Question

(a)At what temperature would the resistance of a copper conductor be double its resistance at${\mathbf{\text{20.0}}}^{\mathbf{0}}\mathbf{\text{C}}$? (Use ${\mathbf{\text{20.0}}}^{\mathbf{0}}\mathbf{\text{C}}$as the reference point in Eq. 26-17; compare your answer withFig. 26-10.)

(b) does this same “doubling temperature” hold for all copper conductors, regardless of shape or size?

Short answer

1. Temperature at which the resistance of copper conductor be doubled its resistance at $20°\text{C}$is $253°\text{C}$.
2. Yes,same doubling temperature holds for all copper conductors.

Step-by-step solution

The given data

1. Temperature,$T_0={20}^0\text{C}$
2. Resistance will be doubled.
3. Coefficient of expansion.$\alpha =4.3\times {10}^{-10}/\text{K}$

Understanding the concept of the resistance

The resistance of a conductor increases with temperature. So, using the equation, we can find the resistance(R) at temperature (T), where the reference temperature is the given temperature. Next, we find the temperature at which the resistance of a copper conductor will double its resistance at the reference temperature. The expansion coefficient used in this is consistent with other units; i.e., the doubling temperature holds for all copper conductors, regardless of shape or size.

Formulae:

The resistivity equation due to change in resistance in changing temperature,

$\rho -{\rho }_0={\rho }_0\alpha (T-T_0)\text{(for}\rho \alpha \text{R,R=}\frac{\rho L}{A}\text{,A,L=constant)}$ (i)

a) Calculation of the temperature at which the resistance doubles

As from given condition $\rho =2{\rho }_0$i.e. resistance is double at the reference point.

Now, using the given data in equation (i), the temperature value at which the resistance value gets doubled can be calculated as follows:

$\begin{array}{l}2{\rho }_0-{\rho }_0={\rho }_0\alpha (T-T_0)\\ {\rho }_0={\rho }_0\alpha (T-T_0)\\ 1=\alpha (T-T_0)\end{array}$

For the given values-

$\begin{array}{c}T=\frac{1}{\alpha }+T_0\\ =\frac{1}{4.3\times {10}^{-3}{/}^0\text{C}}+20{}^0\text{C}\\ =(232.55+20){}^0\text{C}\\ =252.55°\text{C}\end{array}$

Hence, resistance will be doubled at $252.55°\text{C}$.

b) Calculation to know whether the resistance doubling hold for all the conductors

The change in Celsius is equivalent to change on the Kelvin scale. That means, the value of alpha is same for both the units. Also, the change in temperature depends upon resistivity which is the characteristic of the material.Therefore, the doubling temperature holds for all copper conductors, regardless of shape or size.