_{1}

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.

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,

where

where

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:

Λ and a are long and short distance cut-offs respectively. The mutual inductance per length of two wires is

where

We would like to mention that there are similar techniques using stream functions in fluids to understand vortices [

Ampere’s law in SI units in integral form is

where

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.

Consider a complex function

Therefore,

The imaginary part of the complex integral (8) is

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

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,

where t parameterizes the circular path_{2} and t_{1} being the value of the parameter t at the beginning and end of one circle around the wire, and

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

since the integral of

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

The residue theorem for contour integrals is [

where R is the sum of the residues of f enclosed in c. The formula for the residue of a function

For (14) to hold, we must have

A simple form of f which satisfies (14) and (17) for a wire carrying current I in the z-direction at

where

where

We note that in cylindrical coordinates,

The reader can easily see that (18) is not unique in yielding an integral whose residue obeys (17). In fact adding any analytic function to (18) will give an identical result and the condition

and then take the real and imaginary parts of this to find the x and y components of the magnetic field.

Energy density in a magnetic field is

where the integral is over all space. The total energy stored in a magnetic field that is created by a system of currents

where

Suppose we add a current

Let

and

Plugging (25) into (24), then dividing by

Plugging in the definition of f from (20), we find that

Comparing (27) with (23), we arrive at a formulae for both the self inductance and the mutual inductance. The self inductance is

and the mutual inductance is

where

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

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

We first perform the integral over θ. We do this via residues [

integral on the right hand side of (31), write

the integral becomes a contour integral over a unit circle in the complex-u plane, traversed in the clockwise direction, call this contour-c. In the second integral on

the right hand side, we write

integral over a unit circle in the complex-u plane, traversed in the counterclockwise direction. We’ll call this contour c. Performing these contour integrals, we find that the integral over θ in (31) is

In place of

where we introduced a long range cutoff Λ for the integral over

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 [

The author gratefully acknowledges Luis Pauyac and Emma Diextre for helpful conversations about this subject.

Deyo, E. (2017) A Method to Calculate Inductance in Systems of Parallel Wires. Journal of Electromagnetic Analysis and Applications, 9, 1-8. http://dx.doi.org/10.4236/jemaa.2017.91001