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!