Electromagnetic Forces on Parallel Current-Carrying Wires

One ampere is defined as the constant current in two straight parallel conductors of infinite length and negligible circular cross section, placed one meter apart in vacuum, that produces a force of 2 × 10−7 newton per meter of length (N/m). This example shows a setup of two parallel wires in the spirit of this definition, but with the difference that the wires have finite cross sections.

For wires with circular cross section carrying a uniform current density as in this example, the mutual magnetic force is the same as for line currents. This can be understood by the following arguments: Start from a situation where both wires are line currents (I). Each line current is subject to a Lorentz force per meter (I × B), where the magnetic flux density (B) is the one produced by the other wire. Now, give one wire a finite radius. It follows directly from circular symmetry and Maxwell-Ampère’s law that, outside this wire, the produced flux density is exactly the same as before so the force on the remaining line current is unaltered. Further, the net force on the wire with the distributed current density must be of exactly the same magnitude (but with opposite direction) as the force on the line current so that force did not change either. If the two wires exchange places, the forces must still be the same, and it follows from symmetry that the force is independent of wire radius as long as the wire cross sections do not intersect. The wires can even be cylindrical shells or any other shape with circular symmetry. For an experimental setup, negligible cross section is required as resistive voltage drop along the wires and Hall effect may cause electrostatic forces that increase with wire radius but such effects are not included in this example.

The force between the wires is computed using two different methods: first automatically by integrating the stress tensor on the boundaries, then by integrating the volume (Lorentz) force density over the wire cross section. The results converge to 2 × 10−7 N/m for the 1 ampere definition, as expected.

Model Definition

The application is built using the 2D Magnetic Fields interface. The modeling plane is a cross section of the two wires and the surrounding air.

Domain Equations

The equation formulation assumes that the only nonzero component of the magnetic vector potential is Az. This corresponds to all currents being perpendicular to the modeling plane. The following equation is solved:

where μ is the permeability of the medium and

is the externally applied current density.

is set so that the applied current in the wires equals 1 A, but with different signs.

Surrounding the air is an infinite element domain. For details, see the AC/DC Module User’s Guide.

Results and Discussion

The magnetic flux density of the two current carrying coils is shown in Figure 1 where the direction and magnitude of the magnetic flux density are illustrated with streamlines and color scale, respectively.

Figure 1: The magnetic flux density of the two current carrying coils.

As shown in Figure 1, the Maxwell surface stress tensor on the boundaries of conductors is represented by arrow plots. The expression for the surface stress reads

where n1 is the boundary normal pointing out from the conductor wire and T2 the stress tensor of air. The closed line integral of this expression around the circumference of either wire evaluates to −1.99 × 10−7 N/m. The minus sign indicates that the force between the wires is repulsive. The software automatically provides the coordinate components of the force on each wire.

The volume force density is given by

The surface integral of the x component of the volume force on the cross section of a wire gives the result −2.00 × 10−7 N/m.

By refining the mesh and re-solving the problem, you can verify that the solution with both methods converges to −2 × 10−7 (N/m), see Mesh Convergence. The volume force density integral is typically the most accurate one for reasons explained in the COMSOL Multiphysics Reference Manual.

Mesh Convergence

In order to investigate the accuracy of the model, it is recommended to perform a systematic mesh convergence analysis of the desired entity, here the force on the wire. In Figure 2 and Figure 3, the mesh convergence is shown for the absolute errors in the Maxwell surface stress method and the volumetric Lorentz force method, respectively. The Lorentz force is 2–3 orders of magnitude more accurate than the Maxwell stress tensor force for a given mesh density.

Figure 2: Mesh convergence is shown for the force computation using the Maxwell surface stress method.

Figure 3: Mesh convergence is shown for the force computation using the volumetric Lorentz force method.

