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Unformatted text preview: I3 Magnetostatics 13—1 The magnetic ﬁeld The force on an electric charge depends not only on where it is, but also on
how fast it is moving. Every point in space is characterized by two vector quantities
which determine the force on any charge. First, there is the electric force, which
gives a force component independent of the motion of the charge. We describe it
by the electric ﬁeld, E. Second, there is an additional force component, called the
magnetic force, which depends on the velocity of the charge. This magnetic force
has a strange directional character: At any particular point in space, both the
direction of the force and its magnitude depend on the direction of motion of the
particle: at every instant the force is always at right angles to the velocity vector;
also, at any particular point, the force is always at right angles to a ﬁxed direction
in space (see Fig. 13—1); and ﬁnally, the magnitude of the force is proportional to
the component of the velocity at right angles to this unique direction. It is possible
to describe all of this behavior by deﬁning the magnetic ﬁeld vector B, which speci
ﬁes both the unique direction in space and the constant of proportionality with the
velocity, and to write the magnetic force as qv X B. The total electromagnetic
force on a charge can, then, be written as F = q(E + v x B). (13.1) This is called the Lorentz force. The magnetic force is easily demonstrated by bringing a bar magnet close to a
cathoderay tube. The deﬂection of the electron beam shows that the presence of
the magnet results in forces on the electrons transverse to their direction of motion,
as we described in Chapter 12 of Vol. I. The unit of magnetic ﬁeld B is evidently one newton'second per
coulombmeter. The same unit is also one voltsecond per meterz. It is also
called one weber per square meter. 13—2 Electric current; the conservation of charge We consider ﬁrst how we can understand the magnetic forces on wires carrying
electric currents. In order to do this, we deﬁne what is meant by the current density.
Electric currents are electrons or other charges in motion with a net drift or ﬂow.
We can represent the charge ﬂow by a vector which gives the amount of charge
passing per unit area and per unit time through a surface element at right angles to
the ﬂow (just as we did for the case of heat ﬂow). We call this the current density
and represent it by the vector j. It is directed along the motion of the charges.
If we take a small area AS at a given place in the material, the amount of charge
ﬂowing across that area in a unit time is inAS, (13.2) where n is the unit vector normal to AS. The current density is related to the average ﬂow velocity of the charges.
Suppose that we have a distribution of charges whose average motion is a drift
with the velocity v. As this distribution passes over a surface element AS, the charge
Aq passing through the surface element in a time At is equal to the charge contained
in a parallelepiped whose base is AS and whose height is 12 At, as shown in Fig. 13—2.
The volume of the parallelepiped is the projection of AS at right angles to 1) times 131—] 13—1 The magnetic ﬁeld 132 Electric current; the
conservation of charge 13—3 The magnetic force on a
current 134 The magnetic ﬁeld of steady
currents; Ampere’s law 13—5 The magnetic ﬁeld of a
straight wire and of a solenoid;
atomic currents 136 The relativity of magnetic and
electric ﬁelds 13—7 The transformation of currents
and charges 13—8 Superposition; the righthand
rule Review: Chapter 15, Vol. I: The Special
Theory of Relativity Fig. l3—l. The velocitydependent
component of the force on a moving
charge is at right angles to v and to the
direction of B. It is also proportional to
the component of v at right angles to B,
that is, to v sin 0. Fig. 13—2. If a charge distribution of
dénsity p moves with the velocity v, the
charge per unit time through AS is
pv  n AS. SURFACE S Fig. 13—3. The current I through the
surface S is fj  ndS. Fig. 13—4. The integral of j I: over
a closed surface is the rate of change of
the total charge Q inside. 21 At, which when multiplied by the charge density p will give Aq. Thus
Aq = pv  nASAt.
