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Principles of the Magnetic Trapping of Particles

#10.     Principles of the Magnetic
Trapping of Particles

 A full explanation of the motion of trapped particles would be too abstract and mathematical for this level, but some basic ideas may be described on an intuitive level.

Circular motion

The magnetic force on a moving particle is always perpendicular to the motion, as well as to the magnetic field lines. That is why the basic pattern is motion in a circle, around a magnetic field line. In the motion of a satellite circling the Earth above the atmosphere, gravity always balances the centrifugal force; similarly with the charged particle, the magnetic and centrifugal forces are always balanced.

Because the force is perpendicular to the velocity, it can only change the direction of motion, not its speed or energy. Because no energy is needed to keep up the motion, it can (in principle) persist indefinitely.

Mirroring

The fact that the magnetic force is perpendicular to magnetic field lines means that when a particle spirals around a cone of converging field lines, that force is always slightly tilted backwards (drawing).


The motion of ions on
coverging lines.

By the laws of motion, any force can always be resolved into the sum of mutually perpendicularforces, each controlling the motion in its direction. The "radial force" perpendicular to the axis of the cone (drawing) keeps the ion or electron turning in a circle around that axis, and is balanced (as noted above) by the centrifugal force of that rotation.

In addition, however, there will also exist a small force parallel to the axis, repelling the particle away from the tip of the cone. That added force gradually slows down the particle's advance down the axis and finally reverses it, causing it to "mirror" and bounce back.

Throughout all this, the total speed of the particle stays unchanged. Out in space, it usually takes electric forces, not just magnetic ones, to change the total speed and energy of the particles.

Adiabatic Invariants

There exists a different and somewhat more abstract manner of reaching the same result. The period T of rotation, the time required by the particle for one circuit around its guiding field line, becomes shorter as the particle approaches the tip of the cone. After all, the total speed of the particle is unchanged, its rotation speed nearly so, while the distance covered by one circuit gets shorter and shorter near the tip.

In the theory of motions, this is an example of a periodic motion whose period gradually decreases. The best-known periodic motion is the back-and-forth swing of a pendulum, say of a weight suspended by a string (drawing). The shorter the string, the shorter the time of each swing ("period"), which goes like the square root of the length. One can replace the support point with a pulley wheel, over which the string can be pulled back (or let out), changing the period of the swing.

If the string is pulled up while the pendulum is swinging, two things happen. The period T of each swing gets shorter, as noted before. But in addition, the energy E of the pendulum increases, which means the height of each swing becomes larger. For as the pendulum swings, it generates a centrifugal force, and the pull on the string, besides lifting the weight to a higher average position, also has to overcome the resistance of the centrifugal force. That requires an extra input of energy from the force pulling the string, and since energy has to go somewhere, it makes the swing of the pendulum more vigorous.

    Incidentally, this process is related to the way children "pump" a swing to make it go higher. The child moves arms, legs and body in a way that works against the centrifugal force, and the energy invested in overcoming this force ends up producing a more energetic swinging motion.(This is a highly simplified explanation and assumes that from the point of view of the child in the swing, nature behaves exactly the same as anywhere else, only a centrifugal force is added. The actual situation can be more complicated.)

It turns out that the product T x E, the period T times the energy E, is almost a constant. It is not an exact constant, like total energy in a system, but if the rate of change is slow enough, e.g. if the string is pulled rather slowly, it comes very close.

The motion of electrons and ions spiraling around magnetic field lines is also periodic. While the period of a pendulum changes when its string gets longer or shorter, that of a spiraling ion or electron changes as it moves into regions where the magnetic field is weaker or stronger. Just as for a pendulum the product T x E stays very nearly constant, so here too, a certain quality, an "adiabatic invariant," is almost kept at a constant value. From that constancy it is possible to deduce the "mirroring" of particle and many other properties of their motion.

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Last updated March 13, 1999