ELECTROMAGNETISM

11.1 More About the Magnetic Field




fulfill

a / The pair of charged particles, as seen in two different frames of reference.


bdeflects

b / A large current is created by shorting across the leads of the battery. The moving charges in the wire attract the moving charges in the electron beam, causing the electrons to curve.


brelativity

c / A charged particle and a current, seen in two different frames of reference. The second frame is moving at velocity v with respect to the first frame, so all the velocities have v subtracted from them. (As discussed in the main text, this is only approximately correct.)

11.1.1 Magnetic forces


In this chapter, I assume you know a few basic ideas about Einstein's theory of relativity,
as described in section 7.1.
Unless your typical workday involves rocket ships or particle accelerators,
all this relativity stuff might sound like a description of some bizarre
futuristic world that is completely hypothetical. There is, however, a relativistic
effect that occurs in everyday life, and it is obvious and dramatic: magnetism.
Magnetism, as we discussed previously, is an interaction between a moving
charge and another moving charge,
as opposed to electric forces, which act between any pair of charges, regardless of their
motion. Relativistic effects are weak for speeds that are small compared to the speed
of light, and the average speed at which electrons drift through a wire is quite
low (centimeters per second, typically), so how can relativity be behind an impressive
effect like a car being lifted by an electromagnet hanging from a crane? The key is
that matter is almost perfectly electrically neutral, and electric forces therefore
cancel out almost perfectly. Magnetic forces really aren't very strong, but electric
forces are even weaker.


What about the word “relativity” in the name of the theory?
It would seem problematic if moving charges interact differently than stationary charges,
since motion is a matter of opinion, depending on your frame of reference.
Magnetism, however, comes not to destroy relativity but to fulfill it. Magnetic interactions
must exist according to the theory of relativity. To understand how this can be,
consider how time and space behave in relativity. Observers in different frames of reference
disagree about the lengths of measuring sticks and the speeds of clocks, but the laws
of physics are valid and self-consistent in either frame of reference.
Similarly, observers in different frames of reference disagree about what electric and magnetic
fields and forces there are, but they agree about concrete physical events.
For instance, figure a/1 shows two particles, with opposite charges,
which are not moving at a particular moment in time. An observer in this frame of reference
says there are electric fields around the particles, and predicts that as time goes on, the
particles will begin to accelerate towards one another, eventually colliding.
A different observer, a/2, says the particles are moving. This observer
also predicts that the particles will collide, but explains their motion in terms of both
an electric field, E, and a magnetic field, B. As we'll see shortly, the
magnetic field is required in order to maintain consistency between the predictions made
in the two frames of reference.


To see how this really works out, we need to find a nice simple example that is
easy to calculate. An example like figure a is not easy
to handle, because in the second frame of reference, the moving charges
create fields that change over time at any given location. Examples like
figure b are easier, because there is a steady flow of charges, and
all the fields stay the same over time.1
What is remarkable about this demonstration is that there can be no electric fields
acting on the electron beam at all, since the total charge density throughout the wire
is zero. Unlike figure a/2, figure b is purely magnetic.


To see why this must occur based on relativity, we make the mathematically idealized model
shown in figure c. The charge by itself is like one of the electrons
in the vacuum tube beam of figure b, and a
pair of moving, infinitely long line charges has been substituted for the wire. The electrons in a real wire
are in rapid thermal motion, and the current is created only by a slow drift superimposed
on this chaos. A second deviation from reality is that in the real experiment, the protons
are at rest with respect to the tabletop, and it is the electrons that are in motion, but in
c/1 we have the positive charges moving in
one direction and the negative ones moving
the other way. If we wanted to, we could construct a third frame of reference in which the
positive charges were at rest, which would be more like the frame of reference fixed to the
tabletop in the real demonstration. However, as we'll see shortly, frames
c/1 and c/2 are designed so that they are
particularly easy to analyze. It's important to note that even though the two line charges
are moving in opposite directions, their currents don't cancel. A negative charge moving
to the left makes a current that goes to the right, so in frame c/1,
the total current is twice that contributed by either line charge.


Frame 1 is easy to analyze because the charge densities of the two line charges cancel out,
and the electric field experienced by the lone charge is therefore zero:

E1 = 0


In frame 1, any force experienced by the lone charge must therefore be attributed solely
to magnetism.


Frame 2 shows what we'd see if we were observing all this from a frame of reference moving
along with the lone charge.
Why don't the charge densities also cancel in this frame?
Here's where the relativity comes in. Relativity tells us that moving objects
appear contracted to an observer who is not moving along with them.
Both line charges are in motion in both frames of reference, but in frame 1, the
line charges were moving at equal speeds, so their contractions were equal, and their
charge densities canceled out. In frame 2, however, their speeds are unequal. The positive
charges are moving more slowly than in frame 1, so in frame 2 they are less contracted.
The negative charges are moving more quickly, so their contraction is greater now.
Since the charge densities don't cancel, there is an electric field in frame 2, which
points into the wire, attracting the lone charge. Furthermore, the attraction felt
by the lone charge must be purely electrical, since the lone charge is at rest in this
frame of reference, and magnetic effects occur only between moving charges and other
moving charges.2


To summarize, frame 1 displays a purely magnetic attraction, while in frame 2 it is
purely electrical.