The charge per unit time is then pv  n AS, from which we get
j = pv. (13.3) If the charge distribution consists of individual charges, say electrons, each
with the charge q and moving with the mean velocity u, then the current density is i=Mu (no where N is the number of charges per unit volume. The total charge passing per unit time through any surface S is called the
electric current, I. It is equal to the integral of the normal component of the ﬂow
through all of the elements of the surface: I: fsj'ndS (135) (see Fig. 13—3). The current I out of a closed surface S represents the rate at which charge
leaves the volume V enclosed by S. One of the basic laws of physics is that
electric charge is indestructible; it is never lost or created. Electric charges can
move from place to place but never appear from nowhere. We say that charge is
conserved. If there is a net current out of a closed surface, the amount of charge
inside must decrease by the corresponding amount (Fig. 13—4). We can, therefore,
write the law of the conservation of charge as . d
/ J I n dS = _a} (Qinside) ﬁx?
The charge inside can be written as a volume integral of the charge density:
Qinside = / P dV V
inside S If we apply (13.6) to a small volume AV, we know that the lefthand integral
is V  j AV. The charge inside is p AV, so the conservation of charge can also be
written as . 6
vj=£ (mm (Gauss’ mathematics once again!) 13—3 The magnetic force on a current Now we are ready to ﬁnd the force on a currentcarrying wire in a magnetic
ﬁeld. The current consists of charged particles moving with the velocity v along
the wire. Each charge feels a transverse force F=qv><B (Fig. 13—5a). If there are N such charges per unit volume, the number in a small
volume AV of the wire is N AV. The total magnetic force AF on the volume AV
is the sum of the forces on the individual charges, that is, AF = (NAV)(qv X B).
But M11) is just j, so AF = j X BAV (13.9) (Fig. 13—5b). The force per unit volume is j X B.
13—2 If the current is uniform across a wire whose crosssectional area is A, we
may take as the volume element a cylinder with the base area A and the length
AL. Then AF = j X BA AL. (13.10) Now we can call jA the vector current I in the wire. (Its magnitude is the electric
current in the wire, and its direction is along the wire.) Then AF = I X BAL. (13.11) The force per unit length on a wire is I X B. This equation gives the important result that the magnetic force on a wire,
due to the m0vement of charges in it, depends only on the total current, and not on
the amount of charge carried by each particle—or even its sign! The magnetic
force on a wire near a magnet is easily shown by observing its deﬂection when a
current is turned on, as was described in Chapter 1 (see Fig. 16). 134 The magnetic ﬁeld of steady currents; Ampere’s law We have seen that there is a force on a wire in the presence of a magnetic ﬁeld,
produced, say, by a magnet. From the principle that action equals reaction we
might expect that there should be a force on the source of the magnetic ﬁeld, i.e.,
on the magnet, when there is a current through the wire.* There are indeed such
forces, as is seen by the deﬂection of a compass needle near a currentcarrying
wire. Now we know that magnets feel forces from other magnets, so that means
that when there is a current in a wire, the wire itself generates a magnetic ﬁeld.
Moving charges, then, produce a magnetic ﬁeld. We would like now to try to
discover the laws that determine how such magnetic ﬁelds are created. The question
is: Given a current, what magnetic ﬁeld does it make? The answer to this question
was determined experimentally by three critical experiments and a brilliant
theoretical argument given by Ampere. We will pass over this interesting historical
development and simply say that a large number of experiments have demonstrated
the validity of Maxwell’s equations. We take them as our starting point. If we
drop the terms involving time derivatives in these equations we get the equations of
magnetostatics: V ' B = 0 (13.12)
and C2VXB=.'L.
