Helmholtz Coil

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Magnetic Field of a Helmholtz CoilSOLVED WITH COMSOL MULTIPHYSICS 3.5a

helmholtz_coil.book Page 1 Tuesday, November 25, 2008 2:18 PM

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Magne t i c F i e l d o f a He lmho l t z C o i l

Introduction

A Helmholtz coil is a parallel pair of identical circular coils spaced one radius apart and wound so that the current flows through both coils in the same direction. This winding results in a uniform magnetic field between the coils with the primary component ptwco

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helmholtz_coil.book Page 1 Tuesday, November 25, 2008 2:18 PMM A G N E T I C F I E L D O F A H E L M H O L T Z C O

arallel to the axis of the two coils. The uniform field is the result of the sum of the o field components parallel to the axis of the coils and the difference between the mponents perpendicular to the same axis.

he purpose of the device is to allow scientists and engineers to perform experiments d tests that require a known ambient magnetic field. Helmholtz field generation can

e static, time-varying DC, or AC, depending on the applications.

pplications include cancelling the earths magnetic field for certain experiments; enerating magnetic fields for determining magnetic shielding effectiveness or sceptibility of electronic equipment to magnetic fields; calibration of magnetometers d navigational equipment; and biomagnetic studies.

igure 1: The Helmholtz coil consists of two coaxial circular coils, one radius apart along e axial direction. The coils carry parallel currents of equal magnitude.

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Model Definition

The model is built using the 3D Magnetostatic application mode. The model geometry is shown in Figure 2.

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igure 2: The model geometry.

O M A I N E Q U A T I O N S

ssuming static currents and fields, the magnetic vector potential A must satisfy the llowing equation:

here is the permeability, and Je denotes the externally applied current density.he relations between fields and potentials are given by

his model uses the permeability of vacuum, that is, = 4 107 H/m. The external rrent density is zero except in the circular coils, where a current density of 1 A/m2

1 A( ) Je=

B A=H 1 B=

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is specified. This corresponds to a coil current of 2.5 mA. The currents are specified to be parallel for the two coils.

To avoid numerical instability, the application mode by default use an extra equation that sets the divergence of the A field to zero (gauge fixing). This model uses a more efficient approach where the divergence of A is numerically adjusted to zero by using special pre- and postsmoothers called SOR gauge. See Solver Settings for Numerical Gauge Fixing in Magnetostatics on page 114 in the AC/DC Module Users Guide fo

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r more details on this topic.

O U N D A R Y C O N D I T I O N S

he only boundary conditions that you need to specify is for the exterior boundary, at is, the spherical surface, where you apply conditions corresponding to zero agnetic flux:

n A 0=

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Results and Discussion

Figure 3 shows the magnetic flux density between the coils. You can see that the flux is fairly uniform between the coils, except for the region close to the edges of the coil.

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igure 3: The surface color plot shows the magnetic flux density. The arrows indicate the agnetic field (H) strength and direction.

he main property of the Helmholtz coil is that the magnetic flux becomes uniform a reasonably large region with a rather simple coil system. This is illustrated in igure 4 (a), which shows a radial flux density profile for an axial position right etween the coils. Figure 4 (b) shows an axial magnetic flux density profile. The model early demonstrates the highly uniform magnetic field obtained in a Helmholtz coil.

the absence of any test object, this model is fully axisymmetric and could be plemented as a 2D axisymmetric model, which would be much less computationally

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demanding. However, a full 3D model has the advantage that it is possible to include a non-axisymmetric test object in the analysis as a slight modification of the model.

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igure 4: The magnetic flux density profile. The plots show the profile taken along a radial oss-sectional line through the axis right between the coils (a) and a profile taken along e axis (b). A high degree of uniformity is clearly shown.

(a) (b)

Magnetic Field of a Helmholtz CoilIntroductionModel DefinitionResults and Discussion

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Helmholtz Coil