A common source of confusion in this argument is that it seems as though the excess of
negative electric charge must have been stolen from some other part of the wire, leaving that
part with a net positive charge. That wouldn't make sense, because there is no physical
reason why one part of the wire would behave differently than any other. The flaw in this
reasoning has to do with the fact that simultaneity is not well defined in relativity.
In frame c/1, suppose that we label all the positive charges with
integers, and likewise all the negative ones, so that the positive charge labeled 42 is on top of
the negative charge labeled 42, and so on. In this frame of reference, every charge has a partner that cancels it,
and the net charge everywhere is zero. If simultaneity were a valid concept in relativity, then
not only would 42's pairing with 42, and 43's pairing with 43, occur simultaneously in frame c/1,
but these same pairings would occur all at the same time in frame c/2. But observers
in different frames of reference do not agree on simultaneity. For simplicity, let's imagine that
the Lorentz contractions are such that the spacing between the negative charges in frame c/2
is exactly half as much as the spacing between the positive charges. Then we may have negative charge number 42
paired up with positive charge 21, and negative charge 44 paired with positive charge 22, while negative charge 43
has no partner.


Now we can calculate the force in frame 2, and equating it to the force in frame 1, we
can find out how much magnetic force occurs.
To keep the math simple, and to keep from assuming too much about your knowledge
of relativity, we're going to carry out this whole calculation in the approximation
where all the speeds are fairly small compared to the speed of light.3 For instance, if
we find an expression such as (v/c)2+(v/c)4, we will assume that the fourth-order
term is negligible by comparison. This is known as a calculation “to leading order
in v/c.” In fact, I've already used the leading-order approximation twice
without saying so! The first time I used it implicitly was in figure c,
where I assumed that the velocities of the two line charges were u-v and -u-v.
Relativistic velocities don't just combine by simple addition and subtraction like
this, but this is an effect we can ignore in the present approximation. The second
sleight of hand occurred when I stated that we could equate the forces in the two
frames of reference. Force, like time and distance, is distorted relativistically
when we change from one frame of reference to another. Again, however, this is an effect
that we can ignore to the desired level of approximation.


Let ±λ be the charge per unit length of each line charge without relativistic
contraction, i.e., in the frame moving with that line charge.
Using the approximation γ=(1-v2/c2)-1/2≈ 1+v2/2c2 for v<< c, the
total charge per unit length in frame 2 is






λtotal,2
≈λ[1+(u-v)22c2]-λ[1+(-u-v)22c2]



=-2λuvc2.





Let R be the distance from the line charge to the lone charge.
Applying Gauss' law to a cylinder of radius R centered on the line charge,
we find that the magnitude of the electric field experienced by the lone charge
in frame 2 is



E=4kλuvc2R,


and the force acting on the lone charge q is



F=4kλquvc2R.


In frame 1, the current is I=2λ1 u (see homework problem 5),
which we can approximate
as I=2λ u, since the current, unlike λtotal, 2, doesn't
vanish completely without the relativistic effect.
The magnetic force on the lone charge q due to the current I is



F=2kIqvc2R.





vbf

d / The right-hand relationship between the velocity of a positively charged particle, the magnetic field through which it is moving, and the magnetic force on it.


tesla

e / The unit of magnetic field, the tesla, is named after Serbian-American inventor Nikola Tesla.


current-loop-dipole

f / A standard dipole made from a square loop of wire shorting across a battery. It acts very much like a bar magnet, but its strength is more easily quantified.


current-loop-aligns

g / A dipole tends to align itself to the surrounding magnetic field.


arearh

h / The m and A vectors.


squaretorque

i / The torque on a current loop in a magnetic field. The current comes out of the page, goes across, goes back into the page, and then back across the other way in the hidden side of the loop.


inout

j / A vector coming out of the page is shown with the tip of an arrowhead. A vector going into the page is represented using the tailfeathers of the arrow.


adddipoles

k / Dipole vectors can be added.


irregularloop

l / An irregular loop can be broken up into little squares.


iron-filings-around-magnet

m / The magnetic field pattern around a bar magnet is created by the superposition of the dipole fields of the individual iron atoms. Roughly speaking, it looks like the field of one big dipole, especially farther away from the magnet. Closer in, however, you can see a hint of the magnet's rectangular shape. The picture was made by placing iron filings on a piece of paper, and then bringing a magnet up underneath.

