Question

What is the effect of temperature on the resistance of a metal? The resistance of a platinum resistance thermometer at $0{}^\circ C$temperature is$3.0\Omega$ and at$100{}^\circ C$ it is$3.75\Omega$. Its resistance at an unknown temperature is $3.15\Omega$. Find the value of the unknown temperature.

Answer 1

Firstly, we will compute the temperature coefficient of resistance using the values of resistances at the temperature values of $0{}^\circ C$and $100{}^\circ C$. Using this value of the temperature coefficient of resistance, we will find the value of the temperature at the resistance values at $0{}^\circ C$and at unknown temperature.

Formula used:
${{R}_{t}}={{R}_{0}}(1+\alpha T)$

Complete answer:
The formula used to find the resistance of a substance at a given temperature is given as follows.

${{R}_{t}}={{R}_{0}}(1+\alpha T)$

Where ${{R}_{t}}$is the temperature , ${{R}_{0}}$is the resistance at the temperature $0{}^\circ C$, $\alpha$ is the temperature coefficient of resistance and T is the temperature.

Firstly, we will compute the value of the temperature coefficient of resistance using the values of resistance at the temperature values of $0{}^\circ C$and $100{}^\circ C$

The formula used to define the relation between the resistance and temperature is,

${{R}_{t}}={{R}_{0}}(1+\alpha T)$

Rearrange the terms of the above equation to represent the expression in terms of the temperature coefficient of the resistance. So, we have,

$\alpha =\dfrac{{{R}_{t}}-{{R}_{0}}}{{{R}_{0}}T}$

Substitute the values in the above equation.

& \alpha =\dfrac{3.75-3}{3\times 100} \\\ & \Rightarrow \alpha =0.25\times {{10}^{-2}}/{}^\circ C \\\ \end{aligned}$$ Therefore, the value of the temperature coefficient of the resistance is $$0.25\times {{10}^{-2}}/{}^\circ C$$ Now, we will compute the unknown temperature. The formula used to define the relation between the resistance and temperature is, $${{R}_{t}}={{R}_{0}}(1+\alpha T)$$ Rearrange the terms of the above equation to represent the expression in terms of the temperature coefficient of the resistance. So, we have, $$T=\dfrac{{{R}_{t}}-{{R}_{0}}}{\alpha {{R}_{0}}}$$ Substitute the values in the above equation. $$\begin{aligned} & T=\dfrac{3.15-3}{3\times 0.25\times {{10}^{-2}}} \\\ & \Rightarrow T=20{}^\circ C \\\ \end{aligned}$$ **$$\therefore $$ The value of the unknown temperature is $$20\,{}^\circ C$$.** **Note:** In this problem, the units of the temperature are given in degree Celsius, so, no need to change, otherwise, the unit should be converted to Kelvin. The units of the parameters should be taken care of.