IOP PUBLISHING
EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 29 (2008) 879–883
doi:10.1088/0143-0807/29/5/002
Lorentz contraction and
current-carrying wires
Paul van Kampen
Physics Education Group, Centre for the Advancement of Science Teaching and Learning &
School of Physical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
E-mail: Paul.van.Kampen@dcu.ie
Received 28 April 2008, in final form 4 June 2008
Published 26 June 2008
Online at stacks.iop.org/EJP/29/879
Abstract
The force between two parallel current-carrying wires is investigated in the rest
frames of the ions and the electrons. A straightforward Lorentz transformation
shows that what appears as a purely magnetostatic force in the ion frame appears
as a combined magnetostatic and electrostatic force in the electron frame. The
derivation makes use of a reasonably well-known problem of a charged particle
moving parallel to a current-carrying wire, which is often used to illustrate
that what appears as a purely electrostatic force in one frame appears as a
purely magnetostatic force in another. This paper, aimed at undergraduate
electromagnetism and special relativity courses, serves to dispel the notion that
this makes either the electrostatic or the magnetostatic force redundant.
1. Introduction
The case of a charged particle near a current-carrying wire has become a popular vehicle for
showing the relativistic nature of electrostatics and magnetostatics [1–3]. Of these papers,
Gabuzda’s paper is the most general, as it deals with a charged particle moving parallel to
a current-carrying wire at any speed [1]. It contains an in-depth discussion of the literature
on the topic at an introductory university level. Piccioni’s paper takes the special case of
the charged particle moving at the same speed as the charged particles in the wire [2]. It
is primarily written for high-school teaching and contains some additional references. Both
Griffiths [3] and Purcell [4] consider the wire from the somewhat artificial viewpoint of a
reference frame moving at half the drift velocity, so that electrons and ions both move at vd /2.
The storyline unfolds in a similar fashion to the following. The current-carrying wire
is modelled in such a way as consisting of fixed positive ions and free electrons moving at
drift speed vd . As seen in the ion frame the wire is electrically neutral, which means that the
(linear) ion and electron densities must be equal. Since the electrons are Lorentz contracted,
they must have a smaller linear charge density in their rest frame than in the ion frame. In the
c 2008 IOP Publishing Ltd Printed in the UK
0143-0807/08/050879+05$30.00 879
880
P van Kampen
ion frame
+ + + + + + +
- - - - - - -
electron frame
vd
vd
++++++++
- - - - - -
vd
q
q
FB
FE=0
FB=0
FE
+ + + + + + +
- - - - - - -
vd
+ + + + + + +
- - - - - - -
vd
vd
++++++++
- - - - - -
vd
+++++++
- - - - -
l
l’
FB
FE=0
FB
FE
Figure 1. Point charge (top) and current-carrying wire (bottom) in the presence of a long current-
carrying wire in the reference frames of the ions (left) and electrons (right) in the wire. Length
contraction effects are shown explicitly; they are depicted accurately for the unrealistic case of
γ = 1.25. In each case, it is indicated explicitly whether electrostatic and magnetostatic forces are
zero or non-zero.
electron frame, the ions are Lorentz contracted, so their linear charge density must be greater
than that in the ion frame.
This sets the scene for the following argument. Imagine a charged particle q moving
parallel to the current-carrying wire at the electron drift velocity1 . In the ion frame, the
interaction between the wire and the charged particle is described in terms of magnetostatics.
In the electron frame, this interaction is described in terms of electrostatics—see figure 1.
Special relativity is then invoked to show that the two expressions can be transformed into one
another.
Unless warned explicitly, a student following this line of thought may well be left with the
impression that either electrostatics or magnetostatics is redundant. In this paper, I will use
the closely related example of the force between two parallel wires carrying equal currents in
the same direction to illustrate that both electrostatics and magnetostatics need to be retained.
2. Auxiliary problem: a charged particle near a current-carrying wire
The case of a charged particle in the field of a current-carrying wire serves as an auxiliary
problem to the one under consideration. In the solution presented here, which closely follows
that of [1], all variables are taken to be positive.
1
The more general case of a charged particle moving at any speed vd parallel to the wire adopted in [1] contains the
same physics, but is much more cumbersome mathematically and conceptually, since three reference frames must be
considered in that case.
Lorentz contraction and current-carrying wires
881
In the ion frame, both the linear ion charge density λ+ and the linear electron density λ−
are equal to λ0. The electron current is related to the linear charge density and the electron
drift velocity by
I = λ0 vd .