11.1.2 The magnetic field


Definition in terms of the force on a moving particle


With electricity, it turned out to be useful to define an electric field
rather than always working in terms of electric forces. Likewise, we want
to define a magnetic field, B. Let's look at the result of the preceding subsection
for insight. The equation



F=2kIqvc2R


shows that when we put a moving charge
near other moving charges, there is an extra magnetic force on it, in addition to
any electric forces that may exist. Equations for electric forces always have a factor
of k in front --- the Coulomb constant k is called
the coupling constant for
electric forces. Since magnetic effects are relativistic in origin, they end up
having a factor of k/c2 instead of just k. In a world where the speed of light
was infinite, relativistic effects, including magnetism, would be absent, and the
coupling constant for magnetism would be zero. A cute feature of the metric system
is that we have k/c2=10-7 N⋅s2/C2 exactly,
as a matter of definition.


Naively, we could try to work by analogy with the electric field, and define
the magnetic field as the magnetic force per unit charge. However, if we think
of the lone charge in our example as the test charge, we'll find that this
approach fails, because the force depends not just on the test particle's charge,
but on its velocity, v, as well. Although we only carried out calculations for
the case where the particle was moving parallel to the wire, in general this velocity
is a vector, v, in three dimensions. We can also anticipate that the magnetic
field will be a vector. The electric and gravitational fields are vectors, and we
expect intuitively based on our experience with magnetic compasses that a magnetic field
has a particular direction in space. Furthermore, reversing the current I in our
example would have reversed the force, which would only make sense if the magnetic
field had a direction in space that could be reversed. Summarizing, we think there
must be a magnetic field vector B, and the force on a test particle moving
through a magnetic field is proportional both to the B vector
and to the particle's own v vector. In other words, the magnetic force vector F is
found by some sort of vector multiplication of the vectors v and B.
As proved on page 856, however, there is only one physically
useful way of defining such a multiplication, which is the cross product.




We
therefore define the magnetic field vector, B, as the vector that determines
the force on a charged particle according to the following rule:



F=qv×B[definition of the magnetic field]





From this definition, we see that the magnetic field's units are
N⋅s/C⋅m, which are usually abbreviated as
teslas, 1 T=1 N⋅s/C⋅m.
The definition implies a right-hand-rule relationship
among the vectors, figure d, if the charge q is positive, and
the opposite handedness if it is negative.


This is not just a definition but a bold prediction! Is it really true
that for any point in space, we can always find a vector B that successfully
predicts the force on any passing particle, regardless of its charge and
velocity vector? Yes --- it's not obvious that it can be done, but
experiments verify that it can. How? Well for example, the cross product of parallel vectors
is zero, so we can try particles moving in various directions, and hunt for the
direction that produces zero force; the B vector lies along that line, in
either the same direction the particle was moving, or the opposite one.
We can then go back to our data from one of the other cases, where the
force was nonzero, and use it to choose between these two directions and find
the magnitude of the B vector. We could then verify that this vector
gave correct force predictions in a variety of other cases.


Even with this empirical reassurance, the meaning of this equation is
not intuitively transparent,
nor is it practical in most cases to measure a magnetic field this way. For these
reasons, let's look at an alternative method of defining the magnetic field which,
although not as fundamental or mathematically simple, may be more appealing.


Definition in terms of the torque on a dipole


A compass needle in a magnetic field experiences a torque which tends to align
it with the field. This is just like the behavior of an electric dipole in
an electric field, so we consider the compass needle to be a
magnetic dipole.
In subsection 10.1.3 on
page 526,
we gave an alternative definition of the
electric field in terms of the torque on an electric dipole.


To define the strength of a magnetic field, however, we need
some way of defining the strength of a test dipole, i.e., we
need a definition of the magnetic dipole moment. We could
use an iron permanent magnet constructed according to
certain specifications, but such an object is really an
extremely complex system consisting of many iron atoms, only
some of which are aligned with each other. A more fundamental standard
dipole is a square current loop. This could be little
resistive circuit consisting of a square of wire shorting across a battery, f.


Applying F=v×B, we
find that such a loop, when placed in a magnetic
field, g,
experiences a torque that tends to align plane so
that its interior “face” points in a certain direction.
Since the loop is symmetric, it doesn't care if we rotate it
like a wheel without changing the plane in which it lies. It
is this preferred facing direction that we will end up
using as our alternative definition of the magnetic field.


If the loop is out of alignment with the
field, the torque on it is proportional to the amount of
current, and also to the interior area of the loop. The
proportionality to current makes sense, since magnetic
forces are interactions between moving charges, and current
is a measure of the motion of charge. The proportionality to
the loop's area is also not hard to understand, because
increasing the length of the sides of the square increases
both the amount of charge contained in this circular
“river” and the amount of leverage supplied for making
torque. Two separate physical reasons for a proportionality
to length result in an overall proportionality to length
squared, which is the same as the area of the loop. For
these reasons, we define the magnetic dipole moment of a
square current loop as

m = IA ,


where the direction of the vectors is defined as shown in figure h.