60 (13.13) These equations are valid only if all electric charge densities are constant and all
currents are steady, so that the electric and magnetic ﬁelds are not changing with
time—all of the ﬁelds are “static.” We may remark that it is rather dangerous to think that there is such a thing
as a static magnetic situation, because there must be currents in order to get a
magnetic ﬁeld at all—and currents can come only from moving charges. “Mag
netostatics” is, therefore, an approximation. It refers to a special kind of dynamic
situation with large numbers of charges in motion, which we can approximate by
a steady ﬂow of charge. Only then can we speak of a current density j which does
not change with time. The subject should more accurately be called the study of
steady currents. Assuming that all ﬁelds are steady, we drop all terms in aE/at
and aB/at from the complete Maxwell equations, Eqs. (2.41), and obtain the
two equations (13.12) and (13.13) above. Also notice that since the divergence of
the curl of any vector is necessarily zero, Eq. (13.13) requires that V  j = 0. This
is true, by Eq. (13.8), only if 6p/6t is zero. But that must be so if E is not changing
with time, so our assumptions are consistent. " We will see later, however, that such assumptions are not generally correct for electro
magnetic forces! 13—3 (b) Fig. 13—5. The magnetic force on a
currentcarrying wire is the sum of the
forces on the individual moving charges. Fig. 13—6. The line integral of the
tangential component of B is equal to the
surface integral of the normal component ofVXB. The requirement that V  j = 0 means that we may only have charges which
ﬂow in paths that close back on themselves. They may, for instance, ﬂow in wires
that form complete loops—called circuits. The circuits may, of course, contain
generators or batteries that keep the charges ﬂowing. But they may not include
condensers which are charging or discharging. (We will, of course, extend the
theory later to include dynamic ﬁelds, but we want ﬁrst to take the simpler case of
steady currents.) Now let us look at Eqs. (13.12) and (13.13) to see what they mean. The ﬁrst
one says that the divergence of B is zero. Comparing it to the analogous equation
in electrostatics, which says that V  E = p/eo, we can conclude that there is no
magnetic analog of an electric charge. There are no magnetic charges from which
lines of B can emerge. If we think in terms of “lines” of the vector ﬁeld B, they can
never start and they never stop. Then where do they come from? Magnetic ﬁelds
“appear” in the presence of currents; they have a curl proportional to the current
density. Wherever there are currents, there are lines of magnetic ﬁeld making
loops around the currents. Since lines of B do not begin or end, they will often
close back on themselves, making closed loops. But there can also be complicated
situations in which the lines are not simple closed loops. But whatever they do,
they never diverge from points. No magnetic charges have ever been discovered,
so V  B = 0. This much is true not only for magnetostatics, it is always true—
even for dynamic ﬁelds. The connection between the B ﬁeld and currents is contained in Eq. ( 13.13).
Here we have a new kind of situation which is quite different from electrostatics,
where we had V X E = 0. That equation meant that the line integral of E around any closed path is zero:
f E ' ds = 0. loop We got that result from Stokes’ theorem, which says that the integral around any
closed path of any vector ﬁeld is equal to the surface integral of the normal com
ponent of the curl of the vector (taken over any surface which has the closed loop
as its periphery). Applying the same theorem to the magnetic ﬁeld vector and
using the symbols shown in Fig. 13—6, we get fBds = /S (V X B)ndS. (13.14)
1"
Taking the curl of B from Eq. ( 13.13), we have 1 . The integral over j, according to (13.5), is the total current I through the surface S.
Since for steady currents the current through S is independent of the shape of S,
so long as it is bounded by the curve 1‘, one usually speaks of “the current through
the loop I‘.” We have, then, a general law: the circulation of B around any closed
curve is equal to the current I through the loop, divided by eoc2: = IthroughI‘ _
1 6062 This law—called Ampere’s law—plays the same role in magnetostatics that Gauss’
law played in electrostatics. Ampere’s law alone does not determine B from cur
rents; we must, in general, also use V  B = 0. But, as we will see in the next
section, it can be used to ﬁnd the ﬁeld in special circumSIances which have certain
simple symmetries. 13—4 (13.16) 13—5 The magnetic ﬁeld of a straight wire and of a solenoid; atomic
currents We can illustrate the use of Ampere’s law by ﬁnding the magnetic ﬁeld near
a wire. We ask: What is the ﬁeld outside a long straight wire with a cylindrical
cross section? We will assume something which may not be at all evident, but which
is nevertheless true: that the ﬁeld lines of B go around the wire in closed circles.
If we make this assumption, then Ampere’s law, Eq. (13.16), tells us how strong the
ﬁeld is. From the symmetry of the problem, B has the same magnitude at all
points on a circle concentric with the wire (see Fig. 13—7). We can then do the line
integral of B  ds quite easily; it is just the magnitude of B times the circumference.
If r is the radius of the circle, then fBds = B27rr. The total current through the loop is merely the current I in the wire, so B  2117 = —!—2 ’
60C
or
1 21
B — 4711062 —r— (13.17) The strength of the magnetic ﬁeld drops off inversely as r, the distance from the
axis of the wire. We can, if we wish, write Eq. (13.17) in vector form. Remembering
that B is at right angles both to I and to r, we have 1 21Xe,_ B =
41reocz r (13.18) We have separated out the factor l/47reocz, because it appears often. It is
worth remembering that it is exactly 10"7 (in the mks system), since an equation
like (13.17) is used to deﬁne the unit of current, the ampere. At one meter from a
current of one ampere the magnetic ﬁeld is 2 X 10‘7 webers per square meter. Since a current produces a magnetic ﬁeld, it will exert a force on a nearby wire
which is also carrying a current. In Chapter 1 we described a simple demonstration
of the forces between two current—carrying wires. If the wires are parallel, each is
at right angles to the B ﬁeld of the other; the wires should then be pushed either
toward or away from each other. When currents are in the same direction, the
wires attract; when the currents are moving in opposite directions, the wires repel. 1"?) r.‘__.._... iaaaaaraangaammn Illlllllliiiiliillll
IIIIIIIIIIIIIIIIIIIIII Let’s take another example that can be analyzed by Ampere’s law if we add
some knowledge about the ﬁeld. Suppose we have a long coil of wire wound in a
tight spiral, as shown by the cross sections in Fig. 13—8. Such a coil is called a
solenoid. We observe experimentally that when a solenoid is very long compared
with its diameter, the ﬁeld outside is very small compared with the ﬁeld inside.
Using just that fact, together with Ampere’s law, we can ﬁnd the size of the ﬁeld
inside. Since the ﬁeld stays inside (and has zero divergence), its lines must go along
parallel to the axis, as shown in Fig. 13—8. That being the case, we can use Ampere’s
law with the rectangular “curve” I‘ shown in the ﬁgure. This loop goes the distance 13—5 Fig. 1 38.
long solenoid. I\\ Fig. 13—7. The magnetic ﬁeld outside
of a long wire carrying the current I. The magnetic field of a Fig. l3—9. The magnetic ﬁeld outside
of a solenoid. L inside the solenoid, where the ﬁeld is, say, Bo, then goes at right angles to the
ﬁeld, and returns along the outside, where the ﬁeld is negligible. The line integral
of B for this curve is just BOL, and it must be 1/6002 times the total current through
I‘, which is NI if there are N turns of the solenoid in the length L. We have NI
30L = Eoc—z' Or, letting n be the number of turns per unit length of the solenoid (that is, n =
N /L), we get n]
30 = 2—2..
00 (13.19) What happens to the lines of B when they get to the end of the solenoid?
Presumably, they spread out in some way and return to enter the solenoid at the
other end, as sketched in Fig. 13—9. Such a ﬁeld is just what is observed outside of
a bar magnet. But what is a magnet anyway ? Our equations say that B comes from
the presence of currents. Yet we know that ordinary bars of iron (no batteries or
generators) also produce magnetic ﬁelds. You might expect that there should be
some other terms on the righthand side of (13.12) or (13.13) to represent “the
density of magnetic iron” or some such quantity. But there is no such term. Our
theory says that the magnetic effects of iron come from some internal currents
which are already taken care of by the j term. Matter is very complex when looked at from a fundamental point of view—as
we saw when we tried to understand dielectrics. In order not to interrupt our pres
ent discussion, we will wait until later to deal in detail with the interior mechanisms
of magnetic materials like iron. You will have to accept, for the moment, that all
magnetism is produced from currents, and that in a permanent magnet there are
permanent internal currents. In the case of iron, these currents come from electrons
spinning around their own axes. Every electron has such a spin, which corresponds
to a tiny circulating current. Of course, one electron doesn’t produce much mag
netic ﬁeld, but in an ordinary piece of matter there are billions and billions of elec
trons. Normally these spin and point every which way, so that there is no net
effect. The miracle is that in a very few substances, like iron, a large fraction of
the electrons spin with their axes in the same direction—for iron, two electrons from
each atom takes part in this cooperative motion. In a bar magnet there are large
numbers of electrons all spinning in the same direction and, as we will see, their
total eﬂect is equivalent to a current circulating on the surface of the bar. (This is
quite analogous to what we found for dielectrics—that a uniformly polarized di
electric is equivalent to a distribution of charges on its surface.) It is, therefore, no
