A method to calculate inductance in systems of parallel wires

This paper gives a method that maps the static magnetic field due to a system of parallel current-carrying wires to a complex function. Using this function simplifies the calculation of the magnetic field energy density and inductance per length in the wires, and we reproduce well-known results for this case.


Introduction
This paper points out a convenient way to calculate the magnetic field due to parallel, current-carrying wires. Defining a coordinate system such that the wires run along the z-direction, the magnetic field due to a current-carrying wire will be in the x-y-plane. We construct a complex function, ( ) f x iy + , from the components of the magnetic field: x y f B iB = − . The magnetic field due to a constant set of currents is determined from Ampere's law, which we will show is equivalent to the statement that These ideas are based on constant currents. In this magnetostatic case, inductance can be simply calculated. Once inductance is calculated, then it can be used to determine circuit behavior in the case when there are slowly varying currents. To be precise, in this paper, we will define inductance as relating currents to total energy stored in a magnetic field according to: jk L is the inductance matrix (including self-inductance and mutual inductance), U is the total energy stored in the magnetic field, and k I is the current carried in wire-k. We find the self-inductance per length of a wire is Λ and a are long and short distance cut-offs respectively. The mutual inductance per length of two wires is where nj r is the distance from wire n to wire j. These are standard results [1] [2].
Measurements and calculations of the self-inductance of actual thin wires of length l and radius ρ are consistent with (4) with 2l e Λ = and a ρ = . The measured mutual inductance of two thin wires separated by a distance d is also consis- We would like to mention that there are similar techniques using stream functions in fluids to understand vortices [4], and also some magnetic field problems in the absence of currents [5].

Relation between Ampere's Law and a Contour Integral
Ampere's law in SI units in integral form is (6) where enc I is the current puncturing the area enclosed by path c, with the sign of the current given by the right hand rule.
If the path c is in the x-y plane, Ampere's law formally looks like with positive currents I moving in the positive z-direction. x This can be written in terms of a parameter t that parameterizes the contour as The integrand is formally Clearly the real and imaginary parts of contour integral can then be written in the same way, in particular where k c is now interpreted as an infinitesimal circle around a current-carrying wire (the current being the place where the magnetic field becomes non-analytic) Each of the integrals in the sum in (10) can be parameterized and written in the form of (9). We can imagine each of these integrals is then proportional to the time integral of the z-component of the force on a test charge moving around an infinitesimal circle surrounding the current-carrying wire. For a single wire, the magnetic field is either parallel or antiparallel to the circle surrounding the wire, and hence the Lorentz force is zero. In the case of multiple wires, however, this is not the case. Consider two wires, which we can denote as wire-1 and wire-2. Consider a small circle around wire-1, 1 c . Let 1 B be the magnetic field due to the current in wire-1 and 2 B be the magnetic field due to the current in wire-2. The magnetic field along the path 1 c can be written as where t parameterizes the circular path 1 c , with t 2 and t 1 being the value of the parameter t at the beginning and end of one circle around the wire, and This integral can then be written as The first integral on the right hand side of the above equation is zero, by the argument presented for the single wire. Let us consider the second integral on the right hand side of the above equation. If 1 c has a radius which is very small compared to the distance from wire-1 to wire-2, then 2 B is approximately uniform over the circle 1 c . Hence the integral is approximately since the integral of dr is zero over a closed path. A similar argument can be made in the case of many wires. Therefore in the case of a magnetic field due to multiple wires d 0, and this means that in (10), This means that the imaginary part of the complex integral (8) should vanish.
We should note that there is an important condition for (12) to hold. We needed to have the radius of the circles around each of the wires be much smaller than the distance between the wires, so the magnetic field due to one wire could be considered uniform at a different wire. If a is the radius of a wire and d is the shortest distance between wires, then the condition for our theory to hold is that We are left with as our condition on f to be x y B iB − . This proves (1).
The residue theorem for contour integrals is [6] ( )d 2π , For (14) to hold, we must have and then take the real and imaginary parts of this to find the x and y components of the magnetic field.

Energy Density and Inductance
Energy density in a magnetic field is The total energy stored in the magnetic field is then where the integral is over all space. The total energy stored in a magnetic field that is created by a system of currents j I is related to the inductances according to where j I is the current in wire j. jk L is the mutual inductance of wires j and k and is symmetric in its indices. kk L is the self inductance of wire k.
Suppose we add a current n I to the system of currents in the z-direction. Let Plugging (25) into (24), then dividing by n I , we find that Plugging in the definition of f from (20), we find that d . 8π where l is the length of the wires.
The self-inductance, Equation (28), can be directly integrated. We note that Here we introduced long range and short range cutoffs for the integration, Λ and a respectively, and the self-inductance is only written to logarithmic accuracy, as per usual [1]. 2l e Λ = and a ρ = , where ρ is the radius of the wire, gives good agreement with experiment in the case of long, thin wires [3].
The integral for the mutual inductance can also be done, but is a little more involved. Here again, we find it helpful to set the origin of the x-y coordinate system to the position of wire n, and then convert to polar coordinates. Let

Conclusion
We note that these results for inductance are well known, but illustrate our method. In a future paper, we hope to apply this formalism to the calculation of inductance in different systems. We would also like to mention that the similarity between our formalism and the velocity stream function in fluid flow, with currents being replaced by vorticity [4] leads naturally to a nice qualitative picture of the magnetic field around arrays of wires, or around arrays of currents.
In the future, we would to examine the interplay between the energy density of currents in a solid and the force between parallel current carrying wires. It is our belief that the magnetic field may break into an array of vortices similar in structure to the currents around magnetic flux lines in type II superconductors, depending on the solid [7] [8].