Fig. 1
A helpful example: In q's reference frame the positively charged particles are Lorenz contracted to a greater extent than the negatively charged particles. The charge q will thus experience an overall positive electric field & be repelled.
#Direct Proofs of the Lorenz Force law and Lenz's law
for special cases
#Figures 1-3
#All the equations for Lorenz & Lenz (Equation
# in the text refer to these)
#Direct Proof of Coriolis Force
#Direct Proof of Precession
Relativistic Explanations of E&M in Introductory Physics Courses
R. Sivron and N.R. Sivron
Physics Department, Grand Valley State University, Allendale MI 49401
G. A. Mendell
Physics Department, University of Florida, Gainesville, FL 32611
TEXT
Abstract
We present a concise version of E&M that does not include
fields. Magnetism and radiation are derived from Coulomb’s law and
causality.
We assemble old ideas with some new ones. Traditional explanations
on reference frames, time dilation, length contraction, the Lorenz force
law and the Bio-Savart law are combined with new explanations of Lenz’s
law and E&M radiation. Practical everyday applications are used throughout
as “an ax to grind."
Key Words: Electromagnetism, Special relativity
PACS: 41.10F, 03.30
Introduction
In a typical freshman/sophomore two semester introductory course
special relativity is taught towards the end of the 1st or 2nd semesters
as a sort of “intellectual-dessert”. Students in the 60’s and 70’s may
have enjoyed that kind of teaching, but students of the current generation,
coming from a much wider background, find it impractical.
The traditional way of teaching relativity, and electromagnetism,
does not emphasize usefullness. When it comes to magnetism and radiation
students are confused by the traditional methods. Here are some typical
remarks from our evaluation forms and surveys:
· “Electric fields really confuse me”
· “I get A+’s in all my other classes, but I got an F in physics
just because I can’t dig these d... right hand rule”.
· “How does the current KNOW where to go in the secondary loop?”
(Typical remark in an answer to a Lenz’s law question.)
· “I need to understand radiation for my chemistry class, (or:
for my radiation therapy class) but we only spent one day learning about
radiation”
· “Is relativity a mind teaser?”
· “Are time dilation and Lorenz contraction only useful in particle
physics and Star Trek shows?” (Most students don’t think high energy physics
is useful!)
· “In the film on Einstein that was screened in class they said
that Einstein invented relativity to explain magnets. That sounds interesting.
How come we never learned about that?”
(We took the liberty of ‘spicing up’ the second answer .)
Students have a hard time with the apparent uselessness of special
relativity. Students also have a hard time with electric fields, the rules
of thumb associated with the Lorenz Force law, Bio Savart law, and Lenz’s
law. Most troublesome is the confusion surrounding radiation phenomena.
Understanding the reason for radiation sometimes elude the brightest students.
You may be asking: “O.K., so its hard teaching E&M but why
should we fix it?” Here are several reasons:
· Traditionally physics attracts curious students. Students
that want to know ‘what makes the world tick’. Explaining physical phenomena
with rules of thumb annoys such students.
Other students say that “these rules of thumb are the same as
in organic chemistry. A bunch of confusing rules that we have to memorize”.
· The confusion between the rules results in many accidents.
Heard about the guy that pushed the vigorously pushed the patient in and
out of the MRI magnet, ‘so that the patient will radiate’?
There is a way that should work with the current generation of students. The most brilliant physicists of our century gave us explanations of the fundamental concepts in electromagnetism. In this paper we try to translate their ideas to the language of today’s students. Einstein stated that a primary purpose of his first paper on special relativity was to explain the right hand rule associated with the Lorenz force law2. Feynman, in his famous lecture series book, used special relativity to explain the right hand rule associated with the Bio-Savart law3,4,5. In this paper we add similar methods to explain Lenz’s law, and the loss of energy to radiation. We use these explanations to explain technologies that will interest physical sciences and life sciences majors.
2. Electromagnetism is a combination of causality and Coulomb’s
law
Feynman remarked that fields are a fluke of history6. Let us
here try to explain electromagnetism without fields, as the result of two
basic observational laws:
1. Coulomb’s law.
2. The law of causality (you can’t kill your grandfather before he
conceived you.)
3. Analogies with energy
Students today are fascinated when shown the usefullness of special
relativity.
Lenz’s law is used for explaining the direction of the induced
current in generators, transformers, inductors, and many other applications.
