—besides, we believe that it is
probably perfectly right. We should make one further remark about the energy formula. In the ﬁrst
place, the energy per unit volume in the ﬁeld is very simple: It is the electrostatic
energy plus the magnetic energy, ifwe write the electrostatic energy in terms of
E2 and the magnetic energy as 82. We found two such expressions as possible
expreSSions for the energy when we were dOing static problems. We also found a
number of other formulas for the energy in the electrostatic ﬁeld, such as p¢>,
which is equal to the integral of E  E in the electrostatic case However, in an
electrodynamic ﬁeld the equality failed, and there was no obvrous chOice as to
which was the right one. Now we know which is the right one. Similarly, we have
fOund the formula for the magnetic energy that is correct in general The right
formula for the energy dens1ty of dynarmc ﬁelds 15 Eq (27.14) 27—5 Examples of energy ﬂow Our formula for the energy ﬂow vector S is something quite new. We want
now to see how it works in some specral cases and also to see whether it checks
out with anything that we knew before. The ﬁrst example we will take is light.
In a light wave we have an E vector and a B vector at right angles to each other
and to the direction of the wave p...
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 Spring '09
 LeeKinohara
 Physics, Energy, Momentum

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