Chapter 18, Problem 18P

To determine

The temperature at copper will have the same resistivity as tungsten does at $20°C$ .

Answer to Problem 18P

  $363°C$

Explanation of Solution

Given info:

Reference temperature is $20°C$ .

Resistivity of tungsten is $5.6\times {10}^{-8}\Omega \cdot m$

Resistivity of copper is $1.68\times {10}^{-8}\Omega \cdot m$

Formula used:

The resistivity of metal increases linearly with the temperature. Expression for this: ${\rho }_T={\rho }_0\left[1+\alpha \left(T-T_0\right)\right]$

Where,

  ${\rho }_0$ is the resistivity at reference temp $T_0$ ,

  ${\rho }_T$ is the resistivity at reference temp $T$ ,

  $\alpha$ is the true coefficient of resistivity constant.

Calculation:

Formula for temperature-dependent resistivity, ${\rho }_T={\rho }_0\left[1+\alpha \left(T-T_0\right)\right]$

Now, solve for $T$

  $T=T_0+\frac{\frac{{\rho }_T}{{\rho }_0}-1}{\alpha }$

The new resistivity of copper is equal to that of tungsten at room temperature. So, ${\rho }_T={\rho }_W$

Copper’s resistivity is compared with its known value at room temperature. So, ${\rho }_0={\rho }_{Cu}$

Thus, $T=T_0+\frac{\frac{{\rho }_W}{{\rho }_{Cu}}-1}{{\alpha }_{Cu}}$

Now, substitute the given equation and the temperature coefficient $\alpha$ . So,

  $T=20°C+\frac{\frac{5.6\times {10}^{-8}\Omega \cdot m}{1.68\times {10}^{-8}\Omega \cdot m}-1}{\left(0.0068°C^{-1}\right)}=363°C$

Conclusion: The required temperature is $363°C$ .