303 Pages

## 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  .
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