Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

how many volts?



On 4 Apr 1997 13:47:13 John P. Ertel wrote:

As noted in David Jackson's E&M text, the equilibrium charge density is

q 1
sigma(R) = ------ * --------------- , for all R<a
2 pi a Sqrt(a^2 - R^2)

where a is the radius of the disk, R is where you are on the disk, and
the thickness of the disk is assumed to be negligible as compared to a.

I appreciate his effort of typing the formula (and an interesting
paragraph that follows it). This is a good example for those who think
that references they have are easilly available to all phys-L-ers.
We should not forget that the range of our backgrounds, our teaching
loads, and our resources, are very different.

..........................................................................
It is infinitly better to discuss something when we have a formula than to
deal with purely numerical results. At this time I would like to ask a
question about the problem discussed under the "how many volts?" thread
but simplified by making |Q1|=|Q2|.

Disks are initially very far from each other, in an empty space. Positive
disk (upper) is a source of E lines while the negative disk (lower) is a
sink. The number of lines per unit area, according to the above formula,
inreases with the radius of each disk. Note, for example, that at R=0.95*a
the value of sigma is 3.2 times larger than in the center. And it is 7.1
times larger when R=0.99*a.

We start bringing disks toward each other (very slowly). More and more
lines originating on the upper plate are intercepted by the lower plate.
For |Q1|=|Q2|, the generated and intercepted fluxes are identical at any
separation. But the final radial distributions of charge densities change
from non-uniform to uniform. Why does this have to happen? How are the
non-uniform distributions incompatible with E=0 inside plates? I can draw
a picture with the E field lines, parallel to the axis, which are less
crowded near the center and more crowded near the rims. What prevents
nature from arranging charges in that way?

One way to answer this question is to calculate the potential energy (sum
of all the k*dQ1*dQ2/r terms) and to show that TWO uniform distributions
do correspond to the lowest possible energy. (This is not true for the
single plate distribution.) Can the question also be answered in a more
simple, semi-quantitative way?

...................................................................
: Gedanken-ing is not enough; physics is an experimental science! :
: Inspired by thinking about phys-L messages on capacitors :
: Ludwik Kowalski :
...................................................................
kowalskiL@alpha.montclair.edu http://www.csam.montclair.edu/~kowalski