Considering a particle of charge

and mass

, subjected to an electric field

,

,

the resulting force

is given by the Lorentz force:

.

In Cartesian coordinates the position vector is

,

.

In this case, Newton's second law,

,

.

Substituting the vectors, we obtain

.

Doing the cross products and rearranging terms, we obtain:

;

;

.

These are coupled second-order ordinary differential equations, which can be solved by either analytical or numerical methods. In this system, if

, all three equations have the same solutions with constant acceleration. If

, all three solutions have constant velocity. Considering

,

,

,

,

and

nonzero, the equation in the

direction is the simplest to solve, since there are only constants. Rearranging, we obtain

,

,

,

where

and

are constants given by the initial conditions

and

, for velocity and position in the

direction, respectively. The other two equations are coupled and thus must be solved together, which means they require a more complicated method to obtain their solutions.

It is possible to solve these equations by multiplying the one in the

direction by the imaginary unit

and adding it to the equation in the

direction. Doing this leads to

.

Dividing this equation by

and considering

and

, it is possible to write this equation as

,

which is a first-order linear differential equation for

. By using the integrating factor method, its solution is

,

where

is an arbitrary constant that can be determined by the initial condition

. In fact, this constant is

,

.

It is possible to decouple the equations in the

and

directions by introducing the constant

that obeys

.

,

and using the relation

, we obtain

.

From the real part of this equation, we have

and from the imaginary part,

.

These equations can be solved by direct integration, which leads to

,

where

and

are arbitrary constants. By the initial conditions

and

, we obtain

.

Using these constants and remembering that

, we obtain the final solutions

,

,

.

Thus the problem is solved analytically, to give solutions that are plotted in 3D Cartesian coordinates.

[1] J. D. Jackson,

*Classical Electrodynamics*, 3rd ed., New York: Wiley, 1999.