# Helmholtz Coil

Figure 1: B-field distribution around the Helmholtz coils. The wire coils are located inside the rings of vacuum.

## The Physics

The Helmholtz coil refers to the arrangement of two conductor loops which generate an almost homogeneous magnetic field in the center of the structure (Fig. 1). With the assumption of an infinitely thin current path, the theoretical solution can be derived as shown e.g. in [1].

Figure 2: Setup of the Helmholtz Coil, with the parameters labeled.

Using the Biot-Savart law the magnetic field is determined on the z-axis of the arrangement of Fig.2, as

Bz(z)=
 1 2
(
 μ0a2I (a2+(z−d)2)(3∕2)
+
 μ0a2I (a2+(z+d)2)(3∕2)
)

where µ0 is the permeability of free space.

By using a Taylor expansion to analyze the dependence of the function of the z coordinate the optimum arrangement is found to be 2d=a.

## The Model

Two curves are designed and a current is imprinted on them. The boundary conditions should be 'open' with a sufficient distance to the coils. It is possible to activate three symmetry conditions to reduce the mesh requirements – electric in the XZ and YZ planes, and magnetic in the XY plane. If the value for the radius is set (in this example, all given values are in m), the distance of the loops is determined automatically via the parameter list and so the dependencies are correctly considered. By evaluating the resulting fields on the z-axis, the field strengths can be compared to the numerical results. For this purpose the corresponding post-processing templates are used.

Figure 3: The setup of the Helmholtz Coil in CST Studio Suite®, showing the curves (left) and the 3D dummy objects (right).

 Parameter Value Description a 5 m Coil radius d a/2 (2.5 m) Distance of one coil from origin l 1 A Current around each coil

For the evaluation of the field values along the curve one has to ensure that adequate interpolation points are available. It is also helpful to have a dense mesh surrounding the current paths. To allow the mesh to be redefined within the mesh limitations of the CST Studio Suite Student Edition, dummy structures are introduced surrounding the current paths and along the z-axis (Fig. 3). These structures have the same material properties as the background material (vacuum) and ensure a finer mesh generation in this region. These are constructed with around the curves, with a radius of a/10 around each curve.

## Discussion of Results

Figure 4: The Template Based Post Processing dialog, showing the Evaluate Field on Curve option.

Template Based Post Processing (Fig. 4) is used to calculate the BZ field along the line. Set up the template as shown in Fig. 5, and click "Evaluate" in the main Template Based Post Processing window to calculate the field distribution along the curve.

Figure 5: The Evaluate Field on Curve template.

This can be compared directly with the theoretical results. Equation 1 can be represented in a calculable format as:

1/2*mue0*a^2*I*(1/(a^2+(A-d)^2)^(3/2)+1/(a^2+(A+d)^2)^(3/2)), where A is the Z coordinate.

By pasting this into the “Mix Template Results” post processing template (Fig. 6), which is found under "General 1D" in the Template Based Post Processing window, the numerical Bz along the z-axis can be calculated.

Figure 6: The Mix Template Results dialog, with Equation 1 pasted in.
Figure 7: B-Field along z-axis – a comparison of analytical and numerical results.

Comparing the theoretical and numerical results in Fig. 7 shows that a very good agreement is found. To achieve such a good agreement, the described usage of dummy structures is essential.

## Questions

Figure 8: 3D coil definition.

Q1: The coil function (Fig. 8) can be used to define 3D coils, by sweeping a “profile” curve (representing the cross-section of the coil) along a path curve. Define two 3D coils in place of the dummy structures – how do the results change?

Q2: What is the effect of changing the symmetry between the loops to electric?

## References

• [1] H. Henke, Elektromagnetische Felder: Theorie und Anwendung, 3. erweiterte Auflage, Springer-Lehrbuch, pp. 177-179

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