accident that a bar magnet is equivalent to a solenoid. 136 The relativity of magnetic and electric ﬁelds When we said that the magnetic force on a charge was proportional to its
velocity, you may have wondered: “What velocity? With respect to which refer
ence frame?” It is, in fact, clear from the deﬁnition of B given at the beginning of
this chapter that what this vector is will depend on what we choose as a reference
frame for our speciﬁcation of the velocity of charges. But we have said nothing
about which is the proper frame for specifying the magnetic ﬁeld. It turns out that any inertial frame will do. We will also see that magnetism
and electricity are not independent things—that they should always be taken to
gether as one complete electromagnetic ﬁeld. Although in the static case Maxwell’s
equations separate into two distinct pairs, one pair for electricity and one pair for
magnetism, with no apparent connection between the two ﬁelds, nevertheless, in
nature itself there is a very intimate relationship between them that arises from the
principle of relativity. Historically, the principle of relativity was discovered after
Maxwell’s equations. It was, in fact, the study of electricity and magnetism which
led ultimately to Einstein’s discovery of his principle of relativity. But let’s see 1 3—6 Fig. 1310. The interaction of a currentcarrying wire and a particle with the
charge q as seen in two frames. In frame 5 (part a), the wire is at rest, in frame
5' (part b), the charge is at rest. what our knowledge of relativity would tell us about magnetic forces if we assume
that the relativity principle is applicable—as it is—to electromagnetism. Suppose we think about what happens when a negative charge moves with
velocity 00 parallel to a current—carrying wire, as in Fig. l3~10. We will try to under
stand what goes on in two reference frames: one ﬁxed with respect to the wire,
as in part (a) of the ﬁgure, and one ﬁxed with respect to the particle, as in part (b).
We will call the ﬁrst frame S and the second S’. In the Sframe, there is clearly a magnetic force on the particle. The force is
directed toward the wire, so if the charge is moving freely we would see it curve in
toward the wire. But in the S’frame there can be no magnetic force on the particle,
because its velocity is zero. Does it, therefore, stay where it is? Would we see
diﬁ‘erent things happening in the two systems? The principle of relativity would
say that in S’ we should also see the particle move closer to the wire. We must
try to understand why that would happen. We return to our atomic description of a wire carrying a current. In a normal
conductor, like copper, the electric currents come from the motion of some of the
negative electrons—called the conduction electrons—while the positive nuclear
charges and the remainder of the electrons stay ﬁxed in the body of the material.
We let the density of the conduction electrons be p__ and their velocity in S be v.
The density of the charges at rest in S is p+, which must be equal to the negative
of p_, since we are considering an uncharged wire. There is thus no electric ﬁeld
outside the wire, and the force on the moving particle is just F=qUOXB. Using the result we found in Eq. (13.18) for the magnetic ﬁeld at the distance
r from the axis of a wire, we conclude that the force on the particle is directed
toward the wire and has the magnitude
1 qUO F =
4711062 r Using Eqs. (13.4) and (13.5), the current I can be written as p_vA, where A is
the area of a cross section of the wire. Then
1 2qp_Avvo F = “6062 .—r——. (13.20) We could continue to treat the general case of arbitrary velocities for v and 210,
but it will be just as good to look at the special case in which the velocity 00 of
the particle is the same as the velocity v of the conduction electrons. So we write
00 = v, and Eq. (13.20) becomes _ ﬂ __ _. (13.21) Now we turn our attention to what happens in S’, in which the particle is at
rest and the wire is running past (toward the left in the ﬁgure) with the speed I).
The positive charges moving with the wire will make some magnetic ﬁeld B’ at
the particle. But the particle is now at rest, so there is no magnetic force on it!
If there is any force on the particle, it must come from an electric ﬁeld. It must 1 3—7 be that the moving wire has produced an electric ﬁeld. But it can do that only if it
appears charged—it must be that a neutral wire with a current appears to be charged
when set in motion. . We must look into this. We must try to compute the charge density in the
wire in S’ from what we know about it in S. One might, at ﬁrst, think they are the
same; but we know that lengths are changed between S and S’ (see Chapter 15,
Vol. I), so volumes will change also. Since the charge densities depend on the
volume occupied by charges, the densities will change, too. Before we can decide about the charge densities in S’, we must know what
happens to the electric charge of a bunch of electrons when the charges are moving.
