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Dipole Fields Are Complicated

Dipoles are the simplest magnetic objects in physics. Nonetheless, they can cause some surprisingly complex behavior for a charged particle moving nearby.
A charged particle moving according to the Lorentz force about a magnetostatic dipole provides an example of complex behavior that can come about even in simple physical systems. In the case illustrated here, the charged particle spirals along magnetic field lines but reflects in a magnetic "mirror" at "high latitudes" in the field.
The governing equations and definitions for this simulation include:
; ;
.
This Demonstration allows the specification of the initial conditions of the charged particle (Cartesian position and velocity ), the magnetic field parameter, and the duration of motion. The resulting trajectory is shown as a blue curve. The current velocity direction is shown as a short green line segment and the field direction at the current trajectory point is shown by a red segment. The particle trajectory spirals around the field direction.
A sphere and magnetic pole are provided at the middle of the visualization only for conceptual purposes. A magnetic dipole need not have a finite dimension, and the example shown here does not affect the simulation. The sphere is provided for a visual frame of reference. Such a reference is useful in visualizing the motion of a charged particle in the Earth's magnetic field.
  • Contributed by: Jay Farrell (SAIC -- Huntsville, Alabama, USA)

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Increasing the strength of the magnetic field makes this mirroring more evident. For relatively low values of the field, the particle moves in a seemingly more erratic fashion while progressing slowly about the equator. For stronger values, the particle's spiral motion is tighter, and begins more to resemble the field line geometry.
Here, the translucent sphere pictured has nothing to do with the simulation; it is shown to help orient the viewer.
By increasing the time of the simulation until the particle moves around the sphere, a magnetic "shell" is seen. This shell structure resembles that experienced by charged particles moving about the Earth, but in a simplified way. The shape is reminiscent of the Van Allen belts around the Earth.
Varying the initial position and velocity of the particle produces additional, greatly varying behavior, including the possibility of escape from the system.
Snapshot 1: moderate field, showing magnetic reflection clearly along field lines ()
Snapshot 2: same but run to
Snapshot 3: weak field, showing less regular short term behavior
()
Snapshot 4: strong field, showing magnetic reflection clearly along field lines ()
Also, try animating the time evolution very slowly. Follow the progress of the red line segment (along the magnetic field at the current time step) and the green segment (along the velocity vector).
References:
D. P. Stern and M. Peredo. "Principles of the Magnetic Trapping of Charged Particles." (Nov 10, 2003)
D. P. Stern and M. Peredo. "The Inner Radiation Belt." (Nov 25, 2001)
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