9.3 Resistivity and resistance (Page 6/10)

University physics volume 2 Page 6 / 10

Check Your Understanding The resistance between the two conductors of a coaxial cable depends on the resistivity of the material separating the two conductors, the length of the cable and the inner and outer radius of the two conductor. If you are designing a coaxial cable, how does the resistance between the two conductors depend on these variables?

The longer the length, the smaller the resistance. The greater the resistivity, the higher the resistance. The larger the difference between the outer radius and the inner radius, that is, the greater the ratio between the two, the greater the resistance. If you are attempting to maximize the resistance, the choice of the values for these variables will depend on the application. For example, if the cable must be flexible, the choice of materials may be limited.

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View this simulation to see how the voltage applied and the resistance of the material the current flows through affects the current through the material. You can visualize the collisions of the electrons and the atoms of the material effect the temperature of the material.

Summary

Resistance has units of ohms $(Ω)$ , related to volts and amperes by $1 Ω = 1 V/A$ .
The resistance R of a cylinder of length L and cross-sectional area A is $R = \frac{ρ L}{A}$ , where $ρ$ is the resistivity of the material.
Values of $ρ$ in [link] show that materials fall into three groups—conductors, semiconductors, and insulators.
Temperature affects resistivity; for relatively small temperature changes $Δ T$ , resistivity is $ρ = ρ_{0} (1 + α Δ T)$ , where $ρ_{0}$ is the original resistivity and $α$ is the temperature coefficient of resistivity.
The resistance R of an object also varies with temperature: $R = R_{0} (1 + α Δ T)$ , where $R_{0}$ is the original resistance, and R is the resistance after the temperature change.

Conceptual questions

The IR drop across a resistor means that there is a change in potential or voltage across the resistor. Is there any change in current as it passes through a resistor? Explain.

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

In carbon, resistivity increases with the amount of impurities, meaning fewer free charges. In silicon and germanium, impurities decrease resistivity, meaning more free electrons.

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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?

Pictures are a schematic drawing of a resistance object with the long side of the length R and the short side of the length R prime. In the left picture, current flows along the long side; in the right picture, current flows along the short side.

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

Copper has a lower resistivity than aluminum, so if length is the same, copper must have the smaller diameter.

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Problems

What current flows through the bulb of a 3.00-V flashlight when its hot resistance is $3.60 Ω$ ?

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Calculate the effective resistance of a pocket calculator that has a 1.35-V battery and through which 0.200 mA flows.

$R = 6.750 k Ω$

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How many volts are supplied to operate an indicator light on a DVD player that has a resistance of $140 Ω$ , given that 25.0 mA passes through it?

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

$R = 0.10 Ω$

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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 $20.0 ° C$ , how long should it be?

$\begin{matrix} R = ρ \frac{L}{A} \\ L = 3 mm \end{matrix}$

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A lead rod has a length of 30.00 cm and a resistance of $5.00 μ Ω$ . What is the radius of the rod?

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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).

$\frac{\frac{R_{Al}}{L_{Al}}}{\frac{R_{Cu}}{L_{Cu}}} = \frac{ρ_{Al} \frac{1}{π {(\frac{D_{Al}}{2})}^{2}}}{ρ_{Cu} \frac{1}{π {(\frac{D_{Cu}}{2})}^{2}}} = \frac{ρ_{Al}}{ρ_{Cu}} {(\frac{D_{Cu}}{D_{Al}})}^{2} = 1, \frac{D_{Al}}{D_{Cu}} = \sqrt{\frac{ρ_{Al}}{ρ_{C u}}}$

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

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

a. $R = R_{0} (1 + α Δ T), 2 = 1 + α Δ T, Δ T = 256.4 °C, T = 276.4 ° C$ ;
b. Under normal conditions, no it should not occur.

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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 $20.0 ° C$ . Over what temperature range can it be used?

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Of what material is a resistor made if its resistance is 40.0% greater at $100.0 ° C$ than at $20.0 ° C$ ?

$\begin{matrix} R & = & R_{0} (1 + α Δ T) \\ α & = 0.006 ° C^{−1} \end{matrix}$ , iron

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An electronic device designed to operate at any temperature in the range from $−10.0 ° C$ to $55.0 ° C$ contains pure carbon resistors. By what factor does their resistance increase over this range?

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(a) Of what material is a wire made, if it is 25.0 m long with a diameter of 0.100 mm and has a resistance of $77.7 Ω$ at $20.0 ° C$ ? (b) What is its resistance at $150.0 ° C?$

a. $R = ρ \frac{L}{A}, ρ = 2.44 \times 10^{−8} Ω \cdot m$ , gold;
b. $\begin{matrix} R = ρ \frac{L}{A} (1 + α Δ T) \\ R = 2.44 \times 10^{−8} Ω \cdot m (\frac{25 m}{π {(\frac{0.100 \times 10^{−3} m}{2})}^{2}}) (1 + 0.0034 ° C^{−1} (150 ° C - 20 ° C)) \\ R = 112 Ω \end{matrix}$

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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 $20.0 ° C$ ?

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A copper wire has a resistance of $0.500 Ω$ at $20.0 ° C,$ and an iron wire has a resistance of $0.525 Ω$ at the same temperature. At what temperature are their resistances equal?

$\begin{matrix} R_{Fe} = 0.525 Ω, R_{Cu} = 0.500 Ω, α_{Fe} = 0.0065 ° C^{−1} α_{Cu} = 0.0039 ° C^{−1} \\ R_{Fe} = R_{Cu} \\ R_{0 Fe} (1 + α_{Fe} (T - T_{0})) = R_{0 Cu} (1 + α_{Cu} (T - T_{0})) \\ \frac{R_{0 Fe}}{R_{0 Cu}} (1 + α_{Fe} (T - T_{0})) = 1 + α_{Cu} (T - T_{0}) \\ T = 2.91 ° C \end{matrix}$

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Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

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While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

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The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

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Anatomy is the identification and description of the structures of living things

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Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

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Source: OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3

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