ACDC_Module/Introductory_Electromagnetic_Forces/parallel_wires

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
r	0.2[m]	0.2 m	Wire radius
I0	1[A]	1 A	Total current
J0	I0/(pi*r^2)	7.9577 A/m²	Current density
N	1	1	Mesh multiplier

Geometry 1

Add a circle for the main air domain. The outer layer will constitute an infinite element domain to approximate a region extending to infinity.

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 1.5.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	0.5

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type r.

Locate the section. In the text field, type 0.5.

Circle 3 (c3)

In the toolbar, click

In the window for , locate the section.

In the text field, type r.

Locate the section. In the text field, type -0.5.

Click

Definitions

Define an infinite element region in the outer domains.

Infinite Element Domain 1 (ie1)

In the toolbar, click

Select Domains 1–4 only.

In the window for , locate the section.

From the list, choose .

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Magnetic Fields (mf)

By default, the first material you select will apply to your entire geometry. Air is defined with a zero conductivity, and relative permittivity and permeability both equal to 1. These properties are the same as those of vacuum, which is the assumed material in the definition of the ampere. Since the model assumes a given static and uniform current distribution, the electrical conductivity of the wires does not appear in the equations, so it is safe to use the same properties in the wires too.

External Current Density 1

In the window, under right-click and choose .

Select Domain 6 only(the wire on the left).

In the window for , locate the section.

Specify the Je vector as


0	x
0	y
J0	z

External Current Density 2

In the toolbar, click

and choose .

Select Domain 7 only(the wire on the right).

In the window for , locate the section.

Specify the Je vector as


0	x
0	y
-J0	z

The definition of the physics of the system is now complete. Add a Force Calculation feature to make Maxwell’s stress tensor available as a variable.

Force Calculation 1

In the toolbar, click

and choose .

Select Domain 6 only.

In the window for , locate the section.

In the text field, type wire1.

Force Calculation 2

In the toolbar, click

and choose .

Select Domain 7 only.

In the window for , locate the section.

In the text field, type wire2.

The infinite element domain requires some attention when meshing. As it is steeply scaled in the radial direction to model a very large geometry (approximating a geometry extending to infinity), the mesh will effectively be stretched in that direction. A structured mesh is indicated in this case to prevent poor element quality. The Magnetic Fields interface can automatically create an appropriate mesh for this application.

Mesh 1

In the window, under right-click and choose .

The automatically created mesh applies a Mapped operation on the finite element domain. Modify it according to the following instructions.

Size

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click to expand the section. In the text field, type 0.2.

Click

The mesh should look like in the figure.

Study 1

In the toolbar, click

Results

Magnetic Flux Density and Maxwell Stress Tensor

In the window for , click to expand the section.

From the list, choose .

In the text field, type Magnetic Flux Density and Maxwell Stress Tensor.

Locate the section. Select the check box.

The default plot shows the norm of the magnetic flux density. Note that the value inside the infinite element domain has no physical relevance.

Arrow Line 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section.

Select the check box. In the associated text field, type 300000.

In the toolbar, click

Click the

button in the toolbar.

Arrow Line 2

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section.

Select the check box. In the associated text field, type 300000.

In the toolbar, click

The plot shows the Maxwell’s stress tensor distribution on the surface of the wires. The total force on each wire is evaluated as the surface integral of the stress tensor and is available as a postprocessing variable.

Global Evaluation 1

In the toolbar, click

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click

Table 1

Go to the window. The force in the x direction on the first wire evaluates to a value between −2.0×10−7 N/m and −1.9×10−7 N/m. Note that the force is given per meter, since the 2D model has an out-of-plane thickness of 1 m.

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click

Go to the window.

As expected, the force on the second wire has a similar value but the opposite sign.

Proceed to compare the value with those from the Lorentz force distribution.

Results

Surface Integration 1

In the toolbar, click

and choose .

Select Domain 6 only.

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click

Table 2

Go to the window.

This time, the value is expected to be consistently closer to −2×10−7 N/m. When applicable, Lorentz force integrals usually give more accurate results than the Maxwell’s stress tensor.

Select Domain 7 only.

Results

Surface Integration 1