(1)
In the electron frame, we find for the linear ion charge density λ+
λ+ = γ λ0 ,
(2)
for the linear electron charge density
λ−
λ− = λ0 /γ
(3)
and for the net linear charge density λ
λ = λ+ − λ− = γ λ0
vd2
,
c2
(4)
where
γ =
1
1−
(5)
.
vd2
c2
In special relativity, the force exerted by this linear charge distribution on the charged
particle is not instantaneous, as it is in classical physics, but retarded. However, because the
charge distribution does not change with time, we may ignore the finite time it takes for the
electrostatic force due to each element of the linear charge distribution to affect the charged
particle, and Gauss’ law may be applied as in classical electrostatics. In the electron frame, the
charged particle is stationary, so it experiences an electrostatic force due to a long uniformly
charged line, pointing away from the wire, of magnitude
FE =
γ qλ0 vd2
qλ
=
,
2π ε0 r
2π ε0 r c2
(6)
where r is the distance from the charged particle to the wire2 .
In the ion frame, the wire is electrically neutral but the charge moves at a speed vd to
the right. In the ion frame, the distance to the wire is also r. Since the charged particle is
stationary in the electron frame, a force pointing towards the wire transforms as [5]
F = F /γ
(7)
Hence, using equation (7) and the identity c = (ε0 µ0 )
F =
−1/2
, equation (6) becomes
qµ0 λ0 vd2
(8)
2π r
(where the subscript has been dropped deliberately). Using equation (1), we find
qµ0 I vd
F =
,
(9)
2π r
and finally, using Ampere’s law, we arrive at the familiar expression for the Lorentz force:
FB = qvd B,
(10)
where the subscript B is introduced for obvious reasons.
2
Alternatively, equation (6) can be derived by integrating the electric force components perpendicular to the wire due
∞
∞
2
v2
= 4πqε −∞ (r 2rγ+γ 2λx02dx)3/2 · cd2 =
to small segments of linear charge density λ and length dx : FE = −∞ 4π1ε 2qrλ dx
2 3/2
0
vd2 ∞
γ 2 λ0 x
q
4π ε0 r(r 2 +γ 2 x 2 )1/2 c2 |−∞
=
2
γ qλ0 vd
2π ε0 r c2
.
(r +x )
0
882
P van Kampen
3. Two parallel current-carrying wires
In the case of the two parallel current-carrying wires, the charged particle q is replaced by a
long wire carrying an identical current in the same direction. In the ion frame, this second
wire has length l, its proper length. It can be viewed as a line of stationary ions plus a line of
electrons moving at speed vd in the magnetic field due to the first wire; there is no electrostatic
interaction—see figure 1.
In the electron frame, the length l of the second wire is given by
l = l/γ .
(11)
FB
in the electron frame due to the ions
Moreover, there is a magnetostatic interaction
in the second wire moving in the magnetic field caused by the positive ion current in the first
wire, and an electrostatic force FE between the two wires that each have a net linear charge
density λ .
To find FB , first solve the auxiliary problem of finding the magnetostatic force exerted by
the first wire on a single particle of charge q stationary in the ion frame—i.e., moving at vd in
the electron frame. The ion current I in the first wire is given by
I = λ+ vd = γ λ0 vd .
Using Ampere’s law, this current gives rise to a magnetic field
µ0 I γ µ0 λ0 vd
B =
=
,
2π r
2π r
and hence a Lorentz force on the moving particle given by3
(12)
(13)
γ qµ0 λ0 vd2
.
(14)
2π r
To find the magnetostatic force on a line of positive ions of length l moving at speed vd
in the cylindrically symmetric field of the first wire, make the substitution q = λ+ l = λ0 l:
FB = qvd B =
FB =
γ µ0 λ20 vd2 l
.
2π r
(15)
v2
To find the electrostatic force between the two wires, similarly substitute q = λ l = λ0 cd2 l
into equation (6):
FE =
γ λ20 l vd4
γ µ0 λ20 lvd2 vd2
=
.
4
2π ε0 r c
2π r
c2
(16)
Since FB and FE are in opposite directions, the net force F between the wires becomes4
µ0 λ20 vd2 l
vd2
γ µ0 λ20 vd2 l
1− 2 =
.
(17)
F = FB − FE =
2π r
c
γ 2π r
Substitution of equation (1) yields
F =
µ0 I 2 l
.
