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