Download Conductivity

Conductivity
Submitted by: I.D. 123456
The problem:
A resistor is built with two concentric conducting cylindrical shells of a height h and radii R1 > R2 .
The space between the shells is filled with material with constant resistivity ρ.
1. If constant current I flows in the circular direction, what is the resistance of the resistor?
2. The resistor is connected to the voltage source(as in the picture). What is the current through
the source?
The solution:
1. We think of the cylindrical resistor as many resistors of an infinitecimal width connected in
parallel.
Such an infinitecimal resistor is a cylindrical shell at a radius r. Since the current is circular,
2πr
dR = ρ
hdr
Z
Z R2
1
1
hdr
h
R2
=
=
=
ln
R
dR
2πρr
2πρ
R1
R1
(1)
(2)
Then,
R=
2πρ
2
h ln R
R1
(3)
2. By connecting the source in the pictured way, we have actually two resistors connected in parallel
R1 =
R2 =
R
4
3R
4
(4)
(5)
(here R1 , R2 are resistances) since they are built from the respective part of the cylindrical resistor.
Then the total resistance is
Rt =
R1 R2
3R
=
R1 + R2
16
(6)
Then
2
8V h ln R
V
16V
R1
I=
=
=
Rt
3R
3πρ
(7)
1