Chapter 18, Problem 25P

To determine

To Find: The value of each resistor.

Explanation of Solution

Given:

Total resistance, $R_T=4.70\text{kΩ}$

Formula Used:

Relationship between resistance and temperature:

  $R=R_o\left(1+\alpha T\right)$

Here, $R_o$ is the resistance at $0°\text{C}$ and $\alpha$ is temperature coefficient.

Calculations:

Temperature coefficient of carbon, ${\alpha }_C=-0.0005/°\text{C}$

Temperature coefficient of nichrome, ${\alpha }_{ni}=0.0004/°\text{C}$

  $\begin{array}{l}{R}_T=4.70\text{kΩ}=R_1+R_2\\ R_1\left(1+{\alpha }_{C}T\right)+R_2\left(1+{\alpha }_{ni}T\right)=4700\text{Ω}\\ \Rightarrow R_1+R_2+R_1{\alpha }_{C}T+R_2{\alpha }_{ni}T=4700\text{Ω}\\ \Rightarrow R_1{\alpha }_{C}T+R_2{\alpha }_{ni}T=0\\ \Rightarrow R_1\left(-0.0005\right)+R_2\left(0.0004\right)=0\\ \Rightarrow R_1=0.8R_2\\ \because R_1+R_2=4700\\ 0.8R_2+R_2=4700\\ R_2=\frac{4700}{1.8}\\ R_2=2611.11\Omega \\ R_1=4700\Omega -2611.11\Omega \\ R_1=2088.89\Omega \end{array}$

Conclusion:

The value of carbon resistance is $R_1=2088.89\Omega$ and nichrome resistance $R_2=2611.11\Omega$ .