Re: D=2 versus cylinders versus dots on paper.

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

In reading my own message I see something which may
or may not be significant; it has to do with my so-called
"theoretical" curves. I was measuring potentials in various
points with respect to the left silver dot. That is what the
Pasco instruction calls for.

But how do I calculate potentials? For the (a) theory I use

V=Q1/d1 + Q2/d2

where | Q1 | = | Q2 | = arbitrary number. The d1 and d2
are distances from Q1 and Q2 to the point of interest. This
means I am calculating V with respect to infinity, not with
respect to the silver dot on the left side. First I find the
value of V at a starting point on the axis, such as (16,10).
Suppose it is V=12345 volts. Then I use a little True Basic
program to search for V=12345 volts at any given y, for
example, y= 15 cm. If the point found happens to be
x=17 cm then I say that the location (17,15) belongs to the
same equipotential line as (16,10). That is how theoretical
equipotential curves were determined. Was this OK?

I think it was OK; adding an arbitrary constant (JohnD
would probably say "using a different gauge") has no
effect on the SHAPE of a curve. Is this correct?

For the (b) theory I proceed in the same way but used

V=ln(d2/d1)

It means I calculated potentials with respect to the
plane of symmetry parallel to the cylinders, not with
respect to the silver dot on the left side. Is this OK?

The bottom line of all this is that I have experimental
data waiting to be explained. What is the appropriate
theoretical relation V(d1,d2) for equipotential lines on
Pasco sheets? I was not able to answer this question
by reading the message of JohnD. The hints were not
sufficient for me in this case.
*********************************

Ludwik Kowalski wrote:

> Why is it so?
>
> A standard student experiments with Pasco sheets is to
> trace equipotential lines. But how well do these lines
> agree with the theory? And which theory should they
> agree with? I see two options:
>
> a) Filed is like that of a dipole (two spheres in 3-D)
> b) Field is like that of two long cylinders (dots are
>     the cross sections of these cylinders).
>
> The second theory can be objected on the ground that
> there are no long cylinders in this setup.
>
> The first can be objected because the medium in which
> the current is flowing is 2-dimentional.
>
> To proceed I made predictions based on each theory and
> compared them with the shape of the experimental trace.
> Fortunately, the two predictions are very very different.
> The bottom line is that neither (a) not (b) agree with my
> experimental results. Let me illustrate this by providing
> the (x,y) coordinates of the three lines. You have to trace
> them on a graph paper to see what I mean.
>
> But first some experimental details. The two silver circles
> were at (9,10) and (19,10); they were separated by 10 cm.
> The DOP of 80 V was applied and the current was about
> 2 mA. The voltmeter of 10 megaohm was used to measure
> potentials (with respect to the left dot) at various locations.
> As I indicated in the previous posting, the infinite sheet
> approximation is applicable for this geometry.
>
> To illustrate my point (and to minimize your efforts) I will
> focus on only one line, the line passing through the point
> (16,10), on the axis. The axis is at y=10. The point selected
> is 7 cm from the left dot and 3 cm  from the right dot.
>
> The experimental equipotential line (DOP=50 V) is
> symmetrical with respect to the axis (as it should be) and it
> passes through the following points: (22.17), (20,16),
> (18,14), (16.2,12) and (16,10).
>
> The theoretical line (a) passing through (16,10) also passes
> through: (21.5,13), (19,13.6) and (17.6,13).
>
> The theoretical line (b) passing through (16,10) also passes
> through: (21.5,13), (19,13.6) and (17.6,13).
>
> Neither (a) nor (b) are correct. What theory should agree with
> the experimental data? Comments will be appreciated.
> P.S.
> Suppose I am verifying Coulomb's Law using two charged
> pucks on the air table. This is also a 2-dimentional medium.
> I think that the 1/r^2 relation would be observed. On that basis
> I would expect the potential to be proportional to 1/r, as in
> 3-dimentional space. Translating this into what happens on
> the carbon paper I would expect the (a) prediction to be valid.
> But it is not valid. Where am I wrong?
> Ludwik Kowalski