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## Homework Statement

This is a four part problem -- the only issue I have is on part (c) so I'll condense the question:

If the equation of motion of a particle of charge [tex]q[/tex] in an electric field and magnetic field is

[tex]qv_y B \hat{\mathbf{i}} + \left( qE_y - qv_xB \right) \hat{\mathbf{j}} + qE_z \hat{\mathbf{k}} = m \frac{d}{dt} \left( v_x \hat{\mathbf{i}} +v_y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}}\right), [/tex]

obtain expressions for [tex] v_x \left(t\right)[/tex] and [tex] v_y \left(t\right)[/tex]. Show that the time averages of these velocity components are

[tex]\left\langle v_x\left(t\right)\right\rangle = \frac{E_y}{B}[/tex]

and

[tex] \left\langle v_y\left(t\right)\right\rangle = 0[/tex]

(Show that the motion is periodic and then average over one complete period.)

## Homework Equations

## The Attempt at a Solution

Solutions to the DE are

[tex]v_x \left(t\right) = \frac{E_y}{B} + C_1 \cos \left(\omega t\right) + C_2 \sin \left(\omega t\right)[/tex]

and

[tex] v_y \left(t\right) = -C_1 \sin \left(\omega t\right) + C_2 \cos \left(\omega t\right)[/tex],

where [tex]\omega[/tex] is the cyclotron frequency. Now, clearly the velocity functions are periodic with period [tex]2\pi/\omega[/tex]. What do they mean "time averages," and what does the [tex]\left\langle \right\rangle[/tex] notation mean?

Perhaps,

[tex] \left\langle v_x\left(t\right)\right\rangle = \frac{ v_x\left(0\right) + v_x \left(2\pi/\omega\right)}{2}[/tex],

like an arithmetic mean, but I'm just guessing based on the hint (and that doesn't yield the correct answer). Do they mean average velocity?