Fundamentals of Magnetostatics:
A Single Circular Current Loop and
a Loop Pair in Helmholtz Position

A basic loop

©2004 Magneticasoft

The proposed basic loop is made with 25 turns of copper wire, wound in a circular form that is 100 millimeters in diameter. The winding diameter is 25 millimeters and the filling factor is 60%. The loop is activated with a 10 amperes current. The proposed symbolic name of this new magnetic system is loop. This data can be generated by directly typing the input as shown below of by using Magnetica palettes.

Magnetica Initialization

It is assumed that Mathematica and Magnetica have been correctly installed. In order to load the Magnetica application package, use the command Get (<<).

<< Magnetica`

Magnetic System Definition

loop=AXSMagneticSystem[
{AXSCircularLoop[CopperOFHC,100,25,0,24,10,FillingFactor->0.6]},
SystemLabel -> "Basic circular loop"]

Magnetic components are the elementary blocks for building magnetic systems. Under Magnetica there is no built-in limitation to the number of components that a magnetic system can contain. The next example is a magnetic system with two identical loops set in the Helmholtz position; that is, the loops are symmetrically distant from a radial plane in such a way that the distance between them is equal to the radius of the loops. This arrangement of two loops generates a magnetic field of better spatial quality than the single loop.

helmholtz=AXSMagneticSystem[
{AXSCircularLoop[CopperOFHC,100,25,-25,24,10,FillingFactor->0.6],
AXSCircularLoop[CopperOFHC,100,25,25,24,10,FillingFactor->0.6]},
SystemLabel -> "pair of Helmholtz loops"]

Drawing Magnetic Systems

DrawSystem[loop];

DrawSystem2D[helmholtz];

Punctual Magnetic Field Calculation

Magnetic Field (Component Vector and Magnitude)

FieldPoint[loop, {0, 0}]

FieldPoint[helmholtz, {0, 0}]

The implicit units of these examples are millimeter for length and gauss for magnetic field intensity. However, these units can be changed locally (as in the next example) or for a complete session (see the last paragraph on units). The output of these functions can be used for further calculation (using Mathematica).

The verbose function provides a textual, explanatory, and formatted output.

FieldPoint[helmholtz, {30, 10}, 
      FieldUnit -> Milli Tesla]

VerboseFieldPoint[helmholtz, {30, 10}]

Magnetic Potential

PotentialPoint[loop, {30, 10}]

The default unit for field potential in Magnetica is microweber per meter.

Magnetic Flux

FluxArea[loop, {30, 10}]

The default flux unit in Magnetica is microweber.

Magnetic Field Gradient

GradientPoint[helmholtz, {30, 10}] // MatrixForm

The default field gradient unit in Magnetica is gauss per meter.

Field Map Graphics

2D Graphics: Field Functions along a Line

FieldMap[loop, {Bz}, Line[{0, -50}, {0, 50}]];

FieldMap[helmholtz, {Bz}, Line[{0, -50}, {0, 50}]];

FieldMap[helmholtz, {Bz, Flux}, 
        Line[{30, -50}, {30, 50}]];

3D Graphics: Field Functions in a Plane

One interesting feature of Magnetica concerns the ContourFieldMap function applied to the flux: this displays the magnetic field lines such that at any point in the direction of the field is tangent to the displayed lines. This feature is related to the physical laws specific to magnetostatics for axis-symmetric systems.

FieldMap[helmholtz, {Bz}, 
        Plane[{-50, -50}, {50, 50}]];

ContourFieldMap[helmholtz, {Flux}, 
        Plane[{-100, -100}, {100, 100}]];

Spherical Harmonics Field Calculation

Calculating the magnetic field using spherical harmonics is an important feature of Magnetica. The spherical harmonics method is an efficient way to calculate the magnetic field. However, this method is restricted to space without a magnetic source. Consequently, three regions can be distinguished:

1. a central spherical region

2. a forbidden region (due to a fundamental cause: no mathematical convergence)

3. a remote region extending from a limiting outer sphere to infinity

The next graph shows the three regions. The forbidden region (yellow), is bordered by a remote region (brown) where the precision of the calculations becomes insufficient.

DrawSHLimits[loop];

The spherical harmonics method is based on a set of coefficients (the Legendre source coefficients) for Legendre polynomial development. The Legendre source coefficients measure both the intensity of the field, indicated by the zero-order coefficient, and the homogeneity of the field, shown by the coefficients of increasing order. The coefficients of the central and remote regions are two different sets of numbers (expressed by default in the current magnetic field unit).

VerboseLegendreCoefficients[loop]

VerboseLegendreCoefficients[helmholtz]

Observe that the second-order central coefficient is zero for the Helmholtz pair, which indicates a better homogeneity of magnetic field than the single loop.

Inductance Calculations

Inductance is a measurement of the action of the magnetic flux on the electrical circuit generating the field. It is expressed by the ratio of flux to current intensity. Mutual inductance characterizes the interaction between two circuits. Self-inductance characterizes the action of a single circuit on itself. Magnetic energy is directly related to inductance and flux.

The default unit for inductance in Magnetica is microhenry.

SelfInductance[loop, All]

MutualInductance[helmholtz,{1,2}]

This is the interaction flux at the proper current intensity (10 amperes); the default unit is microweber.

SelfFluxInductance[loop, All]

This is magnetic energy at the proper current intensity (10 amperes); the default unit is microjoule.

SelfMagneticEnergy[loop, All]

This displays the computation with detailed information using the verbose version of functions.

VerboseSelfInductance[loop, All]

VerboseSystemInductance[helmholtz]

VerboseInductanceMatrix[helmholtz]

Units

MagneticaCurrentUnits[]

These are the units that are currently taken into account. They can be changed locally or globally for a complete session. A large number of units can be used.

For magnetic field and related data (induction, flux density, magnetization, magnetic field intensity, magnetic polarization), the possible values are: Gauss, Tesla, Ampere/Meter, Oersted; they can be used with any decimal multiplier.