We know that the apparent mass of a particle changes by 1 /\/1 — v2/c2. Does
its charge do something similar? No! Charges are always the same, moving or
not. Otherwise we would not always observe that the total charge is conserved. Suppose that we take a block of material, say a conductor, which is initially
uncharged. Now we heat it up. Because the electrons have a diﬁerent mass than
the protons, the velocities of the electrons and of the protons will change by diﬁer
ent amounts. If the charge of a particle depended on the speed of the particle carry
ing it, in the heated block the charge of the electrons and protons would no longer
balance. A block would become charged when heated. As we have seen earlier, a
very small fractional change in the charge of all the electrons in a block would give
rise to enormous electric ﬁelds. No such effect has ever been observed. Also, we can point out that the mean speed of the electrons in matter depends
on its chemical composition. If the charge on an electron changed with speed, the
net charge in a piece of material would be changed in a chemical reaction. Again,
a straightforward calculation shows that even a very small dependence of charge
on speed would give enormous ﬁelds from the simplest chemical reactions. No
such effect is observed, and we conclude that the electric charge of a single particle
is independent of its state of motion. So the charge q on a particle is an invariant scalar quantity, independent of
the frame of reference. That means that in any frame the charge density of a
distribution of electrons is just proportional to the number of electrons per unit
volume. We need only worry about the fact that the volume can change because
of the relativistic contraction of distances. We now apply these ideas to our moving wire. If we take a length L0 of the
wire, in which there is a charge density p0 of stationary charges, it will contain
the total charge Q = pOLOA 0. If the same charges are observed in a different frame
to be moving with velocity 1), they will all be found in a piece of the material with
the shorter length L = Lox/l — v2/c2, (13.22) but with the same area A0 (since dimensions transverse to the motion are un
changed). See Fig. 13—11. If we call p the density of charges in the frame in which they are moving, the
total charge Q will be pLA 0. This must also be equal to poL 0A, because charge is
the same in any system, so that pL = poLo or, from (13.22), Po «Tm! P = (13.23) (D) Q v=0 Fig. 13—1 1. If a distribution of charged particles at rest has the charge density
p0, the same charges will have the density p = po/x/l — v2/c2 when seen from a
frame with the relative velocity v. 1 3—8 The charge density of a moving distribution of charges varies in the same way as the
relativistic mass of a particle. We now use this general result for the positive charge density p+ of our wire.
These charges are at rest in frame S. In S’, however, where the wire moves with
the speed 0, the positive charge density becomes I p+ p—————.
J’m The negative charges are at rest in S’. So they have their “rest density” p0 in
this frame. In Eq. (13.23) p0 = p’_, because they have the density p’_ when the
wire is at rest, i.e., in frame S, where the Speed of the negative charges is v. For
the conduction electrons, we then have that (13.24) I p
p__ = —————— , (13.25)
V1 — 112/c2
or
p’_ = p_\/1 — v2/c2. (13.26) Now we can see why there are electric ﬁelds in S’——because in this frame the
wire has the net charge density p’ given by p’ = p’+ + pL.
Using (13.24) and (13.26), we have
p' = i..— + p_‘/1 _ vz/cz. V1 — v2/c2
Since the stationary wire is neutral, p__ = —p+, and we have
2 2
v /c
p’ = p ————————— (13.27)
+ \/l — 09/c2 Our moving wire is positively charged and will produce an electric ﬁeld E’ at the
external stationary particle. We have already solved the electrostatic problem of a
uniformly charged cylinder. The electric ﬁeld at the distance r from the axis of the
cylinder is _ p’A _ p+AvZ/c2 .
2750’ linen/m The force on the negatively charged particle is toward the wire. We have, at least,
a force in the same direction from the two points of View; the electric force in S’
has the same direction as the magnetic force in S. The magnitude of the force in S’ is El (13.28) 2 2
F’— q “A “c (13.29) Comparing this result for F’ with our result for F in Eq. (13.21), we see that the
magnitudes of the forces are almost identical from the two points of view. In fact, F
v1 — v2/c2 so for the small velocities we have been considering, the two forces are equal.
