Force on a particle in response to an EM wave

[Dishsoap]

Homework Statement

Consider a particle of charge q and mass m , free to move in the xy plane in response to an electromagnetic wave propagating in the z direction. Ignoring the magnetic force, find the velocity of the particle, as a function of time. Assume the average velocity is zero.

Homework Equations

$E(z,t) = E_0 cos(kz-wt) \hat{x}$

I've omitted the phase constant here since I took it to be zero.

The Attempt at a Solution

Initially, I tried saying dv/dt = F/m = qE/m, where E is given as above. But I'm not quite sure what to make out of the fact that the average velocity is zero. I suppose this means it moves as a closed loop, so I have no idea what to make of this.

 

[mfb]

As force does not depend on the velocity, you can ignore the average motion for now and find "some" solution. Then you can modify that to get the average motion right, if necessary.

 

[Dishsoap]

The fourth part of the question says this, though:

"The problem with this naive model for the pressure of light is that the velocity is 90 degrees out of phase with the fields. FOr energy to be absorbed, there's got to be some resistance to the motion of the charges. Suppose we include a force of the form - gamma m v, for some damping constant gamma. Repeat part a (ignore the exponentially damped transient).

I feel like I'm doing this wrong then. Not only did I not use the pressure of light (since I don't have an area), but I don't get an exponentially damped transient. Even if I include the force term, I then end up with something of the form v' + gamma * v = Eq/m. I tried solving this with the integrating factors method, but I do not get an exponentially damped transient, but an exp(gamma*t) term.

 

[mfb]

Start with the undamped case to see how the motion looks like, you can introduce damping afterwards.

If you get exp(gamma*t) instead of exp(-gamma*t) there is a sign error.