# A charge accelerating while confined to an equipotential surface

Suppose I had the following in Cartesian $$(x,y,z)$$ space, where $$x$$ is horizontal, $$y$$ is vertical, and $$z$$ is depth:

1. A negligibly thin conductive rod extending from $$(0,1,0)$$ to $$(0,+\infty,0)$$.

2. An electric charge at the neighborhood of the origin $$(0,0,0)$$ accelerating on the x-axis in the positive x-direction $$(\frac{1}{2}at^2+v_{\text{initial}}t, 0,0)$$ where $$(\frac{1}{2}at^2+v_{\text{initial}}t) \ll 1$$.

My objective here is to evaluate the consequences that changing the x-velocity of the charge has on the amount of y-work done on the conductive rod.

It appears to me that the faster I move the charge, the amount of work that can be done along the rod increases with the Lorentz factor of the moving charge. By choosing a rod with negligible thickness, we can neglect any work done by the transverse electric field associated with an accelerating electric charge.

This leaves us to evaluate the work done along the y-axis, whose effect on the conducting rod is to polarize it along the y-axis. The electric field acting along this rod's length is subject to change by the accelerating charge near the origin and increases by the charge's Lorentz factor. This induces a changing electric field which the charges in the conducting rod will attempt to screen. This change results in the flow of electrical current along the rod. Due to the thinness of the rod, the electric field produced by the changing current in the rod is cylindrically symmetric around the y-axis. Thus, the x and z components of the electric field of this rod are both negligible near the y-axis. As a result, the electric field due to the induced polarization applies a force on the charge near the origin but without any significant net work being done on it, as this (predominantly) y-force would be applied right angles to the x-velocity of the charge. Furthermore, the charge near the origin would be moving essentially parallel to an electric equipotential surface from the charge distribution induced in the rod. So it appears negligible work is required to change the velocity of the charge near the origin. The work would predominantly be that which depends on the charge's mass.

Given these constraints, how may we generate an x-resistance force on the charge that we try to x-accelerate arbitrarily near the origin that is somehow relative to the y-work done on the charges on the rod whose electric field response is "y only" at the y-axis? Does this ultimately have something to do with the potential's contribution to the mechanical momentum of the charge? And if so, does it mean a charge can experience a "forceless" acceleration due to a changing potential (due to the changing polarization of the rod) which effectively alters the mass of a system charges which it is a part of?