γ 2π r
(18)
3 By symmetry, this force on the moving charge must be equal to that of equation (6), which of course it is. However,
equating the electrostatic force of equation (6) to the magnetostatic force of equation (14) is probably confusing to
students, and hence the simple three-step derivation given here is preferred.
4 It follows that the magnetostatic force is always greater than the electrostatic force by a factor of v 2 /c2 , and the
d
terms only become equal in the limit vd →c. In the more realistic setting of a drift velocity of the order of 10−4 m s−1,
the magnetostatic force is 25 orders of magnitude greater than the electrostatic force. The calculation in the concluding
paragraphs of [2], in which the electrostatic force in the electron frame is equal to the magnetostatic force in the ion
frame, is in error, as can been seen by straightforward substitution of numbers in equations (12) and (15) of [2].
Lorentz contraction and current-carrying wires
883
To transform this force to the ion frame, we need to realize that the force is between the
wires, which move at speed vd within the electron frame. Hence the force transforms, not as
in equation (7), but as5
F = γ F .
(19)
Thus, we find that the force between the wires in the ion frame is given by
µ0 I 2 l
,
(20)
2π r
which is identical to the expression found in the laboratory frame for the attraction between
two wires carrying equal currents in the same direction.
F =
4. Conclusion
This derivation shows that the force between two parallel current-carrying wires is purely
magnetostatic in the rest frame of the wires, but is a combination of electrostatic and
magnetostatic forces in the electron frame. This result should serve to dispel any notion of
redundancy of electrostatics or magnetostatics. While the derivation is primarily aimed at the
introductory undergraduate level, it could also serve to limit the risk of seeing electromagnetic
fields as something that can only be done in the context of antisymmetric second-rank tensors
in higher level courses.
Acknowledgment
The author would like to thank Dr Andrew Crouse of the Physics Education Group at the
University of Washington, Seattle, and Dr Enda McGlynn of the School of Physical Sciences
at Dublin City University, for a fruitful discussion.
References
[1] Gabuzda D C 1987 Magnetic force due to a current-carrying wire: a paradox and its resolution
Am. J. Phys. 55 420–2
[2] Piccioni R G 2007 Special relativity and magnetism in an introductory course Phys. Teach. 45 152–6
[3] Griffiths D J 1999 Introduction to Electrodynamics (Englewood Cliffs, NJ: Prentice-Hall) pp 522–5
[4] Purcell E M 1985 Electricity and Magnetism (New York: McGraw-Hill)
[5] French A P 1984 Special Relativity (Wokingham, UK: Van Nostrand-Reinhold) pp 221–5
5 Note that the force transformation used in equations (7) and (19) differs because, even though the transformation
is between the electron frame and the ion frame in both cases, the form of the transformation of the transverse force
depends on the speed of the particle the force acts on. The derivation of this transformation, which can be found in
[5], is not necessary to grasp the physical principles under discussion, and is therefore not given here.
IOP PUBLISHING
EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 31 (2010) L25–L27
doi:10.1088/0143-0807/31/1/N04
LETTERS AND COMMENTS
Comment on ‘Lorentz contraction and
current-carrying wires’
Dragan V Redžić
Faculty of Physics, University of Belgrade, PO Box 368, 11001 Beograd, Serbia
E-mail: redzic@ff.bg.ac.rs
Received 6 July 2009, in final form 10 August 2009
Published 25 November 2009
Online at stacks.iop.org/EJP/31/L25
Abstract
Some errors are rectified in a recent paper by van Kampen (2008 Eur. J. Phys.
29 879–83) on the force between two parallel current-carrying wires.
In a recent interesting paper, van Kampen [1] discussed the force between two parallel currentcarrying wires, attempting to extend a nice analysis by French [2]. It is perhaps worthwhile to
point out some unhappy errors in [1] which could be a source of confusion.
As is usually done for teaching purposes, van Kampen modelled a current-carrying wire
as consisting of two superposed lines of charge: one moving (that of free electrons moving
at drift speed vd ) and the other, which has an equal but opposite charge density, at rest (that
of fixed positive ions). He considered the force on a charged particle q moving parallel to the
current-carrying wire at the electron drift velocity, first in the electron frame (where the force
is purely electric) and then in the ion frame (where the force is purely magnetic). The author
correctly derived the force in the electron frame given by his equation (6) using the Gauss law.
