Traditional magnetic field lines assume a circular shape when generated by a single conducting wire [2] . These lines are determined using magnetic vectors equivalent to I/r. In Figure 2, our field value coincides with the magnetic vector because only one electrical wire is present. Each circular curve possesses a distinct “field value,” which diminishes as the distance from the wire increases. In certain cases, field values may eventually reach zero as the distance approaches infinity.

To highlight the resemblance between the field curves and topographic maps, one could draw a parallel between a field value of zero (A/m) and sea level on a topographic map. In Figure 2(c), a 3D representation illustrates this concept, where like in topographic maps, field curves cannot intersect with lines representing different levels [12] . The 3D rendering of curve values always maintains a planar arrangement, akin to the contour lines on a topographic map illustrating the relief contour [12] .

In this chapter, we employ two wires with identical electrical current intensities of one ampere (A) and the same direction. At any point on the planar surface, the field value is the result of combining two distinct I/r values. While field curves are not depicted with vectors, it’s worth noting that conventional magnetic field lines are not included; however, certain training courses may introduce such a classical calculation approach [13] .

In Figure 3, the field curves are depicted in various colors to represent distinct physical values. The corresponding colors and their respective values can be found in Table 1. These field curve values are determined using the formula: I1/r1 + I2/r2.

In Table 1, the value at the pass between the two parallel wires remains constant at 2.0 (A/m). This value persists consistently along the black curve, which resembles the infinity symbol.

In this chapter, we introduce the concept of negative field values, which appears to be employed for the first time.

The aim is to illustrate field curves around two parallel electrical-current wires with opposing intensities, as depicted in Figure 4.

In Figure 4, the wire on the left carries a positive electrical current of one ampere (A), while the wire on the right carries a negative electrical current of one ampere (A).

Also, Figure 4 illustrates drawings of field curves at various current field values.

The black linear curve, representing sea level (i.e., a field value of zero A/m), serves as a horizontal symmetrical axis. This means that a 180-degree rotation around this axis does not alter the appearance of the field curves.

When we compare the drawing in Figure 4(a) with the available data from M. Zollner [4] , notable differences emerge. M. Zollner’s work suggests that these curves take on a circular shape, whereas in Figure 4(a), they do not form circles except when very close to the wires on either the left or right side.

For a comprehensive understanding of the values and colors associated with the field curves in Figure 4, please refer to Table 2.

Table 3 introduces the concept of negative field values in a scientific article for the first time. One advantage of this approach is that it facilitates comprehension by drawing parallels between field curves and the contours of a topographic map. In this analogy, positive levels correspond to elevations above sea level, while negative levels represent depths below the zero-sea level.

When considering these two opposing and parallel wires, the representation

of a field curve consistently maintains a planar structure, like the level contours found in topographic maps depicting terrain relief. Furthermore, field curves maintain a uniform field value along their entire length, mirroring the behavior of topographic contours.

With two distinct wires—one carrying a positive current and the other bearing various negative currents—it is possible to compute the geometry of multiple field curves.

As depicted in Figure 5, there are two wires, both with different current in ampere (A). The first wire is situated on the left side at position (−1, 0), while the second wire is on the right side at position (+1, 0). At sea level, where field values equal zero, the resulting equation is expressed as Equation (1):

I1/r1 + I2/r2 = 0 with I1 > 0 and I2 < 0 (1)

By making mathematical adjustments of Equation (1), it yields the conventional polynomial representation of circles for the field curves, as illustrated in Figure 5.

The field curves depicted in Figure 5 exhibit a circular geometry, and the values of their field strengths and centers are provided in detail in Table 3.

Since this paper lacks physical measurements, it is possible to express field values with precision using more than ten significant figures.

In this section, we compare three distinct field curves. Figure 6 illustrates wires with three different currents, resulting in the formation of three distinct passes.

Under alternative electrical conditions, the values for current field passes are computed and presented in Table 4.

This chapter elucidates the process of depicting multiple field values using two dissimilar electrical currents that flow in opposite directions. These configurations are visualized in Figure 7, and they highlight variations in pass positions when compared to the passes between two identical electrical currents with opposing directions, as depicted in Figure 5.