Lenz's law states: “The polarity of the induced emf is such that the induced
current produces an induced magnetic field that opposes the change in flux
causing the emf6." This particular rule of thumb is especially hard to
accept, since it asks the student to GUESS a direction for the current
and then check if that direction agrees with the above statement. Do we
want to leave the impression that one can’t a-priory PREDICT the direction
of the induced current from fundamental electromagnetism?
The direction of the induced current in a loop is traditionally
justified by negation7. It is rightly claimed that if the induced magnetic
flux was not opposing the change energy would not be conserved. This argument
is unsatisfactory because many students want to know WHY something happens
in nature, not WHY IT DOESN’T HAPPEN. Although it is true that there are
many cases in nature in which a proof by negation is the only proof available,
Lenz’s law is not one of these cases.
We show here how to deduce Lenz’s law from the principles of special relativity and Gauss’ law in two simplified cases, and compare the results with the traditional explanation. We show below that the direction of the induced emf in Lenz's law is understood to be the natural result of the properties of the Lorenz transformations.
Review of the relativistic explanation of the Lorenz force law right
hand rule
In order to motivate our reader we begin by briefly repeating
the special relativistic explanation of the magnetic force on moving charges.
This slight variation on the explanation in classical textbooks 2,3,4 is
going to be helpful in explaining Lenz’s law.
In figure 1 we show that a positive charge moving in the same
direction as that of the current will be pulled towards the wire. To determine
the direction of the magnetic field we have used the right hand rule for
magnetic fields around current carying wires. The X shaped backs of arrows
under the wire represent a magnetic field that points into the page, and
the shaped heads of arrows above the wire represent a
field that points out of the page. A second right hand rule, for the Lorenz
force law
, (1)
determine that a moving positive charge is pulled towards the wire.
The force towards the wire can also be explained as the result
of an apparent charge density on the wire in the moving charge reference
frame. In that reference frame S' the positive and negative charges in
the wire move at different speeds. The positive charges seem to move slower
than the negatives. As a result the distance between the charges seems
to undergo different Lorenz contractions for the negative and positive
charges. The distance between the negative charges as measured in S' is
smaller than the distance between positive charges in S’ reference frame.
The negative charge density per unit length is therefore greater than that
of the positives. This net negative charge density attracts a positive
charge q.
The velocities of the positive and negative charges as viewed
in reference frame S' are u’+ and u’- , where
(2)
and
(3)
are the Lorenz transformations of velocities along the x axis. Here
mv0 are the velocities of the negative and positive charge densities
in the lab reference frame, and v is the velocity of q in the lab reference
frame. We chose a situation in which both charge carriers are in
motion for demonstrative purposes.
Because of the different velocities in equation (2) and (3) the
distances in the straight wire as observed in S' are Lorenz contracted
by a different amount. The distances between negative charges in the straight
wire are Lorenz contracted by a factor
, (4)
and the distances between positive charges are Lorenz contracted to
a lesser extent by
. (5)
The observed linear charge densities are therefore
(6)
and
, (7)
where
and l is the charge density per unit length in the straight wire in
the lab reference frame.
Using equations (2) -- (7), the relation , and the
relation , we obtain the charge per unit length in reference frame
S':
(8)
Using Gauss' law for a cylinder enclosing the wire in q's reference
frame we get the electric field in reference frame S’
. (9)
The force on the moving charge is thus
, (10)
in frame S', where the speed of the charge q is v'=0, and ’ has no
effect on the charge.
Transforming the force back to the lab frame we get
, (11)
where we have used , and the Lorenz transformation for forces (with
v'=0) . We can easily identify the last term in (11) as the magnetic
field due to a current carrying wire, and equation (11) is
identical to equation (1).
The above result shows that the magnetic force on the moving
charge can be explained without the need to resort to the magnetic field
in the Lorenz force law. Einstein himself noted that a primary motivation
for the development of special relativity was his desire to get rid of
“fictitious forces” in the Lorenz force law1.
An alternative way of explaining this result is that the form of the
Lorenz force law is the same in all reference frames (see appendix).
Lenz’s law for simplified configuration I
Farady’s law and Lenz's law were originally derived from experiments
with magnetic fields. We show here that Lenz’s law can be derived from
Gauss’ law and the Lorenz transformations, without the use of magnetic
fields.
Lenz’s law gives an intuitive explanation to the negative sign
in Farady's law. Farady’s law can be written as
(12)
in which the induced electromotive force in a loop is
(13)
and the total magnetic flux through the loop is
. (14)
Here and are the electric and magnetic fields, and
G is the closed curve that bounds the surface S.
