Insight needed into electromagnetism problem

[Finestructure]

The problem (From Electromagnetism by Pollack and Stump, exercise 11.34)
A stationary charge of charge e and mass m encounters a electromagnetic wave with vector potential

$$\vec{A}=\vec{j}f(x-ct)$$

where
$$\vec{j}$$

is the unit vector in the y-direction. The scalar potential is zero.
What are the components of the velocity, as a function of time,

$$v_{x}, v_{y},v_{z}$$ ?

What I´m asking for is some insight into why the y-velocity is not a more complicated function. The y-velocity is:

$$v_{y}=\frac{e}{m}f(x-ct)$$

More specifically, when the charge is stationary, it will begin to move in the y-direction because of the changing vector potential but the motion in the y-direction will result in the magnetic field causing a motion in the x-direction which in turn should (?) effect the y-velocity, but apparently it does not. Why?
I wrote the equations of motion but they don´t seem to give any insight into the y-motion. I determined the equations of motion by starting with the following equations:

$$\vec{E}=-\frac{\partial A}{\partial t}$$

(In the equation above, A is a vector)

$$\vec{B}=\nabla X \vec{A}$$

$$\vec{F}=e(\vec{E}+\vec{v}X\vec{B})$$

The function f(x-ct) isn´t a problem if we use the chain rule

w = x -ct

$$\frac{\partial f}{\partial t}=\frac{df}{dw}\frac{\partial w}{\partial t}=-c\frac{df}{dw}$$

likewise,

$$\frac{\partial f}{\partial x}=\frac{df}{dw}\frac{\partial w}{\partial x}=\frac{df}{dw}$$

Note, I´ve been a little sloppy about notation regarding partials and full derivatives.
The resulting equations of motion are:

$$\frac{dv_{x}}{dt}=\frac{e}{m}v_{y}\frac{df}{dw}$$

$$\frac{dv_{y}}{dt}=\frac{e}{m}(c\frac{df}{dw}-v_{x}\frac{df}{dw})$$

$$\frac{dv_{z}}{dt}=0$$

Thus the z-component of the velocity is zero since it started at rest.

The equations do give the right answer for the x-velocity if I assume the value of the y-velocity. But they aren´t helpful in giving me the y-velocity. Note that the two equations involving vy and vx have an additional unknown df/dw. I´ve tried everything, even eliminating df/dw and getting an equation that appears to have just x-dependence on one side and y-dependence on the other which one could solve by setting each to a constant but that doesn´t seem to work either.

 

[Physics Monkey]

You have your vector potential pointing in the z direction so the electric field points in the z direction also, yes? How is the acceleration in the z direction zero then? Also, the magnetic field is in the y direction, so the y velocity can't be affected by either the electric or magnetic field, right?

I think you have some confusion about which way everything is pointing though I suspect this is just a typographical error.

Here is a hint for when you sort it all out: what is $$\frac{d}{dt} f(x(t)-ct)$$?

 

[Finestructure]

Thank you so much for your reply, Physics Monkey.

You are correct about the direction of the vector potential, I made a mistake in its direction. It should be in the y-direction and I've edited my original post to correct it.

I've also gone back and replaced full derivatives which should have been partial derivatives. I think the derivatives for the velocity components should be full derivatives.

I have a feeling that you may be right about considering the x-dependence of time because when the charge does move in the x-direction, it no longer "feels" the same field and this might account for the charge not experiencing further effects in the y-direction due to motion in the x-direction.

Thanks again.