# Magnetic mirrors

1. Feb 22, 2005

### mcciysm3

I'm trying to construct a model for an electron trapped in a magnetic mirror, but i just cant get my head round why the electron stops and changes direction if the field doesnt affect the parallel velocity (lorentz force), anyone got any ideas?

2. Feb 23, 2005

### Davorak

Lets say that we have an electron moving in the direction of the magnetic field and into an area of increasingly strong magnetic field. The electron is orbiting the magnetic field lines because of the vxB force. The electron’s orbit causes a dipole magnetic field. If you work out the direction of orbit and find the dipole you will find that the dipole moment points in the opposite direction of the magnetic field. The force between a magnetic field and a magnetic dipole is:
$$F= \nabla(\vec{m}\cdot\vec{B})$$

The force points in the opposite direction as the gradient of the magnetic field times the magnetic dipole. This is the force that will retard, stop, and reverse the moment of the electron.

Let use define the positive x axes pointing with the magnetic field in the above example.

In the case above the $\vec{m}$ points in negative direction and $\vec{B}$ points in the positive directions, therefore the force is in the negative directions. Away from the increasing magnetic field.

Let us now switch the direction of the magnetic field. Switching the direction of the magnetic field switches the direction of the dipole moment of the orbiting electron. This also leads to the electron moving away from the area of stronger magnetic field.

Does this make sense?:

As the magnetic field gets stronger the electron spins faster and faster in its Larmor orbit. The parallel kinetic energy is transferred into the orbital speed. The increasing magnetic field strength exactly cancels the increased orbital speed so that the Larmor radius stays the same.

3. Feb 23, 2005

### mcciysm3

yes it does, thank you very much