However, in footnote 2 of [1] he presented an alternative way of deriving the electric force
which is erroneous. As can be seen, using the same notation as in [1], the correct starting
equation for deriving the electric force FE reads as
∞
∞
qrλ+ dx 1 − vd2 c2
1 qr(−λ− ) dx 1
FE =
+
(1)
3/2
2
2 3/2
2
v2
−∞ 4π ε0 (r + x )
−∞ 4π ε0 (r 2 + x 2 )3/2 1 − d2 2 r 2
c r +x
The first term in equation (1) gives the contribution to the electric force due to the stationary
line of free electrons; it is obtained by integrating the electric force components perpendicular
to the wire produced by small segments of the stationary line of electrons with linear charge
density −λ− and length dx . The second term in (1) is due to the uniformly moving line of
positive ions; it is obtained by integrating the analogous force components produced by small
segments of the moving line of ions with charge density λ+ and length dx , making use of the
well-known formula for the electric field of a uniformly moving point charge (cf, e.g., [2, 3]
and also [4, 5]).1
1 Everything appears as if a uniformly moving point charge carries its electric and magnetic fields the way a snail
carries its house. However, ‘this is an extraordinary coincidence, since the ‘message’ came from the retarded position’
[6]. (Note that the fields of a uniformly moving particle at a field point at a given instant are unaffected by the particle’s
motion after it passes the corresponding retarded position; that motion need not be uniform.)
c 2010 IOP Publishing Ltd Printed in the UK
0143-0807/10/010025+03$30.00 L25
L26
Letters and Comments
Another point is that the last section of [1] dealing with the force between two parallel
current-carrying wires basically seems to be flawed. Namely, van Kampen considers a
segment of an infinite current-carrying wire of rest length l. He takes it for granted that
the electromagnetic force acting on the wire segment by another infinite current-carrying wire
parallel to the first one transforms from the electron frame to the ion frame according to a
well-known transformation law of a relativistic three-force [2],2 applying the transformation
law to the wire segment moving as a whole at the drift speed vd relative to the electron frame.
However, this is wrong. The well-known transformation law applies to a force acting on a
point particle, such as the Lorentz force (which is a pure relativistic force par excellence). As
can be seen, the same general form of the transformation law (for the transverse force) also
applies to the electromagnetic force acting on the ions of the wire segment considered as well
as to the force acting on the corresponding electrons. However, since the ions move and the
electrons are at rest relative to the electron frame, the general form of the force transformation
reduces to
F+ = γ F+
(2)
in the case of the transverse force on the ions, whereas in the case of the transverse force on
the corresponding electrons it reduces to
F− = F− /γ .
(3)
Consequently, there is no single transformation law for the force acting on the wire segment
as a whole and van Kampen’s argument is erroneous.
Thus, the forces on the ions and on the corresponding electrons must be treated separately.
It can be easily verified that the total force on the line of ions of the wire segment (whose
length is l+ = l/γ in the electron frame) vanishes,
γ μ0 λ20 vd2 l
γ λ0 vd2
−
λ0 l = 0,
2π r
2π ε0 rc2
in the electron frame. Equations (4) and (2) imply that F+ vanishes too:
F+ = FB+
− FE+
=
(4)
F+ = 0,
(5)
as it should. On the other hand, the force on the corresponding line of electrons of the wire
segment (whose length is l− = lγ in the electron frame) is given by
γ μ0 λ20 vd2 l
v 2 λ0
1
γ λ0 d2 γ l
=
,
2π ε0 r
c
γ
2π r
in the electron frame. From equations (6), (5) and (3), we find
=
F− = FE−
(6)
F−
μ0 λ20 vd2 l
μ0 I 2 l
=
=
,
(7)
γ
2π r
2π r
as it should be. Our analysis indicates that the treatment of the current-carrying wire segment
requires the transformation of forces acting on moving electrons from the electron frame to
the ion (wire) frame.
F = F+ + F− =
Acknowledgments
I thank Dr Paul van Kampen for cordial and helpful correspondence and the referee for
suggesting the addition of a few explanatory sentences. My work is supported by the Ministry
of Science and Environmental Protection of the Republic of Serbia under project 141021B.
2
More precisely, a pure relativistic force (cf, e.g., [7]).
