The scalar magnetic field model assigns a field value to each point while incorporating three orthogonal, dimensionless vectors.

Figure 1 focuses on the field induced by a short elementary electrical segment, denoted as dl.

In Figure 1 , three dimensionless vectors originate from p:

These vectors (Au, Cu, and Mu) are mutually perpendicular and dimensionless. By multiplying each of them by the scalar field value Sv at p, the physical vectors dA, dB, and dM are obtained, all with the same physical length and dimension.

The physical field value Sv at p is in A/m and given by product I and dl, divided by the square of the distance between p and the midpoint of the elementary elec-trical segment dl.

The scalar value Sv at point p always corresponds to a non-vector physical quantity. For example, in Figure 1 , this value is given by I dl/D² (A/m).

This paragraph critically examines the Biot-Savart magnetic field model.

In this classical vector model of the magnetic field, a point p—located outside a short elementary segment dl—is considered. At this point as illustrated in Figure 2 , three vectors with different physical units are associated.

The value of dH is determined by the cross product of vectors I dl and D.

In contrast to the previously described model, the three vectors in this case have different physical units, making them incompatible for direct comparison and preventing them from being understood within the same geometrical space.

Let’s now attempt to calculate the field values:

When point p is located on the x-axis, the cross product of D and I dl becomes zero, which physically means that the small segment has no magnetic influence along this axis. Similarly, if p lies directly on the segment dl, dH is also zero.

When point p is located on the y-axis, the cross product of D and I dl evolves from zero when p is very far away and becomes infinite as p approaches the segment dl.

This leads to a clear contradiction: at the same point p (located at (0, 0)), dH cannot simultaneously be zero and infinite. This discontinuity cannot be physically explained by the current model. In contrast, such a phenomenon is absent in the model described in the previous paragraph, which does not involve a cross product given the field value.

With this observation, the author proposes to work with the specific model outlined earlier, which assigns a scalar value to each point and is associated with three dimensionless orthogonal vectors.

Here, the first model described in Figure 1 is applied to determine the iso-magnetic surfaces and field curves around an elementary segment carrying a current, and to represent them in a 3D drawing.

The magnetic field value is the same at all points on the circle centered on the x-axis, where the segment is located as in Figure 3 . This circle represents a magnetic field curve, with the vector Mu being tangent to it.

By extending this concept to the entire segment, it is possible to represent an iso-magnetic field surface, which is associated with circular magnetic field curves. An example of this is shown in Figure 4 .

In Figure 4(a) , the colored surface represents points with the same magnetic field values. In Figure 4(b) , the circles represent magnetic field curves around the segment, each having the same physical value as the iso-magnetic surface in Figure 4(a) .

The scalar magnetic field value induced by a linear, electrically active segment from a to b on the x-axis can be determined by integrating the elementary value I dl/D², as described in Figure 3 . The result is given by Equation (1)

$Sv\left(x,r\right)=\frac{1}{\pi }{\displaystyle {\int }_a^b\frac{dx}{x^2+r^2}}$ (1)

Equation (1) provides the scalar magnetic field value Sv at point p, which is located a distance r from the wire along the x-axis, with current I flow through the segment a b. For example, when point p is on the segment between a and b, the magnetic field scalar value becomes infinite.

If point p is on the x-axis (with r = 0) but outside the segment, the magnetic field decays according to a hyperbolic function, 1/x, and approaches zero as the distance x becomes infinite.

In other words, this model ensures that the field outside the segment remains continuous.

An infinite line is described by setting a = −∞ and b = +∞. In this case, Equation (1) gives Sv = I/r. This expression has been extensively used in our previous articles on infinite straight wires [7] [8] [12] .

A half-line on the x axis of Figure 5 in a x y plane can be described with endpoints extending from a = −∞ and b = 0.

One can observe in Figure 5 that there is no discontinuity in the field value, and that the field becomes infinite everywhere near the half-line.

In Figure 6 , an infinitely long electrical wire is segmented into two half-lines, with a gap between x = 0 and x = 0.6 m.

As shown in Figure 6 , a scalar field value of 2.06 A/m defines a boundary that differentiates between distinct types of equipotential surfaces, separating those near the half-lines from those farther away. This boundary divides the equipotential surfaces into three distinct groups. Close to the electrical half-lines, the surfaces form two separate groups, each associated with one of the half-lines. Beyond the boundary, one equipotential surface encloses both half-lines, as illustrated by the blue surface in Figure 6 .

In the present paragraph and in Figure 7 , however, the currents flow in opposite directions: +1 A on the left side and −1 A on the right side.

In Figure 7 , the magnetic field curves closely resemble the surfaces depicted in Figure 4 when they are near one of the half-lines. However, a key difference arises at the vertical line, which appears as a planar surface in a 3D view and is positioned midway between the two half-lines. On this plane, the magnetic field value is exactly 0 A/m, acting as a natural boundary that separates the regions influenced by each half-line’s current.

The significance of this diagram of Figure 8 , lies in its ability to incorporate additional segments, enabling the representation of more complex geometrical configurations. By extending this concept, one can analyze the interaction of multiple current-carrying segments and their corresponding magnetic field surfaces, providing deeper insights into more intricate electromagnetic structures.

In this article, three dimensionless vectors are used:

The cross product with these three vectors can be applied by using Au and Cu to have Mu, or by using Cu and Mu to have Au and for Mu and Au to have Cu. This is a very important property available in this model.

In this article and in Figure 8 , the magnetic field value at any point around a segment is defined by three orthogonal vectors with the same physical value (not one vector as in the classical model). The key consequence of this approach is that the field values induced by multiple segments can be added together to obtain the total field at any point. This additivity is a fundamental property shared by all physical processes, such as gravitational attraction.

This article is a small part of a long series. It has the number 4 and it uses a field value and three vectors. These three vectors were previously defined and published in 2022 [12] . This first article has been defined like to electrostatic field. Then, this definition was applied to two articles composed of long wire [7] [8] . After these three articles, it was necessary to apply this definition to short segments. During this work, the author found no experimental data about field around electrical segments.

Note that this article is devoted to university students and a future article will be written to be understandable by specialists.