Field lines

Two methods

There are two ways to compute and display magnetic field lines. First, for axially symmetric systems the field lines are contours of a certain function of the magnetic vector potential, and can therefore be plotted using a planar probe. Second, for general systems, the tracer probe may be used to trace a magnetic field line passing through an given point.

Field lines for axially symmetric systems

In axially symmetric geometries field lines are confined to planes containing the z axis. In any such plane the field lines follow the contours of the following expression involving components of the magnetic vector potential:

x A_y -y A_x

To display field lines, a planar probe is used to plot contours of the following expression in BiotSavart's macro language: 

(x*Ay)-y*Ax

An alternative expression, that gives uniformly spaced contours where the magnetic field is uniform, is 

sqrt (x*Ay)-y*Ax

To make field lines for an axisymmetric system:

  1. Make a Planar probe using the Config menu.
  2. Give the probe a name and dismiss the name dialog.
  3. On the Plot tab select "f" as the quantity to plot.
  4. On the Macro tab enter one of the above expressions.
  5. Adjust other probe parameters (like grid resolution nu and nv on the Axes tab). You can move the probe but note that for interpretation of the contours as field lines, the plane of the probe must contain the z-axis.
  6. Press the Apply or Done button.

Field lines for general systems

BiotSavart can trace magnetic field lines for general three-dimensional configurations of conductors using a Runge-Kutta approach. This is accomplished using a tracer probe. The tracer calculates the path of the field line by integrating the fundamental equation

equation of motion

where s is the path length along the field line.

Note that if the conductor geometry is not simple, magnetic field lines can be complicated and not closed.

To trace a field line:

  1. Make a Tracer probe using the Config menu.
  2. Give the probe a name and dismiss the name dialog.
  3. On the Spec tab set the integration step size and path length.
  4. On the Position tab specify the starting point of the field line.
  5. Press the Apply or Done button.