In figure 2 we see a conducting rod sliding on a U shaped conducting
wire towards the right superimposed on the field of figure 1. To
avoid unneeded integration we chose the following configuration: The width
of the U shaped wire l, is much smaller than the distance to the current
carrying wire r, which is in turn much smaller than the length of
the U shaped wire X.
In the standard form of Lenz's law we use Bio-Savart law to get
the direction of the magnetic field. We find that the magnetic field
is unchanged, but the area of the coil grows. The flux in the coil therefore
grows when the rod moves towards the right. The correct choice for the
direction of the current is then be that which will result in the reduction
of the magnetic flux. The appropriate current must therefore flow in the
counter clock-wise direction. Equations (12) and (14) can clearly be written
as
, (15)
where A is the area enclosed by the sliding rod and the U shaped conductor.
In the following we will derive equation (15) from Gauss’ law and the Lorenz transformations for the example of figure (2), without the use of magnetic fields. The direction of current in the wire is the result of the different Lorenz contractions as observed by charges in the rod. Both the negative and positive charges in the conducting rod move towards the right. In the rod's reference frame the negative charges in the wire seem to have a larger relative velocity than the positive charges, and the distance between negative charges in the wire is therefore Lorenz contracted to a greater extent than the distance between positive charges. In the rod's reference frame an overall negative charge per unit length is thus observed. An overall electric field, derived from Gauss’ law, is therefore observed in the rod's reference frame. As a result the positive charges in the rod are attracted to the wire, the negatives are repelled, and the induced current in the coil is flowing in the clockwise direction. The magnetic flux produced by such current will indeed have the opposite direction to that of the magnetic field of the wire. A simple exercise to check that this method works will be the reversing of the direction of motion of the rod.
In writing Farady's law we are interested in the emf, or the potential.
Using equation (10) the electric field at a distance r from the wire
in the moving rod frame is Integrating over the length of the moving
rod yields the potential
. (16)
For simplicity of calculation we assume , and the above integral
becomes
. (17)
To evaluate equation (13) we transform the electromotive force
back to the lab frame, and then integrate the electric field over
the U shaped stationary rod7. This integration does not add to the electro-motive
force, because there are no electric fields or moving charges along the
U shaped conductor in the lab frame11.
We therefore obtain
, (18)
where we have used .
We now notice that the change in area swept by the sliding rod
is , where is the rate of change of the area of the secondary
coil. Since the magnetic field remains the same in the loop, and
the magnitude of the magnetic field is near the straight wire we
get:
, (18)
as in equation (15), and we have used Maxwell’s result: .
One of the questions commonly asked by students regarding Lenz's
law is:
where would you say the source of emf IS in the secondary circuit?
Our method demonstrates that the source of emf in the above example is
in THE SLIDING ROD! We can prove that the electromotive force would have
been produced across the rod even if the U shaped wire was non-conducting
by using the Lorenz force law
in equation (1).
Lenz’s law for simplified configuration II
It is now obvious why the current in the secondary circuit is
produced in the configuration of example 1, but that example may not be
persuade the students that equate Lenz’s law with transformers. The next
example is therefore important due to its resemblance to transformers.
In that example, which is outlined in figure 3, the current in the straight
wire increases, whereas the secondary square coil
remains at rest, as is the case with real transformers.
Using Bio-Savart law we know that the magnetic flux in the coil
increases because the current is on the rise. The current in the coil must
therefore flow in the counter clock-wise direction, according to Lenz's
law.
Using special relativity we show that Lenz’s law is, in this
case, the result of the difference between the electric field near the
two sides of the coil. In our example there are only positive charges in
the “wire” (which could be, for example, a vacuum tube). The increase in
the current is the result of an increase in number of positive charges
pumped into the sphere on the right. The number of charges pumped into
the wire is the same, but the average speed is increasing with increasing
electric field.
The charges in the wire therefore move at a greater speed near
the left hand side of the coil than they do near the right hand of the
coil. This is due to the fact that the changes in the electric field which
drive the current move at the speed of light in the wire. For simplicity
of calculation we assume that the electric susceptibility in the wire and
the coil are , so that changes in the electric field move at the
speed of light. The change in speed of the current are therefore also transmitted
at the speed of light down the wire.
The increase of positive electric field that drives the current
therefore arrives at the left hand side a time before it arrives
near the right hand side of the coil. The induced electric field
in the coil near the right hand side is then traveling at speed c to the
left. Therefore, when the speed near the left hand side of the coil
grows to the electric field arriving near that side from the
right of the coil is due to charges that moved at speed near the
right hand side at time .
