In this paragraph, one point “p” around an electrical wire with a current vector (I) can be placed anywhere around it at distance “r” from the wire.

In Figure 1 , point “p” is associated to a magnetic field value I/r. Also at that place, three vectors M, A and C, are described.

Specificities about points and vectors are:

In Figure 1 , point “p” is at distance r of the wire which has an electrical current “I”. This imposes to have a positive field value of I/r with electronic moving charges. All points and all vectors placed in the space around the wire have the same unit (A/m). This field value is not vector dependent.

In Figure 1 , vector C starting in “p”, is parallel to the wire with the same direction of electrical charges moving in the wire. Its magnitude is equal to I/r.

For the magnetic vector M (red) in Figure 1 , its direction is defined by the classical right-hand rule around the wire [12] . This vector giving the direction of the magnetic field curves will be used to build magnetic curves.

The electrical current induces in point p, an attraction vector A directed toward the wire. Its magnitude is equal to I/r. Letter A stands for: Attraction.

Moreover, Figure 1 is a very general case of moving electrons around a straight wire. In the field value can be calculated and the three vectors giving directions and norms of A, C, and M, are defined. Calculations of these vectors can now be made, and these results are given in Table 1 .

In Table 1 , magnetic field value at point “p” is equal to the identical norms of the three vectors. The present model used in this article, is using cross product with vectors A, R, C, I and Au, Cu and Mu.

Note that formulas given in Table 1 , are different from vector formulas given by Biot et Savart [3] . The main difference is that the three vectors used here are in the same physical space in (A/m).

The concept of “equipotential surface” is frequently used in electrostatic field [13] . For example, a small electrical charge imposes, at a precise voltage, a spherical equipotential surface around itself. As proposed for the first time in our previous article [8] , electrostatic and electromagnetism can be approached in the same way. For example, in magnetism with one conducting wire, points with the same I/r value build an equipotential surface in a cylinder around the wire.

On such a surface, vector C and vector M (see Figure 1 ) belong to a plane tangent to the equipotential surface and parallel to the wire.

As vector A is orthogonal to vectors C and M, it is directly perpendicular to the equipotential surface built around the wire. Then, this vector gives the direction needed to obtain points having any expected magnetic field value. These vectors will be largely used to determine equipotential surfaces in this article and even in the application chapter devoted to 3D drawings.

This chapter shows how points with the same field values around a current wire are marked to form a part of an equipotential surface.

The drawing of five points with the same field value is shown in Figure 2 . The building procedure is detailed below the drawing.

The marking procedure starts at point W where the field value is infinity, then using one line in a chosen direction, the field value decreases when going away from the wire. Then, a marker is plotted where the expected field value I/r is reached. This procedure can be used for the five lines starting at point W. All points S form a circle which is named in this article as an equipotential circle.

As the marking of equipotential circle is possible, an equipotential surface is obtained by using as many circles needed all along the wire.

Note that many other line configurations can be used to draw equipotential surfaces. When vector A is used, it is easier to calculate potential values, but it is not the simplest method.

Field curves will be drawn in equipotential surfaces only by using the direction of vector M, as shown below. Therefore, magnetic field curves having the same field values are always on one equipotential surface.

Figure 3 shows how one field curve is drawn around a single wire in an x-y plane perpendicular to the wire. The drawing procedure is described below the drawing.

These field curves are first, guided by vectors M and second, re-evaluated, in the direction of vector A, the scalar field value.

The drawing procedure of field curves starts a point d0 out of the wire. From d0, and guided by vector A, the field value is calculated and mark p1 is drawn when the expected scalar value I/r of the expected equipotential surface is reached. At p1 and guide by vector M1, point d1 can be marked on this vector M1 direction. As vector M1 is tangent to the field curve, d1 does not have the expected scalar field value. To obtain point p2 with the expected field value, the procedure started previously at point d0 must be repeated. By making five times this procedure, points p1 to p5 are calculated and become visible in Figure 3 . They form a circle centered at point W. These points p have the same field value and they form the same circle as in Figure 2 . It is called “magnetic field curve”. Remind you that magnetic vector M is used as a guide for the magnetic field curve.

This procedure in 2D representations will be extended to 3D representations in the application chapter.

This chapter describes the modifications imposed when adding a second electrical wire.

This is shown in Figure 4 .

