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Re: Flat conductors (was I need help).

John's investigation forced me to collect more accurate data.
What I was trying to do before was to show a discrepancy
between what would be expected for a sheet of infinite
size and what is actually observed on Pasco sheets. For that
purpose my data were good enough. But John is doing
exact calculations (see below) and for this he might need
better data. Here is what I did this morning.

New Pasco sheet, same geometry as before (10 cm between
two small silver-painted circles). This time I applied 300 V
and measured the current of 5.5 mA. Then I quickly measured
the DOPs (with respect to the left circle) along the line between
the silver circles. Keep in mind that coordinates of points to
which the voltmeter lead was applied could not be determined
more accurately than about 1 mm.

d (cm) 2 4 5 6 8 10
DOP (V) 86 129 148 168 212 300

Then I selected four point on the axis and traced the equipotential
lines through them. In terms of Pasco coordinate the center of the
left silver dot was at x=9 cm and y=10 cm. The center of the right
dot was at x=19 cm and y=10 cm.

Point A, chosen at (x=14.5, y=10 cm) DOP=159 volts.
The line passing through this point also passes through:
(15, 4.5) (15.3, 3), (15.5, 2) (15.7, 1) (15.8, 0) and about (16, -1).
The last point is at the paper boundary.

Point B, chosen at (x=15, y=10 cm) DOP=168 volts
The line passing through this point also passes through:
(15,2 8) (15.7, 6), (15.6, 4) (17, 3) (17.5, 2) (17.8, 1) (18, 0)
and about (18.1, -1). The last point is at the paper boundary.

Point C, chosen at (x=16, y=10 cm) DOP=187 volts
The line passing through this point also passes through:
(11,1 1) (16.3, 8), (15.6, 4) (16.8, 5) (17.7, 6) (18.6, 5) (19, 4.7)
(20, 4) (21. 3) (22, 2) (22.8, 1) (23, 0.5) (23.5, 0) and
about (24, -1). The last point is at the paper boundary.

Point D, chosen at (x=17, y=10 cm) DOP=212 volts
The line passing through this point also passes through:
(17,2 1) (18, 8), (19, 7.5) (20.5, 7) 22, 7.2) (23, 7.9)
(24.3, 9) and (24.5, 0). This line closes on itself before
going to the paper boundary. It is not a perfect circle, as
it would be expected for the infinite sheet

Only one quadrant is shown but I did occasionally check
for the symmetry while fishing for equioptential locations.
and each time I checked I was satisfied. It would take me
only 20 minutes or to provide you with an additional
line, or a data point, if needed.
Ludwik Kowalski

John Mallinckrodt wrote:

I told Maple about David's skepticism regarding my 5 image dipole
solution to the conductive paper dipole experiment. See

<http://mailgate.nau.edu/cgi-bin/wa?A2=ind0202&L=phys-l&F=&S=&P=103852>

and it responded with a series of results of the following form

... - - - ...

... + + + ...

-----
| |
... + | + | + ...
-----
... - - - ...

The box shows the region in which the potential is being
determined and the set of image dipoles extends off to the left
and right symmetrically. The arrangement shown above shows "3
pairs" of dipoles, the arrangement used for my previous results.
Maple continued the series up to 11 pairs. This arrangement
guarantees the required boundary condition along the top edge of
the region but only approaches the required condition along the
side as the number of pairs of image dipoles increases.

In each case the simulation region is 14 units wide (x) by 20
units high (y) and the positive line source is at (0,5). I
normalized each result by multiplying it by a factor and then
subtracting a constant with the factor and the constant chosen in
order to force V(0,5.5)=10 and V(0,0)=0. Then I included
potential contours at 1/2 unit steps up to 10. The results are
shown at

<http://www.csupomona.edu/~ajm/special/condsheet2.pdf>

I think you'll agree that the series appears to converge quite
well to a result that satisifes the required boundary conditions.

I also include revised 3-d views comparing the "11 pair" image
solution to the standard unbound 2-d dipole.

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm