Magnetic Field from an Infinite Conductor

This introduction model creates a simple model of the magnetostatics problem with a coaxial cable of infinite length carrying current, which is commonly found in textbooks. Since there is an analytical solution to this problem, the model can be used to compare theory with numerical results from the simulation. The region outside of the coaxial cable is not part of this model.

Model Definition

First, consider the analytical solution to this problem, which can be used for comparison. Let ri and ro be the radii of the inner and outer conductors, respectively. In this case, the latter is also equivalent to the radius of the modeling domain. In the air region between the two conductors, — that is, for ri < r < ro — the magnetic flux density, B, satisfies

where μ0 is the permeability of vacuum, I0 is the current through the conductor, and

is a unit vector in the azimuthal direction. Inside the center conductor, where r < ri, the solution instead reads

where J0 is the current density. These expressions can, for example, be derived by using the integral formulation of Ampère’s law. From the symmetry of the problem it follows that the magnetic field is orientated purely in the azimuthal direction and that its strength depends only on the radial distance from the conductor.

The geometry of the conductor will be created in two different ways:

•

By using a 2D component, where the Magnetic Fields interface solves for the out-of-plane magnetic vector potential. The 2D component then corresponds to the cross section of the infinite conductor.

•

By using a 2D axisymmetric component, where the Magnetic and Electric Fields interface solves for the in-plane magnetic vector potential as well as the electric potential. This component, when revolved around the axis, represents the full conductor.

Even though the outer conductor of the coaxial cable is seemingly not part of the model geometry, which only contains the inner conductor and an air domain, the return currents are still present in the model. This is achieved by using the Magnetic Insulation boundary condition, which is the default boundary condition in the Magnetic and Electric Fields physics interface. This condition applies the constraint n × A = 0, where A is the magnetic vector potential. Current can therefore flow along this boundary, and as shown in Figure 5, the current forms a closed loop through the inner conductor.

Results

The magnetic flux density norm in the 2D component is plotted in Figure 1, while the magnetic flux density norm in the 2D axisymmetric component is shown in Figure 2 (symmetry cross section) and Figure 3 (revolved geometry). Figure 4 shows a plot of the magnetic flux density as a function of the radial coordinate for the two numerical solutions as well as for the analytical solution discussed above. There, it can be seen how well the numerical solutions corresponding to the two different approaches agree with the analytical solution. Finally, Figure 5 shows the return currents along the outside of the air domain in the 2D axisymmetric case.

Figure 1: Magnetic flux density norm in the 2D component.

Figure 2: Magnetic flux density norm in the 2D axisymmetric component.

Figure 3: Magnetic flux density norm in the 2D axisymmetric component, shown in the full revolved geometry.

Figure 4: The magnetic flux density norm plotted as a function of the radial coordinate. Here, the numerical solutions from the two different components are shown, along with the analytical solution.

Figure 5: The surface currents plotted on the outside of the modeling domain, along with the currents and the electric potential in the inner conductor.

ACDC_Module/Introductory_Magnetostatics/magnetic_field_infinite_conductor

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

First, define some parameters that will be used when building the model.

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
ri	1[cm]	0.01 m	Radius of the conductor
ro	10[cm]	0.1 m	Radius of the computation domain
I0	1[A]	1 A	Conducting current
J0	I0/(pi*ri^2)	3183.1 A/m²	Current density in the conductor

Geometry 1

One way of constructing the geometry is to use a 2D component. Since the Magnetic Fields physics interface by default only solves for the out-of-plane component of the magnetic vector potential, this is equivalent to having a conductor of infinite length, where the 2D geometry corresponds to the cross section.

Circle 1 (c1)

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type ro.

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


Layer name	Thickness (m)
Layer 1	ro-ri

Point 1 (pt1)

In the toolbar, click

Line Segment 1 (ls1)

In the toolbar, click

and choose .

On the object , select Point 1 only.

In the window for , locate the section.

Find the subsection. Click to select the

toggle button.

On the object , select Point 7 only.

In the toolbar, click

Materials

Air

In the window, under right-click and choose .

In the window for , type Air in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Conductor

Right-click and choose .

In the window for , type Conductor in the text field.

Select Domain 4 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	1e6	S/m	Basic

The only modification that needs to be done for the physics, is adding a current density.

Magnetic Fields (mf)

External Current Density 1

In the window, under right-click and choose .

Select Domain 4 only.

In the window for , locate the section.

Specify the Je vector as


0	x
0	y
J0	z

Study 1

In the toolbar, click

Now, add a second, 2D axisymmetric, component that will be used to illustrate a different approach to modeling the same problem. It will contain an equivalent geometry and the same materials.

Add Component

In the window, right-click the root node and choose .

Geometry 2

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type ro.

In the text field, type 2*ro.

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


Layer name	Thickness (m)
Layer 1	ri

Select the check box.

Clear the check box.

In the toolbar, click

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Air

In the window, under right-click and choose .

In the window for , type Air in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic