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Applying the science practices: examining resistance

Using the PhET Simulation “Resistance in a Wire”, design an experiment to determine how different variables – resistivity, length, and area – affect the resistance of a resistor. For each variable, you should record your results in a table and then create a graph to determine the relationship.

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Which of the following affect the resistivity of a wire?

  1. length
  2. area of cross section
  3. material
  4. all of the above
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The lengths and diameters of four wires are given as shown.

The figure shows four cylinders representing sections of wire. The length and diameter of the wires are indicated graphically and with labels below. The wires labeled 2L are the same length and twice as long as the wires labeled L. The wires labeled 2D are twice as thick as the wires labeled D. Wire 1: 2L, D (long, thin). Wire 2: L, D (short, thin). Wire 3: 2L, 2D (long, thick). Wire 4: L, 2D (short, thick).

If the four wires are made from the same material, which of the following is true? Select two answers.

  1. Resistance of Wire 3>Resistance of Wire 2
  2. Resistance of Wire 1>Resistance of Wire 2
  3. Resistance of Wire 1<Resistance of Wire 4
  4. Resistance of Wire 4<Resistance of Wire 3

(b), (d)

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Suppose the resistance of a wire is R Ω. What will be the resistance of another wire of the same material having the same length but double the diameter?

  1. R /2
  2. 2 R
  3. R /4
  4. 4 R
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The resistances of two wires having the same lengths and cross section areas are 3 Ω and 11 Ω. If the resistivity of the 3 Ω wire is 2.65 × 10 −8 Ω∙m, find the resistivity of the 1 Ω wire.

9.72 × 10 −8 Ω∙m

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The lengths and diameters of three wires are given below. If they all have the same resistance, find the ratio of their resistivities.

Wire Length Diameter
Wire 1 2 m 1 cm
Wire 2 1 m 0.5 cm
Wire 3 1 m 1 cm
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Suppose the resistance of a wire is 2 Ω. If the wire is stretched to three times its length, what will be its resistance? Assume that the volume does not change.

18 Ω

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Section summary

  • The resistance R size 12{R} {} of a cylinder of length L size 12{L} {} and cross-sectional area A size 12{A} {} is R = ρL A size 12{R = { {ρL} over {A} } } {} , where ρ size 12{ρ} {} is the resistivity of the material.
  • Values of ρ size 12{ρ} {} in [link] show that materials fall into three groups— conductors, semiconductors, and insulators .
  • Temperature affects resistivity; for relatively small temperature changes Δ T size 12{DT} {} , resistivity is ρ = ρ 0 ( 1 + α Δ T ) size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) } {} , where ρ 0 size 12{ρ rSub { size 8{0} } } {} is the original resistivity and α is the temperature coefficient of resistivity.
  • [link] gives values for α size 12{α} {} , the temperature coefficient of resistivity.
  • The resistance R size 12{R} {} of an object also varies with temperature: R = R 0 ( 1 + α Δ T ) size 12{R = R rSub { size 8{0} } \( "1 "+ ΔαT \) } {} , where R 0 size 12{R rSub { size 8{0} } } {} is the original resistance, and R is the resistance after the temperature change.

Conceptual questions

In which of the three semiconducting materials listed in [link] do impurities supply free charges? (Hint: Examine the range of resistivity for each and determine whether the pure semiconductor has the higher or lower conductivity.)

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Does the resistance of an object depend on the path current takes through it? Consider, for example, a rectangular bar—is its resistance the same along its length as across its width? (See [link] .)

Part a of the figure shows a voltage V applied along the length of a rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the length of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the length is shown as R and the current is shown as I. Part b of the figure shows a voltage V applied along the width of the same rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the width of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the width is shown as R prime, and the current is shown as I prime.
Does current taking two different paths through the same object encounter different resistance?
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If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?

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Explain why R = R 0 ( 1 + α Δ T ) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {} for the temperature variation of the resistance R size 12{R} {} of an object is not as accurate as ρ = ρ 0 ( 1 + α Δ T ) size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) } {} , which gives the temperature variation of resistivity ρ size 12{ρ} {} .

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Problems&Exercises

What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm diameter?

0.104 Ω

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The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.

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If the 0.100-mm diameter tungsten filament in a light bulb is to have a resistance of 0.200 Ω at 20 . C size 12{"20" "." 0°"C"} {} , how long should it be?

2 . 8 × 10 2 m size 12{2 "." "82"´"10" rSup { size 8{-2} } " m"} {}

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Find the ratio of the diameter of aluminum to copper wire, if they have the same resistance per unit length (as they might in household wiring).

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What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when 1.00 × 10 3 V is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)

1 . 10 × 10 3 A size 12{1 "." "10"´"10" rSup { size 8{-3} } " A"} {}

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(a) To what temperature must you raise a copper wire, originally at 20.0ºC , to double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?

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A resistor made of Nichrome wire is used in an application where its resistance cannot change more than 1.00% from its value at 20 . C size 12{"20" "." 0°"C"} {} . Over what temperature range can it be used?

C to 45ºC

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Of what material is a resistor made if its resistance is 40.0% greater at 100º C size 12{"100"°"C"} {} than at 20 . C size 12{"20" "." 0°"C"} {} ?

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An electronic device designed to operate at any temperature in the range from –10 . C to 55 . C size 12{"10" "." 0°"C to 55" "." 0°"C"} {} contains pure carbon resistors. By what factor does their resistance increase over this range?

1.03

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(a) Of what material is a wire made, if it is 25.0 m long with a 0.100 mm diameter and has a resistance of 77 . 7 Ω size 12{"77" "." 7 %OMEGA } {} at 20 . C size 12{"20" "." 0°"C"} {} ? (b) What is its resistance at 150º C size 12{"150"°"C"} {} ?

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Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at 20 . C size 12{"20" "." 0°"C"} {} ?

0.06%

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A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?

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A copper wire has a resistance of 0 . 500 Ω size 12{0 "." "500 " %OMEGA } {} at 20 . C size 12{"20" "." 0°"C"} {} , and an iron wire has a resistance of 0 . 525 Ω size 12{0 "." "525 " %OMEGA } {} at the same temperature. At what temperature are their resistances equal?

17º C size 12{-"17"°"C"} {}

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(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has α = 0 . 0600 / ºC size 12{α"=-"0 "." "0600"/°"C"} {} ) when it is at the same temperature as the patient. What is a patient's temperature if the thermistor's resistance at that temperature is 82.0% of its value at 37 . C size 12{"37" "." 0°"C"} {} (normal body temperature)? (b) The negative value for α size 12{α} {} may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: Resistance can't become negative.)

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Integrated Concepts

(a) Redo [link] taking into account the thermal expansion of the tungsten filament. You may assume a thermal expansion coefficient of 12 × 10 6 / ºC size 12{"12"´"10" rSup { size 8{-6} } /°"C"} {} . (b) By what percentage does your answer differ from that in the example?

(a) 4 . 7 Ω size 12{4 "." 7 %OMEGA } {} (total)

(b) 3.0% decrease

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Unreasonable Results

(a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent?

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Questions & Answers

a submersible pump is dropped a borehole and hits the level of water at the bottom of the borehole 5 seconds later.determine the level of water in the borehole
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the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
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What is the far point of a person whose eyes have a relaxed power of 50.5 D?
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the speed of something in a given direction.
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Practice Key Terms 2

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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