Question 26.68: The copper windings of a motor have a resistance of 50 Ω at ......
The copper windings of a motor have a resistance of 50 Ω at 20°C when the motor is idle. After the motor has run for several hours, the resistance rises to 58 Ω. What is the temperature of the windings now? Ignore changes in the dimensions of the windings. (Use Table 26-1.)
Table 26-1
Resistivities of Some Materials at Room Temperature (20°C)
| Material | Resistivity, ρ (Ω.m) | Temperature Coefficient of Resistivity, \alpha\left({K}^{-1}\right) |
| Typical Metals | ||
| Silver | 1.62\times10^{-8} | 4.1\times10^{-3} |
| Copper | 1.69\times10^{-8} | 4.3\times10^{-3} |
| Gold | 2.35\times10^{-8} | 4.0\times10^{-3} |
| Aluminum | 2.75\times10^{-8} | 4.4\times10^{-3} |
| Manganin^a | 4.82\times10^{-8} | 0.002\times10^{-3} |
| Tungsten | 5.25\times10^{-8} | 4.5\times10^{-3} |
| Iron | 9.68\times10^{-8} | 6.5\times10^{-3} |
| Platinum | 10.6\times10^{-8} | 3.9\times10^{-3} |
| Typical Semiconductors | ||
| Silicon,pure | 2.5\times10^{3} | -70\times10^{-3} |
| Silicon,n-type^b | 8.7\times10^{-4} | |
| Silicon,p-type^C | 2.8\times10^{-3} | |
| Typical Insulators | ||
| Glass Fused | 10^{10}-10^{14} | |
| quartz | \sim10^{16} | |
^aAn alloy specifically designed to have a small value of α.
^bPure silicon doped with phosphorus impurities to a charge carrier density of 10^{23}\,{\mathrm{m}}^{-3}.
^cPure silicon doped with aluminum impurities to a charge carrier density of 10^{23}\,{\mathrm{m}}^{-3}.
We use Eq. 26-17: \rho-\rho_{0}=\rho\alpha(T-T_{0}), , and solve for T:
\rho-\rho_{0}=\rho_{0}\alpha(T-T_{0}). (26-17)
T=T_{0}+{\frac{1}{\alpha}}\left({\frac{\rho}{\rho_{0}}}-1\right)=20^{\circ}{\mathrm{C}}+{\frac{1}{4.3\times10^{-3}/\,{\mathrm{K}}}}\left({\frac{58\Omega}{50\Omega}}-1\right)=57^{\circ}{\mathrm{C}}.
We are assuming that \rho/\rho_{0}=R/R_{0}.