We can say that for low velocities, at least, we understand that magnetism and
electricity are just “two ways of looking at the same thing.” But things are even better than that. If we take into account the fact that
forces also transform when we go from one system to the other, we ﬁnd that the
two ways of looking at what happens do indeed give the same physical result for
any velocity. Fl H , (13.30) 1 39 (b) Fig. l3—l 2. In frame 5 the charge
density is zero and the current density is
i. There is only a magnetic field. In 5',
there is a charge density p', and a differ
ent current density 1". The magnetic field B’ is different and there is an electric
field 5'. One way of seeing this is to ask a question like: What transverse momentum
will the particle have after the force has acted for a little while? We know from
Chapter 16 of Vol. I that the transverse momentum of a particle should be the same
in both the S— and S’frames. Calling the transverse coordinate y, we want to
compare Apu and Apg. Using the relativistically correct equation of motion,
F = dp/dt, we expect that after the time At our particle will have a transverse
momentum Apy in the Ssystem given by Apy = FAt. (13.31)
In the S’system, the transverse momentum will be
Ap; = F’ At’. (13.32) We must, of course, compare Apy and Apf, for corresponding time intervals At and
At’. We have seen in Chapter 15 of Vol. I that the time intervals referred to a
moving particle appear to be longer than those in the rest system of the particle.
Since our particle is initially at rest in S’, we expect, for small At, that At’
At = —— , (13.33)
V1 — 122/c2
and everything comes out OK. From (13.31) and (13.32),
AL; __ F’At’
Ap, " FAt ’ which is just = 1 if we combine (13.30) and (13.33). We have found that we get the same physical result whether we analyze the
motion of a particle m0ving along a wire in a coordinate system at rest with respect
to the wire, or in a system at rest with respect to the particle. In the ﬁrst instance,
the force was purely “magnetic,” in the second, it was purely “electric.” The two
points of View are illustrated in Fig. 13—12 (although there is still a magnetic ﬁeld
B’ in the second frame, it produces no forces on the stationary particle). If we had chosen still another coordinate system, we would have found a
different mixture of E and B ﬁelds. Electric and magnetic forces are part of one
physical phenomenon—the electromagnetic interactions of particles. The separa
tion of this interaction into electric and magnetic parts depends very much on the
reference frame chosen for the description. But a complete electromagnetic de
scription is invariant; electricity and magnetism taken together are consistent
with Einstein’s relativity. Since electric and magnetic ﬁelds appear in diﬁerent mixtures if we change our
frame of reference, we must be careful about how we look at the ﬁelds E and B.
For instance, if we think of “lines” of E or B, we must not attach too much reality
to them. The lines may disappear if we try to observe them from a different co
ordinate system. For example, in system S’ there are electric ﬁeld lines, which we
do not ﬁnd “moving past us with velocity v in system S.” In system S there are no
electric ﬁeld lines at all! Therefore it makes no sense to say something like: When
I move a magnet, it takes its ﬁeld with it, so the lines of B are also moved. There
is no way to make sense, in g? 1eral, out of the idea of “the speed of a moving ﬁeld
line.” The ﬁelds are our way of describing what goes on at a point in space. In
particular, E and B tell us about the forces that will act on a moving particle. The
question “What is the force on a charge from a moving magnetic ﬁeld?” doesn’t
mean anything precise. The force is given by the values of E and B at the charge,
and the formula (13.1) is not to be altered if the source of E or B is moving (it is
the values of E and B that will be altered by the motion). Our mathematical de
scription deals only with the ﬁelds as a function of x, y, z, and t with respect to
some inertial frame. We will later be speaking of “a wave of electric and magnetic ﬁelds travelling
through space,” as, for instance, a light wave. But that is like speaking of a wave
travelling on a string. We don’t then mean that some part of the string is moving 13—10 in the direction of the wave, we mean that the displacement of the string appears
first at one place and later at another. Similarly, in an electromagnetic wave, the
wave travels; but the magnitude of the ﬁelds change. So in the future when we—or
someone else—speaks of a “moving” ﬁeld, you should think of it as just a handy,
short way of describing a changing ﬁeld in some circumstances. 137 The transformation of currents and charges You may have worried about the simpliﬁcation we made above when we
took the same velocity v for the particle and for the conduction electrons in the
wire. We could go back and carry through the analysis again for two different
velocities, but it is easier to simply notice that charge and current density are the
components of a fourvector (see Chapter 17, Vol. I). We have seen that if p0 is the density of the charges in their rest frame, then
in a frame in which they have the velocity v, the density is p = _&__.
