Problem of the Block
Block 7
Resistors
(problem idea from Dr. Sherman in the Physics Department)
Resistors are circuit elements that are used to modify the current or voltage in a circuit. The basic unit of resistance
is the ohm, Ω. When 2 resistors, R1 and R2 are placed in series (see Figure 1) the resistance adds R = R1 + R2 .
1
1
1
=
+
, or equivalently,
When 2 resistors are placed in parallel (see Figure 2) the resistance behaves as
R
R
R
1
2
−1
1
1
+
R=
.
R1
R2
Problem 1. Consider having 10 resistors, each with value 1Ω. What is the resistor setup you can make that has an
equivalent resistance closest to π Ω? (hint: think about fractional approximations to π)
Problem 2. What is the smallest number of 1Ω resistors that will let you create a circuit with a value within one
part per million of π?
Problem 3. Imagine having a countably infinite supply of 1Ω resistors. Can you design a circuit element that has
equivalent resistance exactly equal to π Ω?
1Ω
1Ω
1Ω
1Ω
Figure 1: The circuit piece above has a resistance of
1 + 1 = 2Ω.
Figure 2: The circuit piece above has a resistance of
−1
1 1
1
+
= Ω.
1 1
2
1Ω
1Ω
1Ω
1 1
Figure 3: The circuit piece above has a resistance of
+
1 2
−1
=
2
Ω.
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Turn in solutions to Dr. S in Law 204 or by email at tskorczewski@cornellcollege.edu by April 10. Multiple submissions are allowed. Solutions for only one/some questions or partial solutions will receive credit (and are encouraged!).
Turned in solutions can earn culture points toward the math major. Dr. S and other faculty are known to give generous hints to those who ask. For more information about the Problem of the Block and to print off your own copy
visit http://www.cornellcollege.edu/mathematics/problem-of-the-block/index.shtml.
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