Another look at the current-carrying wire in special relativity

In this paper we examine two models for a current-carrying wire in special relativity. The first model assumes an infinitely long wire and does not specify the source or return path for the current. This model is simple and easy to analyze, so it is often used in introductory textbooks to show the significance of special relativity for electromagnetism. The second model is a coaxial line with a battery at one end and a short circuit at the other. The center conductor of the line represents the wire of interest, and the outer conductor forms the return path for the current. The analysis for this model is more involved than that of the first model, but it provides information not available from that model. This new information answers questions a student might have when first encountering the simpler model.


Introduction
Introductory textbooks on special relativity often show the significance of relativity for electromagnetism by considering a simple model for a current-carrying wire observed in different inertial reference frames.This model dates from the early days of special relativity, and it was described by von Laue in his book Das Relativitätsprinzip from 1911 [1].Since then, it has appeared in various forms in many other textbooks on special relativity [2][3][4][5][6].Now it is probably best known because of Purcell's discussion in his famous book 'Electricity and Magnetism' from the Berkeley Physics Course [7].This model is the motivation for the analysis presented in this paper, so we will begin with a very brief review of the model.
As shown in figure 1(a), the straight wire is modeled by two infinitely-long arrays of equally-spaced charges.The array of positive charges, = + q q, represents the lattice of positive ion in the metallic conductor, while the array of negative charges, = --q q, represents the electrons.In the inertial reference frame K, the spacing between the charges is the same for both arrays, = = + -∆ ∆ ∆ z z z, and the positive charges (ions) are at rest = +  u 0, while the negative charges (electrons) are moving to the left with the velocity = --- u u z.Let the cross-sectional area of the wire be A, then the volume densities of charge and current for the wire are Now we will consider the wire as observed in the inertial reference frame ¢ K , which moves with constant velocity =  v vz with respect to the inertial reference frame K, see figure 1(a).At time = ¢ = t t 0 the origins, O and ¢ O , of the rectangular coordinates systems in these two Schematic drawings showing a small section of the infinitely long wire.The array of positive charges, = + q q, represents the lattice of positive ion in the metallic conductor, while the array of negative charges, = - -q q, represents the electrons.The wire as observed (a) in the inertial reference frame K, where the wire is at rest, and (b) in the inertial reference frame ¢ K , which moves with the velocity =  v vz with respect to frame K.For clarity, in these figures the positive and negative charges are shown on opposite sides of the wire; in an actual wire they are intermingled and spaced at microscopic distance.frames coincide.The values of the charges are the same in both frames, ¢ = q q (charge invariance), as are the cross-sectional areas, ¢ = A A (coordinates transverse to  v ).After using the Lorentz transformations, we find the spacings and the velocities for the charges in the two arrays in frame ¢ K are Notice from (3) and figure 1(b), the spacings between the charges in both arrays are reduced from their values in frame K (Lorentz-FitzGerald contraction), and the spacing for the positive charges is greater than for the negative charges.Using these results, we can now determine the volume densities of charge and current in frame ¢ K : When we compare these results, (6) and (7), for frame ¢ K with those for frame K, (1) and (2), we see an important difference: in frame K the wire is uncharged and carries a current; whereas, in frame ¢ K the wire is charged (negative) and also carries a current.So, simply changing the inertial frame of reference for the observation has introduced a charge in the wire.Stated differently, in frame K there is a magnetic field produced by the current in the wire; whereas, in frame ¢ K there are both an electric field and a magnetic field produced by the charge and current in the wire.This observation, or something equivalent to it, is discussed in introductory textbooks on special relativity [1][2][3][4][5][6][7].
Recently, there has been some debate in this journal concerning the validity of this microscopic model for a wire [8][9][10].The purpose of this paper is not to directly enter into that debate.Rather, it is to provide an analysis that can alleviate some of the misgivings a student might have on first encountering this model.Specifically, we will address the following questions: i.Any finite length of the wire is uncharged in frame K; whereas, it carries a net negative charge in frame ¢ K .What happened to the missing positive charge?ii.A straight, current-carrying wire is usually part of a closed circuit.Current does not just start at one end of a wire and end at the other.In this model, where is the structure that returns the current?iii.What and where is the source causing the current in the wire?
Of course, the answers to all of these questions could be said to depend on what is at infinity, since the wire is assumed to be infinitely long.But this is a not a very satisfactory answer, particularly for a student.
In the remainder of the paper, we will analyze the finite structure shown in figure 2, and use the results of the analysis to answer the questions outlined above.This is a coaxial transmission line of finite length L in the rest frame K, fed at one end by a battery and terminated at the other end by a conducting disk that functions as a short circuit.The inner conductor of the transmission line is equivalent to the wire in the simple model discussed earlier.