The field values for curves presented in Figure 7 are provided in detail in Table 5.

When considering two opposing currents with varying intensities, the pass curves do not resemble the horizontal Figure 8 as depicted in Figure 2. Instead, these distinct configurations for six different current values are illustrated in Figure 8.

Figure 8 displays six distinct pass values corresponding to six different current values. In addition to these six values, further evaluations of pass values and their respective positions are documented in Table 6.

Domains can be established when field curves are constructed with at least one pass. In the instance of a single electrical current wire, where no pass can be created, only one domain can be identified.

With several wires, domains are surrounded near and around wires by planar field curves. By adding electrical wires, the number of domains and the number of passes changes. When adding a new wire, in Table 7, the change in the number of domains and the number of passes is evaluated.

Table 7 provides the most common value for the number of passes or domains. For instance, the first domain (n = 2) includes one pass and three domains. When a third electrical current wire is introduced, a new pass is automatically generated, resulting in five domains (n = 3, following the pattern 2 * n − 1).

However, when additional wires are added, the possibility of creating more than one pass can occur. In such cases, the number of domains does not adhere to the guidelines outlined in Table 7. Consequently, when a loop traverses multiple passes, determining the number of passes and domains must be estimated manually, as described in the following chapter.

In Figure 9, the arrangement is depicted where multiple passes exhibit field values of 5.34521 A/m in black and 5.73295 A/m in red.

In Figure 9, the arrangement is depicted where multiple passes exhibit field values of 5.34521 A/m in black and 5.73295 A/m in red.

Also in Figure 9, there are five electrical current wires positioned at the center of five small green points. The electrical wire placed in the middle of these five wires is represented by the red line, which is the closest and has the highest field value of 5.73295 A/m. The black line is farther away and has a lower field value of 5.345208 A/m. As anticipated, these current field curves do not intersect. Surprisingly, the number of domains observed is seven, which is fewer than the expected number of nine based on the direct calculation from Table 7.

In Gauss’s law of magnetism, it is asserted that magnetic field lines neither have a distinct starting point nor an endpoint; they extend infinitely [15] . Every 3D illustration in this current article adheres to this principle. These field lines are nearly circular, with only one extending to infinity, as seen in Figure 4.

In our model, each curve maintains a uniform field value throughout its length. Therefore, when certain positions of the curves are at an infinite distance from the wires, all positions must exhibit a field value of zero (A/m).

This characteristic, exemplified by the linear black curve in Figure 4, represents an enhancement of Gauss’s law by introducing the concept of zero field value in straight lines.

M. Zollner conducted experiments involving magnetic fields with two parallel current wires. When the currents were in opposite directions, he represented the field lines as eccentric circles [4] . In Figure 4 of this article, the field lines may indeed resemble circles when they are near one of the current wires. However, as the field values approach zero, the lines progressively adopt a more linear configuration around the midpoint between the two conducting wires. Through this comparison, it becomes evident that both theoretical and experimental lines are not significantly discrepant.

One notable advantage of the theory proposed in this article is its applicability to highly intricate electrical scenarios, offering a valuable tool for analyzing complex electrical configurations.

Regarding magnetic field lines, Feynman remarked that “The field lines, however, are only a crude way of describing a field, and it is very difficult to give the correct, quantitative laws directly in terms of field lines”. He also noted that “the ideas of the field lines do not contain the deepest principle of electrodynamics, which is the superposition principle”, Furthermore, he pointed out that “we don’t get any idea about what the field line patterns will look like when both sets are present together” [16] .

In our paper, we introduce a novel paradigm for field values, employing a distinct theoretical approach to construct curves based on electrical current fields. In Figure 1(b), the addition of field values (without vectors) enables the representation of highly intricate structures. As a result, it appears that our article adheres to the superposition principle, as complex field configurations can be effectively delineated using this approach.

At this juncture in the article, the reader might find that a clear and comprehensible presentation has been provided, substantiating the author’s claims effectively.