# I Particle movement in a non-static magnetic field

Tags:
1. Jan 11, 2019

### Javier Lopez

Particles follows accordingly the general equation:

$$m*\overrightarrow{a}=q*\overrightarrow{E}+q*\overrightarrow{v}\wedge \overrightarrow{B}$$

But in the case of two coupled coils the time varying current at primary coil (and its magnetic field variation) creates corresponding varying current in the secondary coil

To calculate it I use the magnetic flux conservation like in transforrmers where the plasma is the secondary coil but it does not work well to calculate the movement of each particle

Then, what formula is better to use to calculate the particle movement?

Accordingly Lorentz applied to a loop (where the ion describes a circled current):
$$Fem=\oint \overrightarrow{E}\overrightarrow{dl}=-\frac{d\overrightarrow{\phi }}{dt}\\\\ 2\pi r*\overrightarrow{E}=-\pi r^2*\frac{d\overrightarrow{B}}{dt}\\\\ \overrightarrow{E}=-0.5*r*\frac{d\overrightarrow{B}}{dt}$$
So:
$$m*\overrightarrow{a}=q*\overrightarrow{E}+q*\overrightarrow{v}\wedge \overrightarrow{B} - \frac{1}{2}*q*r* \frac{d\overrightarrow{B}}{dt}$$

Where r is the Cyclotron radius. That is only true for slow varying magnetc fields because the ion does not describes a circle but an helix

Is it correct?

Last edited: Jan 12, 2019
2. Jan 15, 2019

### Staff: Mentor

Fun question, but unfortunately, for other reasons this user will not be returning to the PF. Thread is closed.