Inertial frame K
A theoretical model was developed for the coaxial line at rest in frame K, and figure 3 shows a longitudinal cross section of that model along with the cylindrical coordinates j r z , , used for the analysis.The length of the line is L, the radius of the center conductor is a, and the innerradius of the outer conductor is b.The constitutive parameters of the center conductor, which is the wire of interest, are s e m , , , 0 0 where s is the electrical conductivity, and the region between the conductors is free space (e m , 0 0 ).All of the other conductors are assumed to be perfect electric conductors, in the sense that the electric and magnetic fields in their interior are assumed to be zero, and they can support electric charge and current on their surfaces.Longitudinal cross section of the theoretical model used for the analysis of the coaxial transmission line.All of the conductors, except the center conductor, are assumed to be perfect electric conductors, and the magnetic surface current placed in front of the conductor on the left produces the impressed electric field of the battery.The electromagnetic field in the line is described in terms of the cylindrical coordinates j r z , , .
The cylindrical battery is modeled by a magnetic surface current with the density placed in front of the perfect conductor on the left.This produces the impressed electric field The voltage of the battery V 0 and the total conduction current I through the center conductor are related by the resistance R of the center conductor: The model described above represents a well-defined electromagnetic boundary-value problem.Maxwell's equations are to be solved in the closed cylindrical cavity containing free space and the wire of interest (center conductor).The tangential component of the electric field is specified on the interior surfaces of this cavity.On the surfaces of the outer conductor and right end, which are perfect electrical conductors, it is zero.On the surface of the left end, it is zero, except on the annulus   a r b, where it jumps to the impressed electric field (9) due the presence of the adjacent magnetic surface current (8).These boundary conditions are sufficient for obtaining a unique solution to Maxwell's equation in the cavity.
For those unfamiliar with this use of magnetic surface current, we offer the following additional comments.The magnetic surface current is a fictitious quantity introduced to enforce a boundary condition in a theoretical electromagnetic model.When this current is placed in front of a perfect electric conductor, the electric field jumps from zero inside the conductor to a prescribed value at its surface.If desired, the magnetic current can be thought of as due to moving magnetic monopoles, another fictitious quantity, but this is not necessary.For more information about this technique, we recommend the textbooks [11][12][13].
In our model, the combination of the magnetic surface current and the perfect electric conductor acts as an ideal voltage source does in circuit theory.It produces the voltage V 0 across the left end of the coaxial line, no matter what the current is through the source (through the perfect electric conductor).
Models like the one shown in figure 3 have frequently been used to study the transfer of electromagnetic energy between a source (battery) and a resistive conductor (wire) [14][15][16][17].Because detailed analyzes for these models are readily available, we will only summarize the results for our case and use these results to discuss points relevant to our problem.In Appendix A, we give expressions for the steady-state electric and magnetic fields,  E and  B, volume densities of charge and current, r V and  J , V and surface densities of charge and current, r S and  J .

S
We will confine our attention to the charge and current on the conductors of this model, which are shown in the schematic drawings of figure 4. The center conductor, figure 4(a), has a uniform volume density of axial current throughout and a surface density of charge that varies linearly with axial position ( l Surface charge like this has been shown many times to be necessary on wires of a circuit for proper behavior of the current [18][19][20][21]. In figure 4(b) we indicate the total charge on each conductor as well as the total current passing through a cross section of each conductor.For example, for the center conductor, the total charge is obtained by integrating (12) over the surface: S r a z L S r a 0 0 2 l and the total conduction current is simply I.
When we consider the charge on all surfaces, as indicated in figure 4(b), we see that the net charge on the transmission line is zero -the charges on the center and outer conductors cancel as do the charges on the right and left ends.The current I is the same in all conductors and circulates counter-clockwise in the structure.For this case, the current in the center conductor (wire) clearly returns to the source (battery) though the outer conductor and the two ends.

Inertial frame ¢
K moves with constant velocity =  v vz with respect to inertial frame K, see figure 3, and at time = ¢ = t t 0 the origins, O and ¢ O , of the rectangular coordinates systems in these two frames coincide.The results for frame ¢ K are obtained from those for frame K (appendix A) by applying the Lorentz transformations (standard configuration) for the kinematical variables and the electromagnetic field [22,23].This is a straight-forward but tedious procedure that we will not reproduce here; instead, we will simply state the results in

appendix B. Because frame ¢
K is moving, the field in this frame is continually changing with time, and we have to pick a single time for our comparison, which is ¢ = t 0. This step was not necessary for frame K, because the field in that frame is steady state; hence, any time will do.
As we did for frame K, we will confine our attention to the charge and current on the conductors of the model, which are shown for frame ¢ K in the schematic drawings of figure 5.The center conductor, figure 5(a), has the same two terms we saw for frame K: a uniform volume density of axial current and a surface density of charge r g e s p gr Notice that both are greater by the factor g than the same terms in frame K.There are also two additional terms that were not present in frame K: a volume density of charge (negative) In problems like this, it is sometimes instructive to divide the total current in frame ¢ K into two parts [24][25][26]: a convection current (CV) that involves the motion of the volume density of charge CV (Here we have - v , because this is the velocity of the coaxial line with respect to frame ¢ K .), and a conduction current (CD) associated with the electrical properties of the wire As an example of this division, consider the volume density of current in the center conductor: In figure 5(b) we indicate the total charge on each conductor.Notice that, as in frame K, the net charge on the transmission line is zero -the charges on and within the center conductor cancel those on the outer conductor, and the charges on the right and left ends cancel.In this figure, we also indicate the total conduction current ¢ I CD passing through a cross section of each conductor.The conduction current is the same in all conductors and circulates counterclockwise in the structure.As in frame K, the conduction current in the center conductor (wire) clearly returns to the source (battery) though the outer conductor and the two ends.

