(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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,

[tex] E_x(t)=E_xe^{-i \omega t} [/tex]

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

[tex] v_x(t)=v_{x0}e^{-i \omega t} [/tex]

and

[tex] v_y(t)=v_{y0}e^{-i \omega t} [/tex]

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

[tex] \omega_c = qB/m [/tex]

2. Relevant equations

[tex] \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B}) [/tex]

[tex] \mathbf{F}=m \mathbf{a} [/tex]

3. The attempt at a solution

[tex] m \frac{d \mathbf{v}}{dt}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B}) [/tex]

[tex] \frac{dv_x}{dt}=-i \omega v_{x0}e^{-i \omega t} [/tex]

[tex] \frac{dv_y}{dt}=-i \omega v_{y0}e^{-i \omega t} [/tex]

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

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# Lorentz force and Newton's law

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