Lorentz force and Newton's law

[v_pino]

Homework Statement

This problem asks you to work out the dielectric function of a gas of particles with number density n, charge q, and mass m, with a steady magnetic field applied in the z direction.

Assume an electric field in the x direction,

$$E_x(t)=E_xe^{-i \omega t}$$

is applied. Write down the x and y components of the Newton’s Law using the Lorentz force equation and no damping. Assume a solution for the velocity of the form,

$$v_x(t)=v_{x0}e^{-i \omega t}$$

and

$$v_y(t)=v_{y0}e^{-i \omega t}$$

Solve for v_x0 and v_y0 in terms of E_x and the cyclotron frequency,

$$\omega_c = qB/m$$

Homework Equations

$$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$

$$\mathbf{F}=m \mathbf{a}$$

The Attempt at a Solution

$$m \frac{d \mathbf{v}}{dt}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$

$$\frac{dv_x}{dt}=-i \omega v_{x0}e^{-i \omega t}$$

$$\frac{dv_y}{dt}=-i \omega v_{y0}e^{-i \omega t}$$

I am having trouble pulling all these equations to write out the components of Newton's law.

 

[darkys]

I am not sure how to solve for v_x0 and v_y0 in terms of E_x and the cyclotron frequency. Any help would be greatly appreciated!