Summary and comparison of the two models for the wire
In the Introduction we discussed a popular, simple model (figure 1) for a current-carrying wire used for studying the relativistic behavior of the electromagnetic field, and the analysis of this model raised three questions.Next, we constructed a model (coaxial line) for a similar wire with a source and return conductors, and the electromagnetic analysis of this model provided the following answers to these three questions: i.The net negative charge on the center conductor (wire) when observed in moving frame ¢ K is compensated for by a net positive charge on the outer conductor.So, the overall structure is neutral (zero net charge) in both the K and ¢ K frames.ii.The elements of the coaxial line (center conductor, outer conductor, and the two ends) form a closed circuit.In frame K, the total current through any cross section of the circuit is the same (continuous) with the value I.The situation is the same for the moving frame ¢ K , only it is the conduction current ¢ I CD that is the same through any cross section.iii.The battery at the left end of the line is the source causing the current in the center conductor (wire).The battery provides the energy dissipated in the resistance of the center conductor.
When we compare these two models, we see similarities and differences.Both models predict the same volume densities of charge and current for the wire in the moving frame ¢ K as related to those in frame K: (6) and ( 16), and g ¢ =  J V  J V (7) and (14).However, the more detailed model also includes a surface density of charge (12) on the wire in frame K, which becomes both a surface density of charge (15) and a surface density of current (17) in the moving frame ¢ K .Such surface densities have been shown, again and again, to be required for the correct analysis of current-carrying wires in circuits [14][15][16][17][18][19][20][21].
Of course, the above observation raises the following question: Are there any situations in which these surface densities don't overshadow the desired relativistic effects?The volume density of charge for the center conductor is a relativistic effect, as it only occurs in frame ¢ K .Whereas, the surface density of charge occurs in both frames K and ¢ K .To compare these two quantities in frame ¢ K , we take the ratio of their values per unit length, indicated by L, for the center conductor: where V m e = / 0 0 0 is the wave impedance of free space.Because the surface charge is a function of axial position, we used its average value over length in (22), indicated by the overbar.For the surface charge to be negligible compared to the volume charge, this ratio must be large.When the velocity is relativistic, b is of order one, and the logarithmic function is expected to be also of order one.This means the resistance of the wire R must be much less than W 120 for (22) to be large.This is not an unreasonable requirement for a short piece of metallic wire.
We end by offering the following suggestion for how the material in this paper might be used in a class on special relativity or electromagnetism in which the simple model for the current-carrying wire is discussed.The main results from the study of the coaxial line, as presented in figures 4 and 5, could be included as a supplement to the traditional presentation.These graphical results, without the details for the full electromagnetic field (appendices A and B), then can be used to answer questions like the those posed in this paper.
Appendix A. Results for the analysis in frame K (steady state)

Electromagnetic field
Center conductor,   r a 0 : Between conductors,   a r b:

Charge and current
Center conductor,   r a 0 : Center conductor,   r a 0 :  Center conductor,   r a 0 : ) can be expressed in terms of the voltage V , 0 rather than the current I, by using equation (10).

Figure 1 .
Figure 1.Schematic drawings showing a small section of the infinitely long wire.The array of positive charges, =

Figure 2 .
Figure 2. A coaxial transmission line connected to a battery on the left and terminated by a short circuit on the right.The center conductor represents the current-carrying wire of interest.

Figure 3 .
Figure 3. Longitudinal cross section of the theoretical model used for the analysis of the coaxial transmission line.All of the conductors, except the center conductor, are assumed to be perfect electric conductors, and the magnetic surface current placed in front of the conductor on the left produces the impressed electric field of the battery.The electromagnetic field in the line is described in terms of the cylindrical coordinates j r z , , .

Figure 4 .
Figure 4. Charge and current in frame K: (a) cross section of the center conductor showing the surface density of charge varies linearly with position (indicated by the gray shaded area), and the volume density of current (arrows) is in the axial direction and uniform throughout the conductor.(b) The total charge (Q) on each conductor and the total current (I) passing through a cross section of each conductor.

Figure 5 .
Figure 5. Charge and current in frame ¢ K : (a) cross section of the center conductor showing the volume and surface densities of charge and current.Notice that both surface densities (charge and current) vary linearly with position (indicated by the gray shaded area), and the densities of current (arrows) are in the axial direction.(b.)The total charge ( ¢ Q ) on each conductor and the total conduction current ( ¢ I CD ) passing through a cross section of each conductor.