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Fermi method (was Calculating resistance)



What is R between two identical silver spots (each of
diameter 2*a=1 cm), separated by the distance L=20 cm,
on an infinitely large sheet of the carbon-impregnated
paper? Suppose the paper thickness d=0.2 mm and its
resitivity (rho) is 0.00001 ohm*m.

Let me first estimate R by Fermi Method. Then I will
show what the predicted range includes the R calculated
by David's formula.

1) In my mind I cut a strip between two identical dots
and calculate the resistance of the strip. According to the
common textbook formula R'=rho*L/(2*a*d)]=1 ohm.
I would expect the resistance of the entire sheet to be about
ten to twenty times less. Why? Because R of the entire
sheet looks (more or less) like R of ten or twenty single
strips in parallel .(Parts of paper which are far away do
not contribute much; most current flows through the
central area of about 20*20 cm^2). So I expected the
entire sheet to have R between 0.1 and 0.05 ohms.

2) David's formula: R = (rho/(pi*d))*arccosh(L/(2*a)).

L/(2*a)=20/1=20. Arccosh(20)=3.688

rho/(pi*d)=0.00001/(pi*0.0002)=0.0159

R=0.0587ohms

This is within the expected range from 0.05 to 0.10 ohms.
Nothing profound; only an illustration of Fermi method.
Ludwik Kowalski