In Figure 4 , vectors have different colors. Brown vectors determine electrical current in yellow wires. The three other colors are for vectors in the same physical space with field unit in (A/m). Green colored vectors are parallel to the wires, blue vectors have attractive directions to the wires, and red vectors are the cross product of other green and blue vectors.

It is possible to add vectors having the same physical units. With a larger number of wires, additions are given by the following formulas. Formula 1 is an addition of green vectors C. Formula 2 is addition of attractive blue vectors. Formula 3 is an addition of magnetic red vectors.

$\stackrel{\to }{\text{C}}={\displaystyle \underset{\text{i}}{\sum }\left(\frac{{\text{I}}_{\text{i}}}{{\text{r}}_{\text{i}}}\ast \stackrel{\to }{{\text{Cu}}_{\text{i}}}\right)}$ (1)

$\stackrel{\to }{\text{A}}={\displaystyle \underset{\text{i}}{\sum }\left(\frac{{\text{I}}_{\text{i}}}{{\text{r}}_{\text{i}}}\ast \stackrel{\to }{{\text{Au}}_{\text{i}}}\right)}$ (2)

$\stackrel{\to }{\text{M}}={\displaystyle \underset{\text{i}}{\sum }\left(\frac{{\text{I}}_{\text{i}}}{{\text{r}}_{\text{i}}}\ast \stackrel{\to }{{\text{Mu}}_{\text{i}}}\right)}$ (3)

This particularity is applicable to the scalar field value which is the addition of every field values of each wire. This addition of scalar field values allows to calculate equipotential surface. This addition capability will be now used for 3D drawings in the following application chapter.

In Figure 5 , equipotential surface value is chosen to be one (A/m) with a current of one ampere in the wire. These results in a cylindrical equipotential surface is located at one meter away and centered around the wire. On this cylinder, magnetic field curves are drawn using the previous procedure which starts in the vector M direction. Then a correction of the scalar field value is calculated in the vector A direction.

Note: In Figure 5 , black field curves drawn on the equipotential cylinder are fully compatible with results obtain with classical field lines when only one wire is present [14] .

Every point on the colored equipotential surface in Figure 6 , has the same field value of one (A/m). Consequently, none of these points depend on any vector. For black field curves, only vector directions are used.

In Figure 6 , black points are calculated with one starting point which is followed by more than 1000 points. The calculation stops when the point becomes very near the starting point.

Any field curves in Figure 6 , never touch another curve. This observation agrees with the proposition of Gauss which says that magnetic fields is a vector field with a zero divergence i.e. it has neither source nor end [15] .

In Figure 7 , field curves seem to be nearly like in Figure 6 , but it will be shown in Figure 8 , that some curves have a very particular drawing.

As previously for two wires, each starting points have a return point equal to the starting point after it is drawn on the equipotential surface.

Details about crossings of black field curves are in the discussion chapter.

In our previous publication [8] , definition of borders between domains has been given. To summarize, a border is a magnetic field curve which cross itself separating two domains. In Figure 5 , as curves are circular, they never cross themselves. In this case, the physical space around the wire is made of one domain.

When three wires cross as in Figure 8 , the field curve looks like a three-leaf clover. This is a multi-domain case: Its center seen to be the center of the drawing. In that case, each leaf depicts the clover has one domain, to which one can add the external, and main. By symmetry and on the hidden side of the equipotential surface, there are also three internal domains. The total number of domains is thus seven.

In the case of two wire crossing, as in Figure 6 , the borders have not been drawn but one could reasonably suppose that five domains exist.

In the present article, studied field lines are about ten meter long. Mathematically, the iterative process implies that a calculation error adds up at each step of the process. The precision of the software thus plays a crilical role in the proximity of the last point with the starting point.

For the curves drawn in the present article, the distance between successive points is chosen to be 5 * 10^(−3) m. This gives about several thousand points on each field curves. The distance between the starting point and end point is around this value. At our drawing resolution, end points are blended with starting points and positions are not visible on the figures.

For the clover of Figure 8 , the crossing position of leaves is mathematically known to be the square root of 9/2. To reach this point, precision of the starting point must be 10^(−8) m. With it, the lower distance of point of the curve to the pass connection point is 14 * 10^(−3) m. This is small enough so they cannot be seen on curves drawn in the equipotential surfaces.