We try to find the induced fields from the electric fields . The
charges on the left side of the coil observe a high line density
(19)
where .
The electric field on the left is therefore
(20)
The electric field on the right at time is
, (21)
and evaluating the integral (13) with the same care as before7 we get
, (22)
where we have assumed that . Equation (22) is the equivalent
of (18), if we takes and remember that the current in the case of single
charge cariers is instead of in equation
(18).
The above result can interpreted as follows: If one transforms
to a reference frame that moves at speed to the right then the left
side of the loop looks like a rod moving at to the right. If one
transforms to a reference frame that moves at speed to
the left, then the right side of the loop looks like a rod moving to the
right. Integrating over three reference frame then yields the same results
as in (18) and (22).
The negative sign in Farady's law both examples is therefore the result of the properties of the Lorenz contraction AND the principle of special relativity: Motion in one reference frame is equivalent to motion in the other reference frame WITH A NEGATIVE SIGN.
For the second example the classical equations are not as straight
forward to derive. We use equations (12) and (14) to get the classical
form of Farady’s law. The current, however, is not changing at the same
time along the whole wire. We therefore have to establish I, then use Biot-Savart
law to calculate B(x,t), and then calculate and take its derivative.
The result agrees with equation (22).
We chose to show here a simplified version: using an averaged
current at and at . Using
Farady’s law we therefore get for the induced electromotive force
, (23)
where we have used the area of a rectangular loop , the
magnetic field of a wire , the current difference and
. The factor difference from equation (22) is the result of our assumption
of constant current along the wire.
Appendix
We should not be surprised about the above results. This is a
simplistic way of proving the fact that the general form of the Lorenz
force law,
(24)
remains the same in all reference frames.
Equation (24) can also be written as
, (25)
where we have used the definitions of the current , current density
and charge density . In the lab reference frame equation (25) contains
no charges, the first term disappears and we recover equation (11).
In the rod’s reference frame the non-vanishing magnetic field component
transforms as and cancels out, and equation (25) becomes equation
(10). We have therefore shown that the fields, charge density, current
density transform as 4-vectors, leaving the form of the Lorenz
force law invariant. This result holds for Lenz’s law as well.
References
Electronic mail: sivronr@gvsu.edu
1Sivron, R., in preparation.
2Einstein, A., 1905, translated in: The principle of relativity, 1952,
Dover publications.
3Purcell, 1963 Electromagnetism (Berkley series).
4France
5Feynman, R., Physics lecture series
6Feynman, R., Q.E.D.
6Cutnell, J.D. and Johnson, K.W., Physics, (John Wiley & Sons,
New York, 1995), p 717.
7Jones, E.R. and Childers, R.L. Contemporary College Physics, (Addison-Wesley,
New York, 1992), p 560.
8See page 211, Jackson, J.D., Classical Electromagnetics, 2nd edition,
1975, John Wiley & Sons.
Serway, R.A., Physics for Scientists and Engineers, 4th edition, 1996, Saunders College Publishing, pg 915.
Wangness, R.K., Electromagnetic Fields, 2nd edition, 1986, John Wiley
\&
Sons.
Figures
Fig. 1
A helpful example: In q's reference frame the positively charged particles
are Lorenz contracted to a greater extent than the negatively charged particles.
The charge q will thus experience an overall positive electric field &
be repelled.
Fig. 2
In the reference frame of the sliding conducting rod that slides upwards
positive charges are repelled from the straight wire and negative charges
are attracted to the wire. As a result an electromotive force pushes current
through the loop in the clock-wise direction. The direction can also
be obtained with Lenz's law.
Fig. 3
The current in the wire is increasing by an increase in the speed of
the charges. This increase depends on the distance from the power source
(e.g. a capacitor), and is therefore due to a retarded potential. Another
way of looking at it is described in Fig 4.
Fig 4.
In the above figure the charges are acclerated. As a result a kink
forms in the electric field. The radial fields from the charges cancel
because of the equal number of positives and negatives. The "kink" does
not cancel.
All the equations (for a better quality equations
& graph please e-mail me
at SivronR@gvsu.edu
Equations:
A direct proof for the direction
in the Coriolis effect. The direction of the Coriolis force in the Coriolis
effect is determined from the motion of a test particle near the equator
northward, and a test particle near the tropic of cancer southwards. Details
soon.This is our own work.
The direction of precession is
determined for a tilted dumbell around two possible axes. Details soon.