Re: Capacitor and Force


"Jonathan Kirwan" <jkirwan@xxxxxxxxxxxxxx> wrote in message
news:62r8h3hsbc8cdc5mvb5fmg3kbr4jlculnd@xxxxxxxxxx
On Tue, 16 Oct 2007 04:51:13 GMT, "Jon Slaughter"
<Jon_Slaughter@xxxxxxxxxxx> wrote:

<snip>
But the main point is, I'd like to see you find the closed form solution
to
it by hand since, again, you seem to think computing integrals by hand is
easy. Remember, its not the same on that page because they did not
compute
any integeral for the general case of two 2d parallel plate... or, if you
think they did, then do it for arbitrary charge densities.. doesn't quite
matter because you say they are easy.

Jon, I'd already thought just a little bit about the integral before
looking at the web page that The Phantom pointed at and had already
come to exactly the same approach that the web page mentions, using
annular rings with an infinitesimal width and integrating, as the
right simplifying approach. I didn't go further, as their description
about the approach matched my own initial impression about it and the
solution they gave appeared, without spending time, to be correct.

Do you have any reason to imagine that the author of the page didn't
actually solve the integral correctly? Why also do you bring up the
complicating idea of 'arbitrary charge densities' when, in the case of
idealized parallel plates I'd imagine that the charge was distributed
to minimize the required work and thus uniformly spread (except
perhaps at the edges of the plates, where closer thinking may be
required, but probably wouldn't much impact the final result?)


Yes I do. Becuase it says it there that it assumes E is always perpendicular
to the plate... and the integral is probably impossible to solve in closed
form. By making assumptions you can get away with a lot. Integrals, in
general, are impossible to solve in closed form. If the geometry has some
symmetry involved then you can usually simplify a great deal.

quoting from that site...(the paragraph nex tot the "field and surface
charge" figure,

"since we assume that the only variation is in x, d2?/dx2 = 0."

This means they are neglecting fringe effects. Why? Because then it makes
the problem possible to actually do. It takes 2 integrals instead of 4.

I am not saying he solved it correctly or not. I'm saying he made
assumption, just like I originally made, to make the problem easier... which
is exactly what I tried to do at the start knowing how complicated the
integrals are(actually I tried to integrate it using maple but since I got
no result I assumed it was probalby impossible). My assumption was for a
very rough estimate based on a very weak assumption.

All I'm getting at is that he made an assumption and didn't carry out the
full analysis and there is a very good reason for that. Phantom is trying to
make it seem like its an easy thing to do the full blown volume or surface
integrals and its not... its most likely impossible and one can only get a
reasonable answer if they make some valid assumption. I made an assumption
but I make a mistake in applying it. We are not really even talking about my
original mistake any more but the fact that assumptions have to be made to
do most of the work. In fact, almost all of physics is based on
assumptions... even mathematics uses assumptions constantly.

I am not at all arguing with the method the site uses to prove anything.
Hell, that site doesn't do what phantom claims it does. It doesn't carry
about the full blown general case without simplifying assumptions. But
Phantom claims one doesn't need to make assumptions(at least he claims that
the volume integrals are what one should really work with and has said that
I was wrong for not using them(in fact I did used them when I first
approached this problem and quickly realized it was either impossible to to
hard so I tried to make some assumptions for an estimate).


You mentioned you were off by a factor of two (I'm not accepting or
rejecting that.) The web page that The Phantom mentioned said that
folks often jump initially to imagine a result that is, in fact, a
factor of two off the actual mark. Is that coincidence?


Possible, I don't know. I might have been wrong in that too. I've been
making so many mistakes lately that I don't know what is right or wrong any
more ;/

But he says its from reasoning about the electric field which I do not
do(ok, I guess I do implicitly but he does the same in his following
derivation). Even a factor of 2 is not bad though ;) I wasn't looking for
exact answer(because no one will find it)... I was looking at a way to get
some idea of the force. My mistake was telling me it was much larger than it
was and I knew something was wrong because I did some experiments and it
didn't follow. A factor of 2 isn't going to be that big a difference for the
experiement because I would have seen some results... even a factor of 10
would have given something.






.

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