Re: Motion within an electrical potential

Andrew,

As you correctly deduce at the end of your second paragraph, any electrical force on an object in a static electric potential is related to the gradient (or, if you like, the "3D derivative") of the scalar electrical potential. This gradient (also called the electric field) is a vector, having both magnitude and direction, and the actual force vector would be:

F = q*E = -q*grad(V)

where the force F and electric field E are vectors, q is the charge on the object you're considering, V is the electrical potential function (or "voltage"), and grad() is a shorthand way of referring to the vector gradient of whatever is in the parentheses. These equations can be found in most any college physics textbook, and certainly in any calculus-based general physics text.

Back to your first question, though: what would be the effect for a non-charged object drifting through a uniform electrical potential? And the answer is, exactly nothing: the local electric field will be zero, since the potential is uniform (gradient=0), and the charge q=0 by hypothesis.

Second question: what if the moving object were charged? As far as electrostatics is concerned, there would still be no effect, since there is no electric field. So, there is no electrostatic force. However, the moving charge does count as an electric current (in a sense), so there may be image currents induced in the surrounding conductor, which in turn could put a force on the charged object.

Similarly, in your third question, what if the charged object accelerated? Then, as described in your college physics text, there will be electromagnetic radiation produced, in addition to the peculiar joy of figuring out the magnetic forces due to image currents, etc etc etc. A detailed description of the motion of the charged object will depend strongly upon the details of the problem: what is the shape of this surrounding conductor? How far away is the conductor, relative to the space and time scales of interest? Is the conductor "perfect", or does it have finite resistivity? Is there a vacuum magnetic field penetrating the "hollow" region? Depending upon the answers to these questions, the effects upon the accelerating charge could be either immense or negligible.

I encourage you to keep reading, and to read more carefully: the "thought experiment" of a charged particle in a hollow conductor (charged or uncharged) is standard fare in college physics texts, in the chapters explaining electrostatics and the electrostatic potential (or "voltage"). The problem of a moving charged particle in a hollow conductor is NOT standard fare, precisely because it is a very hard problem to describe, and setting up and solving the problem can depend entirely upon the "boundary conditions" I outlined above (shape of the hollow volume, time and length scales, and so on).

Good luck!

Aaron