The Radiation Belts!
Forces and Motions:
How do particles move in a magnetic field?
The answer is rather complicated, and is usually covered in college-level physics courses. Entire books have been written about this subject alone, and physicists have studied this technical problem for over 100 years. A deep understanding of how particles move in magnetic fields requires advanced mathematics such as calculus to really understand the details.
Charged particles (electrons and protons for example) experience magnetic forces very intensely. In fact, near Earth, the magnetic forces are far more important for charged particles than gravity!
Charged particles will tend to move in at least one of three ways, and very often they will do all three at once!
Gyration -
This is similar to the particle moving along the loops in a Slinky toy, with the Slinky stretched along a line of magnetism. That's the motion you see in the above figure. As the strength of the field increases, the gyrations decrease in radius making smaller and smaller circles (orbits) around the magnetic line of force. This motion is very fast. Electrons in the Inner Van Allen Belt, for example, can travel at more than 200,000 kilometers per second in their tiny orbits. The orbits can have a radius from 10 to 100 kilometers. The more energy a particle has, the faster it will travel, and the larger will be its 'gyro-radius'. Typically, the electrons gyrate about 30,000 cycles per second.Bounce -
As the particle gyrates, it will also move up or down along the magnetic field line. It will then turn-around and travel back the other way. It can do this many times until it eventually collides with an oxygen or nitrogen atom in the atmosphere and is absorbed. This motion is much slower than gyration. For example, an electron in the Inner Belt can take a second or so to travel one full bounce. The bounce frequency is about 10 cycles per second and the speed is about 10,000 kilometers per second.Drift -
As the particles gyrate and bounce from pole to pole in the magnetic field, they will also drift to the west around the Earth if they are positively-charged (protons and ions), and eastward if they are negatively-charged (electrons). This motion is very slow compared to gyration. It takes electrons in the Inner Belt about 20 minutes to drift once around Earth's equator in the Inner van Allen Belt. This speed is only about 40 kilometers per second.When trillions of these particles are trapped by Earth's magnetic field, they individually gyrate, bounce and drift and through their collective motions form the donut-shaped Inner and Outer Belt systems. This is why artists show them as smooth clouds with this shape!
If you want to explore the details of how particles actually move in magnetic fields, here are several technical references:
SPENVIS General discussion at an advanced level. No math.
General Introduction - College-level, no math
Particle Drifts - General, no math.
Dr. Chris Owen's Introduction to Space Plasma Physics Illustrated slide show.