**Problem**

Consider a uniform magnetic field directed in the x-direction, and a uniform electric field directed in the z-direction. A particle with positive charge and mass is released from the origin, and initially at rest. Determine the trajectory of the particle over time.

**Solution**

The magnetic force is calculated using the cross-product

- .

Subsequently the Lorentz force, , in vector form is

Divide the mass of the particle on both sides of the equation. Let and , then

- .

To solve this system of coupled differential equations, let . Thus

- .

Substitute the middle equation, into the bottom equation to obtain

- .

The above differential equation is non-homogeneous, thus the solution is the sum of the homogeneous solution and particular solution.

(1) Homogeneous solution

The characteristic equation is , where .

The solution is therefore

or

- .

(2) Particular solution Since is constant, if , then . Then the differential equation

reduces to

- .

Hence .

Consequently, the solution of the differential equation is

- .

Since the particle is initially released at rest,

To obtain the trajectory,

Since the particle is initially released at the origin,

Replacing and with its original quantities

- .

The solution of displacement presents a cycloid trajectory, which is usually given in parametric form as

- .