Question

A carbon resistor is to be used as a thermometer. On a winter day when the temperature is 4.00 C, the resistance of the carbon resistor is 217.3 Ω. What is the temperature on a spring day when the resistance is 216.0 Ω?

 

Take the temperature coefficient of resistivity for carbon to be α = −5.00×10⁻⁴ C^-1.

 

Answer 1

Concepts and reason

The concept required to solve this problem is the variation of resistance with respect to temperature. Rearrange the equation of variation of resistance with respect to temperature to solve for temperature and plug all values in the equation to calculate the temperature.

Fundamentals

Variation of resistance with temperature is given by the following formula. \(R=R_{0}\left[1+\alpha\left(T-T_{0}\right)\right]\)

Here, \(R\) is the temperature of resistance at temperature \(T, R_{0}\) is the resistance of resistance at some reference temperature \(T_{0}\), and \(\alpha\) is the coefficient of resistivity.

Variation of resistance with temperature is given by the following formula. \(R=R_{0}\left[1+\alpha\left(T-T_{0}\right)\right]\)

Rearrange, variation of temperature equation for temperature. \(T=\frac{1}{\alpha}\left[\frac{R}{R_{0}}-1\right]+T_{0}\)

Resistance of the resistor changes with temperature and that variation is given by the following expression. \(R=R_{0}\left[1+\alpha\left(T-T_{0}\right)\right]\)

The expression for temperature is, \(T=\frac{1}{\alpha}\left[\frac{R}{R_{0}}-1\right]+T_{0}\)

Substitute, \(-5.00 \times 10^{-4 \circ} \mathrm{C}^{-1}\) for \(\alpha, 216.0 \Omega\) for \(R, 217.3 \Omega\) for \(R_{0}\) and \(4.00^{\circ} \mathrm{C}\) for \(T_{0}\).

$$ \begin{array}{c} T=\frac{1}{-5.00 \times 10^{-4} \mathrm{C}^{-1}}\left[\frac{216.0 \Omega}{217.3 \Omega}-1\right]+4.00^{\circ} \mathrm{C} \\ =15.96^{\circ} \mathrm{C} \end{array} $$

The reference temperature is \(4.00^{\circ} \mathrm{C}\) and resistance at this temperature is \(217.3 \Omega\) which is \(R_{0}\).

The temperature was \(15.96^{\circ} \mathrm{C}\) when resistance was \(216.0 \Omega\).