# 20.3 Resistance and resistivity  (Page 4/6)

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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] .)

If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?

Explain why $R={R}_{0}\left(\text{1}+\alpha \Delta T\right)$ for the temperature variation of the resistance $R$ of an object is not as accurate as $\rho ={\rho }_{0}\left(\text{1}+\alpha \Delta T\right)$ , which gives the temperature variation of resistivity $\rho$ .

## Problems&Exercises

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

$\text{0.104 Ω}$

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.

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

$2\text{.}\text{8}×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{m}$

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).

What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when ${1.00 × 10}^{\text{3}}\phantom{\rule{0.25em}{0ex}}\text{V}$ is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)

$1\text{.}\text{10}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{A}$

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

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

$-5º\text{C to 45ºC}$

Of what material is a resistor made if its resistance is 40.0% greater at $\text{100º}\text{C}$ than at $\text{20}\text{.}0º\text{C}$ ?

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

1.03

(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 $\text{77}\text{.}7\phantom{\rule{0.25em}{0ex}}\Omega$ at $\text{20}\text{.}0º\text{C}$ ? (b) What is its resistance at $\text{150º}\text{C}$ ?

Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at $\text{20}\text{.}0º\text{C}$ ?

0.06%

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

A copper wire has a resistance of $0\text{.}\text{500}\phantom{\rule{0.25em}{0ex}}\Omega$ at $\text{20}\text{.}0º\text{C}$ , and an iron wire has a resistance of $0\text{.}\text{525}\phantom{\rule{0.25em}{0ex}}\Omega$ at the same temperature. At what temperature are their resistances equal?

$-\text{17º}\text{C}$

(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has $\alpha =–0\text{.}\text{0600}/\text{ºC}$ ) 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 $\text{37}\text{.}0º\text{C}$ (normal body temperature)? (b) The negative value for $\alpha$ may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: Resistance can’t become negative.)

Integrated Concepts

(a) Redo [link] taking into account the thermal expansion of the tungsten filament. You may assume a thermal expansion coefficient of $\text{12}×{\text{10}}^{-6}/\text{ºC}$ . (b) By what percentage does your answer differ from that in the example?

(a) $4\text{.}7\phantom{\rule{0.25em}{0ex}}\Omega$ (total)

(b) 3.0% decrease

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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