The resistance of a flat resistor whose terminals are concentric
circles (silver-painted on Pasco paper) can be shown to be:
R" = (rho/(2*PI*t))*ln(r2/r1)
where rho is the resistivity and t is the thickness of the paper.
The r2 and r1 are the radii of two silver boundaries facing each
other. This is an unusual resistor because the gradient of the
potential is not constant, as in a wire. Students can be asked
to derive the above formula (a trivial calculus problem) and
to check that the experimentally measured resistance, R', is not
very different from the expected R."
Here are the data from a one successful attempt to do this.
Let me add that on two occasions I was not successful and I
do not know why.
Part 1
A 15.6 mm wide strip of Pasco paper was cut and its ends
were silver-painted. The distance between the silver lines
was 298 mm. I measured the strip resistance (471 kohms)
and its thickness (0.13 mm). This allowed me to calculate
rho=3.2 ohm*m.
Part 2
A circular resistor was prepared with r1=30 mm and
r2=57 mm. The expected R" of this radial resistor was 2420
ohms. The measured resistance turned out to be 2410 ohms.
But, as I indicated above, two other attempts were not
successful. In one case the discrepancy was 10% and in
another it was 40%. Perhaps somebody else will be
more successful; it could be a good activity.
P.S. I also tried this with an electrolyte (salty water) but
quickly discovered my lack of competence. (One has to
to know how to deal with surface phenomena near the
electrodes.) In principle the experiment should be simple
(salty water of known depth, t, a short metallic tube (outer
circle, r2) and a metallic cylinder (inner circle, r1). Perhaps
somebody will perform such experiment and share the result.
Ludwik Kowalski