V1 — 2)2/c2
In that frame their current density is
. pol)
J = W = _. (13.34)
V1 — 1)2/c2 Now we know that the energy U and momentum p of a particle moving with
velocity v are given by
2 U=__mic__ p=__’L‘L
V1 — 112/62 v1 —— 112/c2 where m0 is its rest mass. We also know that U and [1 form a relativistic fourvector.
Since p and j depend on the velocity v exactly as do U and p, we can conclude that p
and j are also the components of a relativistic fourvector. This property is the key
to a general analysis of the ﬁeld of a wire moving with any velocity, which we
would need if we want to do the problem again with the velocity v0 of the particle
different from the velocity of the conduction electrons. If we wish to transform p and j to a coordinate system moving with a velocity
u in the xdirection, we know that they transform just like I and (x, y, 2), so that
we have (see Chapter 15, Vol. I) x,_ x—ut j,_ jx—up
— '—_'— ' , a; — —_———’
V1 — u2/c2 v1 — uZ/c2
y’ = y, j; = jy»
z, = z, = jzg
_ 2 _  2
,I = w, p: = ML. (1335)
V1 — u2/c2 V1 — uZ/c2 With these equations we can relate charges and currents in one frame to those
in another. Taking the charges and currents in either frame, we can solve the
electromagnetic problem in that frame by using our Maxwell equations. The
result we obtain for the motions of particles will be the same no matter which frame
we choose. We will return at a later time to the relativistic transformations of the
electromagnetic ﬁelds. 138 Superposition; the righthand rule We will conclude this chapter by making two further points regarding the
subject of magnetostatics. First, our basic equations for the magnetic ﬁeld, VB = 0, V X B =i/c260,
1311 are linear in B and j. That means that the principle of superposition also applies
to magnetic ﬁelds. The ﬁeld produced by two diﬁerent steady currents is the sum
of the individual ﬁelds from each current acting alone. Our second remark con
cerns the righthand rules which we have encountered (such as the righthand
rule for the magnetic ﬁeld produced by a current). We have also observed that the
magnetization of an iron magnet is to be understood from the spin of the electrons
in the material. The direction of the magnetic ﬁeld of a spinning electron is related
to its spin axis by the same righthand rule. Because B is determined by a “handed”
rule—involving either a cross product or a curl—it is called an axial vector. (Vec
tors whose direction in space does not depend on a reference to a right or left hand
are called polar vectors. Displacement, velocity, force, and E, for example, are'
polar vectors.) Physically observable quantities in electromagnetism are not, however, right
(or left) handed. Electromagnetic interactions are symmetrical under reﬂection
(see Chapter 52, Vol. 1). Whenever magnetic forces between two sets of currents are
computed, the result is invariant with respect to a change in the hand convention.
Our equations lead, independently of the righthand convention, to the end result
that parallel currents attract, or that currents in opposite directions repel. (Try
working out the force using “lefthand rules.”) An attraction or repulsion is a
polar vector. This happens because in describing any complete interaction, we
use the righthand rule twice—once to ﬁnd B from currents, again to ﬁnd the force
this B produces on a second current. Using the righthand rule twice is the same
as using the lefthand rule twice. If we were to change our conventions to a left
hand system all our B ﬁelds would be reversed, but all forces—or, what is perhaps
more relevant, the observed accelerations of objects—would be unchanged. Although physicists have recently found to their surprise that all the laws of
nature are not always invariant for mirror reﬂections, the laws of electromagnetism
do have such a basic symmetry. 13—12 ...
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This note was uploaded on 03/26/2010 for the course PHYSICS V85.0093.0 taught by Professor Chaikin during the Spring '10 term at NYU.
 Spring '10
 Chaikin
 Physics

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