A Paul trap that uses an oscillating quadrupole field to trap ions radially and a static potential to confine ions axially. The quadrupole field is realized by four parallel electrodes laying in the -axis positioned at the corners of a square in the -plane. Electrodes diagonally opposite each other are connected and an a.c. voltage  is applied. Using Maxwells equations, the electric field produced by this potential is electric field . Applying Newtons second law to an ion of charge  and mass   in this a.c. electric field, we can find the force on the ion using . We wind up with
Assuming that the ion has zero initial velocity, two successive integrations give the velocity and displacement as
where  is a constant of integration. Thus, the ion oscillates with angular frequency  and amplitude proportional to the electric field strength and is confined radially.
Working specifically with a linear Paul trap, we can write more specific equations of motion. Along the -axis, an analysis of the radial symmetry yields a potential
The constants  and  are determined by boundary conditions on the electrodes and  satisfies Laplace's equation . Assuming the length of the electrodes  is much greater than their separation , it can be shown that
Since the electric field is given by the gradient of the potential, we get that
Defining , the equations of motion in the -plane are a simplified form of the Mathieu equation,