# Particle Motion in a Penning Trap

A Penning trap is a device that confines charged particles using an ideal quadrupole electric field and a transverse magnetic field. In this Demonstration, a small (∼10 cm) Penning trap confines a single antiproton. With this setting, the stability condition is roughly , where is a magnetic field strength in and is the potential difference in V between two electrodes (the blue and orange surfaces).

### DETAILS

Electric quadrupole potential energy for a particle with mass and charge is given by
where .
Under this potential and transverse magnetic field , the equations of motion for a charged particle are given by
,
,
, where .
The third equation is decoupled, thus the particle's motion is sinusoidal in the direction. We can exactly solve the coupled , equations by introducing the complex variable . In terms of , we can write the first two equations as a single equation involving only :
.
The general solution to this equation is
, where and .
From this solution, we can see that the motion of the particle is stable (confined in a finite spatial region) when .
For an antiproton, this condition corresponds roughly to , where is measured in and in V.
References
[1] C. Gignoux and B. Silvestre-Brac, Solved Problems in Lagrangian and Hamiltonian Mechanics, London: Springer, 2009.
[2] R. L. Tjoelker, "Antiprotons in a Penning Trap: A New Measurement of the Inertial Mass," Ph.D. thesis, Harvard University, Cambridge, 1990.

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