# 9.6 Superconductors  (Page 5/11)

 Page 5 / 11

## Conceptual questions

What requirement for superconductivity makes current superconducting devices expensive to operate?

Very low temperatures necessitate refrigeration. Some materials require liquid nitrogen to cool them below their critical temperatures. Other materials may need liquid helium, which is even more costly.

Name two applications for superconductivity listed in this section and explain how superconductivity is used in the application. Can you think of a use for superconductivity that is not listed?

## Problems

Consider a power plant is located 60 km away from a residential area uses 0-gauge $\left(A=42.40\phantom{\rule{0.2em}{0ex}}{\text{mm}}^{2}\right)$ wire of copper to transmit power at a current of $I=100.00\phantom{\rule{0.2em}{0ex}}\text{A}$ . How much more power is dissipated in the copper wires than it would be in superconducting wires?

$\begin{array}{}\\ \\ \\ \\ {R}_{\text{copper}}=0.24\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}\hfill \\ P=2.377\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\text{W}\hfill \end{array}$

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

Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has $\alpha =-0.06\text{/}\text{°}\text{C}$ ) 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 $37\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ (normal body temperature)?

$\begin{array}{c}R={R}_{0}\left(1+\alpha \left(T-{T}_{0}\right)\right)\hfill \\ 0.82{R}_{0}={R}_{0}\left(1+\alpha \left(T-{T}_{0}\right)\right),\phantom{\rule{0.8em}{0ex}}0.82=1-0.06\left(T-37\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\right),\phantom{\rule{0.8em}{0ex}}T=40\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\hfill \end{array}$

Electrical power generators are sometimes “load tested” by passing current through a large vat of water. A similar method can be used to test the heat output of a resistor. A $R=30\phantom{\rule{0.2em}{0ex}}\text{Ω}$ resistor is connected to a 9.0-V battery and the resistor leads are waterproofed and the resistor is placed in 1.0 kg of room temperature water $\left(T=20\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\right)$ . Current runs through the resistor for 20 minutes. Assuming all the electrical energy dissipated by the resistor is converted to heat, what is the final temperature of the water?

A 12-guage gold wire has a length of 1 meter. (a) What would be the length of a silver 12-gauge wire with the same resistance? (b) What are their respective resistances at the temperature of boiling water?

a. ${R}_{\text{Au}}={R}_{\text{Ag}},\phantom{\rule{0.8em}{0ex}}{\rho }_{\text{Au}}\frac{{L}_{\text{Au}}}{{A}_{\text{Au}}}={\rho }_{\text{Ag}}\frac{{L}_{\text{Ag}}}{{A}_{\text{Ag}}},\phantom{\rule{0.8em}{0ex}}{L}_{\text{Ag}}=1.53\phantom{\rule{0.2em}{0ex}}\text{m}$ ;
b. ${R}_{\text{Au,20 °C}}=0.0074\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.8em}{0ex}}{R}_{\text{Au,100 °C}}=0.0094\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.8em}{0ex}}{R}_{\text{Ag},100\phantom{\rule{0.2em}{0ex}}\text{°C}}=0.0096\phantom{\rule{0.2em}{0ex}}\text{Ω}$

What is the change in temperature required to decrease the resistance for a carbon resistor by 10%?

A coaxial cable consists of an inner conductor with radius ${r}_{\text{i}}=0.25\phantom{\rule{0.2em}{0ex}}\text{cm}$ and an outer radius of ${r}_{\text{o}}=0.5\phantom{\rule{0.2em}{0ex}}\text{cm}$ and has a length of 10 meters. Plastic, with a resistivity of $\rho =2.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{13}\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}·\text{m}$ , separates the two conductors. What is the resistance of the cable?

$\begin{array}{ccc}\hfill dR& =\hfill & \frac{\rho }{2\pi rL}dr\hfill \\ \hfill R& =\hfill & \frac{\rho }{2\pi L}\text{ln}\phantom{\rule{0.2em}{0ex}}\frac{{r}_{\text{o}}}{{r}_{\text{i}}}\hfill \\ \hfill R& =\hfill & 2.21\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{11}\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}\hfill \end{array}$

A 10.00-meter long wire cable that is made of copper has a resistance of 0.051 ohms. (a) What is the weight if the wire was made of copper? (b) What is the weight of a 10.00-meter-long wire of the same gauge made of aluminum? (c)What is the resistance of the aluminum wire? The density of copper is $8960{\phantom{\rule{0.2em}{0ex}}\text{kg/m}}^{3}$ and the density of aluminum is $2760{\phantom{\rule{0.2em}{0ex}}\text{kg/m}}^{3}$ .

A nichrome rod that is 3.00 mm long with a cross-sectional area of $1.00{\phantom{\rule{0.2em}{0ex}}\text{mm}}^{2}$ is used for a digital thermometer. (a) What is the resistance at room temperature? (b) What is the resistance at body temperature?

a.
${R}_{0}=3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{Ω}$ ; b.
$\begin{array}{ccc}\hfill {T}_{\text{c}}& =\hfill & 37.0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\hfill \\ \hfill R& =\hfill & 3.02\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}\hfill \end{array}$

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