Chapter 18, Problem 17P

To determine

The temperature of a copper wire to increase its resistance by $15\%$ .

Answer to Problem 17P

  $42.1°C$

Explanation of Solution

Given info:

Original temperature of copper wire is $20°C$ .

Resistance Increased by $15\%$

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

The problem asks for the temperature at which the resistance of a copper wire is $15\%$ higher than at $20°C$ . So,

  ${\rho }_T=1.15{\rho }_0$

Substitute the value, ${\rho }_T=1.15{\rho }_0$ , in the formula ${\rho }_T={\rho }_0\left[1+\alpha \left(T-T_0\right)\right]$

So,

  $1.15=\left[1+\alpha \left(T-T_0\right)\right]$

Now, solve for $T$

  $T=T_0+\frac{0.15}{\alpha }$

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

  $T=20°C+\frac{0.15}{\left(0.0068°C^{-1}\right)}=42.1°C$

Conclusion: The required temperature is $42.1°C$ .