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Magnetic field on axis of a current loop Application


Physics

The Biot–Savart law makes it possible to determine magnetic field produced by electric current. The law is completely general and can in principle be used for any configuration of current paths.
The law states that an infinitesimally small current carrying path space I d l spaceproduces magnetic flux density space stack d B with rightwards arrow on top at a distance r:

space stack d B with rightwards arrow on top equals fraction numerator mu subscript 0 space I over denominator 4 pi end fraction space fraction numerator space space stack d l with rightwards arrow on top cross times space r with hat on top space over denominator r squared space end fraction

Where mu subscript 0 equals 4 pi cross times 10 to the power of negative 7 end exponent space H divided by m space is the vacuum permeability and r with hat on top is the unit vector in the direction of the distancespace r space left parenthesis r with hat on top equals fraction numerator r with rightwards arrow on top over denominator r end fraction right parenthesis. In symmetric problems, it is possible to simplify the analysis and obtain a closed form solution. The field on the axis of a current carrying loop can be easily computed using the Biot-Savart law, due to the fact that only z axis component d stack B subscript z with rightwards arrow on top equals d B with rightwards arrow on top sin theta of the  stack d B with rightwards arrow on top spacevector contributes to the resultant field intensity (Fig. 1):

d B subscript Z equals space fraction numerator mu subscript 0 I over denominator 4 pi space end fraction space fraction numerator d l over denominator r squared end fraction sin capital theta equals fraction numerator mu subscript 0 space I over denominator 4 pi end fraction fraction numerator d l over denominator r squared end fraction R over r equals fraction numerator mu subscript 0 space I over denominator 4 pi end fraction space fraction numerator R over denominator left parenthesis R squared plus z squared right parenthesis to the power of 3 divided by 2 end exponent end fraction space d l

Biot-Savart law and field on centerline of a current loop
Figure 1- Biot-Savart law and field on centerline of a current loop

The total flux density at a point on the centerline at a distance z is found by integrating the expression for  over the circumference of the loop:

stack B subscript z with rightwards arrow on top equals space fraction numerator mu subscript 0 space I over denominator 4 pi end fraction space fraction numerator R over denominator left parenthesis R squared plus z squared right parenthesis to the power of 3 divided by 2 end exponent end fraction contour integral d l equals fraction numerator mu subscript 0 space I over denominator 4 pi end fraction space fraction numerator R over denominator left parenthesis R squared plus z squared right parenthesis to the power of 3 divided by 2 end exponent end fraction 2 pi space R equals fraction numerator mu subscript 0 space I R squared over denominator 2 left parenthesis R squared plus z squared right parenthesis to the power of 3 divided by 2 end exponent end fraction

For a current I equals 100 Aand loop radius R equals space 100 space m m, the axial magnetic field is B subscript Z equals fraction numerator mu subscript 0 over denominator 2 left parenthesis 10 to the power of negative 2 end exponent plus z squared right parenthesis to the power of 3 divided by 2 end exponent end fraction T.

Model

A thin toroid, with a cross-section area radius 5mm, and a loop radius 100mm is simulated with Magnetostatic study1m in EMS.  Copper is prescribed as a material to the Toroid, while air covers the rest of the assembly. To get accurate magnetic field results, it is necessary to create sufficiently large air domain.

To see how to assign material in EMS, see the “Computing capacitance of a multi-material capacitor” example.
To learn how to define the air domain in EMS, consult the “Electric field inside the cavity of a charged sphere” example.

Solidworks model of the studied example
Figure 2 - Solidworks model of the studied example

Solid Coil

To prescribe the EMS Coil feature to the Toroid, it is necessary to have access to its cross-section surface. Therefore, the Toroid part should be split in two bodies. To do so:
 

  1. Select the Toroid part in the Solidworks feature manager
  2. Click Edit component 2m in the Solidworks Assembly tab
  3. In Solidworks menu click Insert/Molds/Split
  4. In the Split feature manager, select the Top Plan of the Toroid in the Trim Tools Tab and Click Cut Part
  5. In the Resulting Bodies tab, Click Select all.
  6. Click OK 3m 
To add a solid coil to a Magnetostatic study:

In the EMS feature tree, Right-click on the Coils4mfolder, select Solid Coil 5m .
Click inside the Components or Bodies for Coils box 6m.
Click on the (+) sign in the upper left corner of the graphics area to open the components tree.
Click on the Toroid icon.  It will appear in the Components and Solid Bodies list.
Click inside the Faces for Entry Port box 7m then select the entry port face.
In the Exit Port Tab, Check “Same as Entry Port“. (Figure 3)

General Properties:

  1. Click on General properties tab.
  2. Keep default Coil Type as a Current driven coil.
  3. Type 100 in the Net current mmfield.
  4. Click OK3m .

Entry and Exits ports of the Solid Coil
Figure 3 - Entry and Exit ports of the Solid Coil

Results

To be able to display the variation of the magnetic field along the axis of the Toroid, before running the simulation:

  1. In the Assembly, Select the ZX plane and sketch a line 9mgoing from the center of the toroid along the z axis, with a length of 100mm.
  2. Then Insert/Reference geometry/Point and add a Reference point for each end of the line.
  3. In the EMS feature tree, right click study 10m and select Update geometry 11m.
  4. Mesh and run the study. 

Once the simulation is complete:

  1. In the EMS feature tree, Under Results12m, right click on the Magnetic Flux Density folder and13m select 2D Plot then choose Linear.         
  2. The 2D Magnetic Flux Density Property Manager Page appears.
  3.  In the Select points tab, select the start and the end points.
  4. Click OK3m

The theoretical and EMS result of the magnetic flux density along the axis of the toroid are plotted in Figure 4.The agreement between the two solutions is very good.

Comparison of EMS and theoretical results for magnetic flux density along the axis of a toroid

Figure 4 - Comparison of EMS and theoretical results for magnetic flux density along the axis of a toroid

To plot magnetic flux density in the space around the current loop (Figure 5):
  1. Under Results 12m, right click Magnetic flux density13m in the EMS feature tree
  2. Select 3D Vector Plot 14m, Section Clipping15m .
  3. Under Section Clipping tab, select one of the sections that contain the system centerline.
  4. Under Vector Options tab, define size, density and shape of the vectors in the plot.                                                                                                                                                                         

Section plot of the magnetic flux density
Figure 5 - Section plot of the magnetic flux density