Problem

Em4-1.gif

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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