Motion of Test Charge in EM Field [closed]

Question

I have a negative infinite sheet of charge moving at a velocity $v$ in the $+x$ direction. A test charge $Q$ with mass $m$ moves at a constant velocity $v$. My Question is simple: How will the test charge $Q$ move? Will it keep going in the $+x$ direction at a constant velocity $v$, as if no electromagnetic forces act on it? Will it go in a diagonal motion? Did I even get the Force-Body Diagram right? With Gauss' Law and a Cylinder as my Gaussian Surface & Ampere's Law and a Rectangle as my Amperian Loop, I've found that $$\begin{array}{l} \vec{F}=Q(\vec{E}+\vec{v} \times \vec{B}) \\ \vec{F}=Q\left(\frac{\sigma}{2 \varepsilon_{0}}-\vec{v} \times \frac{\mu_{0} \sigma_{s}}{2}\right) \rightarrow \vec{F}=Q\left(\frac{\sigma}{2 \varepsilon_{0}}-\frac{\mu_{0} v^{2} \sigma}{2}\right). \end{array}$$ Taking the integral gives me $\vec{r}(t)=\vec{y}(t)=\hat{j} \frac{Q \sigma t^{2}}{4 \varepsilon_{0} m}\left(1-\frac{v^{2}}{{{(\mu}_{0}{\varepsilon}_{0})}^{2}}\right)$

My equations don't meet my intuition -- shouldn't the positive test charge just slam into the negative sheet of charge due to the electrostatic attraction? Again, my main question is: How will $Q$ actually move?

Answer 1

You are actually thinking of two reference frames, not one frame of reference:

Reference Frame 1: where the charge and the plane are moving with velocity $\vec v$, as an observer, you are static with respect to the coordinate axes you drew in your diagram. In this frame, the charge experiences two forces, an electric force and a magnetic force.

Reference Frame 2: where the charge and the plane are static, as an observer you are moving with respect to the coordinates axes you drew in your diagram such that the charge and the plane look still...in better phrasing: you are moving with the charge and plane. In this frame, the charge experiences only one force; the electric force, there is no magnetic force in this frame.

The magnetic and electric forces are NOT the same in all reference frames. To calculate these forces in different reference frames, you need to study Special Relativity. Using special relativity, you will find that there is no ambiguity in the motion of the particle. The physical path described by the particle motion is the same, regardless of your frame of reference.

I haven't practiced relativity in a while, so I'd rather leave it to someone else who is better prepared. Please do not accept this as an answer, this is just clarification. I'd leave it someone else to fully answer your question.

Answer 2

I don't know why this is not consistent with your analysis, but it occurs to me that if you go to a coordinate system which is moving in the x direction with velocity v, then there is no magnetic field, and the test charge will accelerate toward the sheet.