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*From*: Carl Mungan <mungan@usna.edu>*Date*: Sun, 24 Feb 2013 18:08:14 -0500

I find the following exercise to be interesting without being overly

complicated.

Let's start with a slightly messy real-world application, and then

simplify it. My motivation was that I wanted an explanation of how

an electrophorus works. I found a whole bunch of bogus explanations.

There may be some good explanations out there, but they seem to be

hopelessly outnumbered by bogus explanations. This version is typical:

http://en.wikipedia.org/wiki/Electrophorus

In particular, the diagram shows an unbalanced negative charge on

the dielectric ... but it does not show the corresponding positive

charge. There must be some electric field lines that end on the

dielectric, but it is impossible to guess how those lines are

arranged because we have no idea where the counterelectrode is.

For today, let's not attack the full real-world problem. Instead,

let's do a warm-up exercise. Consider the situation in the following

diagram:

http://www.av8n.com/physics/img48/pre-electrophorus.png

There is a room with conducting walls (shown in blue). The floor

of the room is X by X. The height of the room is Y. Centered over

the middle of the floor is a thin metallic disk of radius r (shown

in red). The disk is horizontal at a height h above the floor.

There is a charge Q on the disk ... and correspondingly a charge -Q

on the walls of the room. The disk is not very large, and the room

is much wider than it is tall, so that X ≫ Y ≫ r.

Your mission, should you decide to accept it, is to find an algebraic

expression for the voltage on the disk, as a function of h. Voltage

is measured relative to the walls, which we take to be effectively

"chassis ground".

The disk has an insulating handle, so you can change h arbitrarily

without changing Q.

Make whatever reasonable simplifying assumptions you like. (Replacing

the conducting walls with non-conducting walls would not be considered

reasonable.)

As a specific numerical instance, set

X = 10 m

Y = 2 m

r = 0.1 m

h = 1e-5 m initially

Charge up the disk to 1.5 volts using a battery, and then disconnect

the battery. Raise the disk (maintaining constant Q during phase of

the process). What happens to the voltage?

The usual jsd puzzle rules apply: Everything I have said is (to the

best of my knowledge) true and non-misleading, although some details

may be irrelevant. OTOH I have not told you everything I know. I

have not told you the answer or even the method of solution ... and

more importantly, the problem may be underspecified.

Again, the point is that after doing this warm-up exercise you should

have a fighting chance of figuring out how an electrophorus works.

Wow, that's absolutely delightful. Please, please consider writing this all

up for The Physics Teacher once all is said and done here on PHYS-L.

I'll take a stab at starting your exercise. (As usual, an advantage of the

Digest mode is I don't get biased by seeing what anyone else tries until

tomorrow.)

Initially there's -Q on the floor right below the +Q plate and the

potential difference increases linearly as V = Q*h/A*eps0 where A = pi*r^2.

Physically, this voltage increase happens because I have to do work to pull

the positively charged plate away from the negatively charged floor.

However, as the plate moves away from the floor, the negative charges there

start moving away from the region directly below the plate and start going

up the walls. Electric field lines spread out and we have an electric

dipole, with an image charge beneath the floor.

Eventually, when the plate reaches half height, h = Y/2, then V must level

off by symmetry. Thereafter, V will decrease as h increases because it gets

attracted to the greater amount of charge on the ceiling than the floor.

So one might expect something like a second-order Taylor series, resulting

in the parabola:

1 - V/V0 = (1-2h/Y)^2

where V0, the maximum potential at h=Y/2, is Q*Y/4*A*eps0. (The factor of 4

was chosen to properly match dV/dh at h=0.) -Carl

--

Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)

Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363

mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/

**Follow-Ups**:**Re: [Phys-L] amusing electrostatics exercise***From:*John Denker <jsd@av8n.com>

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