Re: Resistor paper

[kowalskil <kowalskil@MAIL.MONTCLAIR.EDU>]

Roger Haar wrote:

> HI,
>         We do this with the terminals at (7,10)  and
> (23,10) and our equipotentials look similar to
> yours.

Your left dot is 7 cm from the center of the paper and
your right dot is 9 cm from the center of the paper.
You are probably correct that the size of the paper
may become significant for that geometry.

In my geometry each dot is 5 cm from the center of
the paper and I know (see below) that, for this geometry,
the paper already behaves like an infinite sheet. What I
might do is to reduce the distance between the dots from
10 to 8 or 6 cm. If the size of the paper does have an
effect of the shape of equipotential lines than new geometry
should reduce the discrepancy between the data and the
theory.

Meanwhile I would very much appreciate if you could
make measurements with my geometry and describe the
outcome for the line passing through DOP=(10,16).
By the way, I just repeated the experiment on another
sheet and got the same line. This time I was using an
electrostatic voltmeter (R>10^10 ohms).

The reason I am confident that the sheet size is not a factor
is described below; it was posted several days ago.
Ludwik Kowalski
**********************************************

****    Do you remember this question?
What is R between the two identical silver circles, each
of diameter 2*a=1 cm) separated by the distance of 20 cm?
Suppose the carbon paper thickness d=0.2 mm and its
rho=0.00001 ohm*m.

****   It was answered by David Bowman:

           R = (rho/(pi*d))*arccosh(L/(2*a)).

In this example:
rho/(pi*d)=0.00001/(pi*0.0002)=0.0159
L/2*a=20/1=20 --> Arccosh(20)=3.688
R=0.0587ohms
[ By the way, if x is larger than ~5 then
Arccosh(x) is nearly the same as ln(2*x).]

It turns out that the value of rho (for the carbon impregnated
paper from Pasco) is much larger than in the above hypothetical
illustration; is 0.29 ohm*m. I measured it today. Let me tell you
what I did; a student project can be conceived along these lines.

1) two circles of diameter 7.5 mm were silver-painted; the
distance between their centers was 10 cm. The measured
resistance turned out to be 27,800 ohms. Using the micrometer
the thickness was found to be very uniformly equal to 0.013 mm.

2) But can a 35 cm by 23 cm sheet can be considered as infinite
when the midpoint between the silver circles is the in the center
of the sheet? To answer this question experimentally I cut 2 cm
strips along each margin and remeasured R. No significant
(larger than ~2%) change in  R was observed. Then I cut
additional 2 cm along each margin and found R=29,400 ohms.
Another cut of 2 cm along each margin resulted in R=32,200
ohms. These numbers convinced me that the full size sheet is
large enough to use David's formula. This formula produced
rho=0.33 ohm*m.

3) The carbon paper was finally reduced to a narrow strip of
7.5 mm between the centers of two silver circle. The resistance
of this strip was 260,000 ohms. The value of rho calculated
from this R turned out to be 0.315 ohm*m. I consider this to
be a good experimental confirmation of David's formula.