Letters and Comments
References
[1]
[2]
[3]
[4]
[5]
van Kampen P 2008 Lorentz contraction and current-carrying wires Eur. J. Phys. 29 879–83
French A P 1968 Special Relativity (London: Nelson) pp 221–5, 251–4, 262–4
Purcell E M 1985 Electricity and Magnetism 2nd edn (New York: McGraw-Hill)
Jefimenko O D 1995 Retardation and relativity: the case of a moving line charge Am. J. Phys. 63 454–9
Rosser W G V 1996 Comment on ‘Retardation and relativity: the case of a moving line of charge’
by Oleg D Jefimenko (Am. J. Phys. 63(5) 454–9 (1995)) Am. J. Phys. 64 1202–3
[6] Griffiths D J 1999 Introduction to Electrodynamics 3rd edn (Upper Saddle River, NJ: Prentice-Hall) p 439
[7] Rindler W 1991 Introduction to Special Relativity 2nd edn (Oxford: Clarendon) pp 90–3
L27
IOP PUBLISHING
EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 31 (2010) L29–L30
doi:10.1088/0143-0807/31/1/N05
LETTERS AND COMMENTS
Reply to ‘Comment on “Lorentz contraction and
current-carrying wires”’
Paul van Kampen
Physics Education Group, Centre for the Advancement of Science Teaching and Learning &
School of Physical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
E-mail: paul.van.kampen@dcu.ie
Received 16 September 2009, in final form 19 October 2009
Published 25 November 2009
Online at stacks.iop.org/EJP/31/L29
Abstract
This reply answers the issues raised in the Comment on my paper (van Kampen
2008 Eur. J. Phys. 29 879–83). The error of applying a single Lorentz
transformation to a wire segment is discussed in some detail.
1. Introduction
The Comment by Redžić [1] on my paper on the Lorentz transformation of current-carrying
wires [2] raises an interesting point.
To remind the reader, my paper concerned the force exerted on a point charge by a currentcarrying wire. The paper was written for an introductory physics audience that may not even
be familiar with calculus. The lucid introduction by French [3] was therefore at too high a
level, and I had to make some simplifying assumptions. Thus, Redžić’s assertion that one
must look at the retarded field is incontestable in a more advanced course on special relativity,
but it would render the treatment inappropriate for its intended audience.
Therefore, I argue that, within the confines of the model used, my derivation is fine1 .
The model of an unchanging ion current in an infinitely long wire implies that the ions must
have been in uniform motion for an infinitely long time, and therefore the charge distribution
of the line of charges can be treated as stationary. Hence, we can assume that dρ/dt = 0,
and classical electrostatics can be used to describe the effect of the full line of charge. Any
effects of retardation drop out of the equations; indeed, when one performs the integral in
equation (1) of Redžić’s comment, the classical electrostatics result emerges. Footnote 2 of
my paper merely intended to give an alternative calculational method under the same classical
electrostatics assumption that the charge distribution is stationary; it is possible that the text
did not make this point sufficiently clear.
1
Likewise, Redžić’ final point, that the assumption of charge neutrality is problematic, is an interesting idea worthy
of further exploration, but it is outside the confines of the model used in [2].
c 2010 IOP Publishing Ltd Printed in the UK
0143-0807/10/010029+02$30.00 L29
L30
Letters and Comments
2. Transformation of a wire segment
My derivation given in section 3 of [2] is indeed incorrect, and Redžić’s correct derivation is
appropriate for the intended audience. There is a very interesting lesson here, even if I would
have preferred a less public forum to learn it. (It is scant consolation that, when it comes to
messing up transformations between reference frames, I am in good company [4].) The lesson
is this: wire segments cannot be reified. Electromagnetic forces do not act on wire segments;
they act on ions and electrons. The difference becomes apparent when these are in motion
with respect to each other. To my frustration, I pointed out aspects of this issue in another
paper [5].
References
[1]
[2]
[3]
[4]
Redžić D V 2009 Comment on ‘Lorentz contraction and current-carrying wires’ Eur. J. Phys. 31 L25
van Kampen P 2008 Lorentz contraction and current-carrying wires Eur. J. Phys. 29 879–83
French A P 1968 Special Relativity (London: Nelson) pp 221–5, 251–4, 262–4
Tiersten M S and Soodak H 1998 Propagation of a Feynman error on real and inertial forces in rotating systems
Am. J. Phys. 66 810–2
[5] McGlynn E and van Kampen P 2008 A note on linking electrical current, magnetic fields, charges and the pole
in a barn paradox in special relativity Eur. J. Phys. 29 N63–7