We can now give an alternative definition of the magnetic
field:




The magnetic field vector, B, at any location in space is
defined by observing the torque exerted on a magnetic test
dipole mt consisting of a square current loop. The
field's magnitude is



|B|=τ|mt|sinθ,


where θ is the angle between the dipole vector and the field.
This is equivalent to the vector cross product

τ=mt×B

.





Let's show that this is consistent with the previous definition, using the
geometry shown in figure i. The velocity vector that point in
and out of the page are shown using the convention defined
in figure j.
Let the mobile charge carriers in the wire have linear
density λ, and
let the sides of the loop have
length h, so that we have I=λ v, and
m=h2λ v. The only nonvanishing torque comes from the forces on the
left and right sides. The currents in these sides are perpendicular to the field,
so the magnitude of the cross product F=qv×B is simply
|F|=qvB. The torque supplied by each of these forces
is r×F, where the lever arm r has length h/2,
and makes an angle θ with respect to the force vector. The magnitude of the total torque
acting on the loop is therefore






|τ|
=2h2|F|sinθ



=h qvB sinθ,


and substituting q=λh and v=m/h2λ, we have


|τ|
=h λh mh2λBsinθ



=mBsinθ,








which is consistent with the second definition of the field.


It undoubtedly seems artificial to you that we have discussed dipoles only in
the form of a square loop of current. A permanent magnet, for example, is made
out of atomic dipoles, and atoms aren't square! However, it turns out that the
shape doesn't matter. To see why this is so, consider the additive property of
areas and dipole moments, shown in figure k. Each of the square
dipoles has a dipole moment that points out of the page. When they are placed
side by side, the currents in the adjoining sides cancel out, so they are equivalent
to a single rectangular loop with twice the area. We can break down
any irregular shape into little squares, as shown in figure l,
so the dipole moment of any planar current loop can be calculated based on its area,
regardless of its shape.


Example 1: The magnetic dipole moment of an atom


Let's make an order-of-magnitude estimate of the magnetic dipole moment of an atom.
A hydrogen atom is about 10-10 m in diameter, and the electron moves at speeds
of about 10-2 c. We don't know the shape of the orbit, and indeed it turns out that
according to the principles of quantum mechanics, the electron doesn't even have a well-defined
orbit, but if we're brave, we can still estimate the dipole moment using the
cross-sectional area of the atom, which will be on the order of
(10-10 m)2=10-20 m2.
The electron is a single particle, not a steady current, but again we throw caution to
the winds, and estimate the current it creates as e/Δ t,
where Δ t, the time for one orbit, can be estimated by dividing the
size of the atom by the electron's velocity. (This is only a rough estimate,
and we don't know the shape of the orbit, so it would be silly, for instance,
to bother with multiplying the diameter by π based on our intuitive visualization
of the electron as moving around the circumference of a circle.)
The result for the dipole moment is m∼10-23 A⋅m2.


Should we
be impressed with how small this dipole moment is, or with how big it is, considering
that it's being made by a single atom?
Very large or very small numbers are never very interesting by themselves. To get a
feeling for what they mean, we need to compare them to something else. An interesting
comparison here is to think in terms of the total number of atoms in a typical object,
which might be on the order of 1026 (Avogadro's number). Suppose we had this
many atoms, with their moments all aligned. The total dipole moment would be on the
order of 103 A⋅m2, which is a pretty big number. To get
a dipole moment this strong using human-scale devices,
we'd have to send a thousand amps of current through a
one-square meter loop of wire! The insight to be gained here is that, even in
a permanent magnet, we must not have all the atoms perfectly aligned, because that
would cause more spectacular magnetic effects than we really observe. Apparently, nearly
all the atoms in such a magnet are oriented randomly, and do not contribute to the
magnet's dipole moment.



Discussion Questions


◊
The physical situation shown in figure c on page 604
was analyzed entirely in terms of forces. Now let's go back and think about it in terms of fields.
The charge by itself up above the wire is like a test charge, being used to determine the magnetic
and electric fields created by the wire. In figures c/1 and
c/2, are there fields that are purely electric or purely magnetic? Are there
fields that are a mixture of E and B? How does this compare with the forces?




◊
Continuing the analysis begun in discussion question A, can we come up
with a scenario involving some charged particles such that the fields are purely magnetic
in one frame of reference but a mixture of E and B in another frame?
How about an example where the fields are purely electric in one frame, but mixed in
another? Or an example where the fields are purely electric in one frame, but purely
magnetic in another?