A resistance thermometer reads R- 20 -0- -27-5- - and 50 -0- at the ice point 0 -C - the steam point 100 -C - and the zine point 420 -C - respectively- Assuming that the resistance varies with temperature as R - R 0 1- - 2 - find the values of R 0 - - and - Here - represents the temperature on Celsius scale-

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A resistance ...

Question

A resistance thermometer reads $R = 20^{\circ} 0 Ω, 27.5 Ω,$ and $50^{\circ} 0 Ω$ at the ice point $(0^{\circ} C),$ the steam point $(100^{\circ} C),$ and the zine point $(420^{\circ} C),$ respectively. Assuming that the resistance varies with temperature as $R_{θ} = R_{0} ({1 + α θ + β θ}^{2}),$ find the values of $R_{0},$ $α$ and $β$ . Here $θ$ represents the temperature on Celsius scale.

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Solution

$R$ at ice point $(R_{0}) = 20 Ω$
$R$ at steam point $(R_{100}) = 27.5 Ω$
$R$ at Zinc point $(R_{420}) = 50 Ω$

The resistance at particular value of temperature is given as:
$R_{θ} = R_{0} (1 + α θ + β θ^{2})$
$\Rightarrow R_{100} = R_{0} + R_{0} α θ + R_{0} β θ^{2}$

$\Rightarrow \frac{R_{100} - R_{0}}{R_{0}} = α θ + β θ^{2}$

$\Rightarrow \frac{27.5 - 20}{20} = α \times 100 + β \times 10000$

$\Rightarrow \frac{7.5}{20} = 100 α + 10000 β$

$R_{420} = R_{0} (1 + α θ + β θ^{2}) \Rightarrow \frac{50 - R_{0}}{R_{0}} = α θ + β θ^{2}$

$\Rightarrow \frac{50 - 20}{20} = 420 \times α + 176400 \times β$

$\Rightarrow \frac{3}{2} = 420 α + 176400 β$

$\Rightarrow \frac{7.5}{20} = 100 α + 10000 β$

$\Rightarrow \frac{3}{2} = 420 α + 176400 β$

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A resistance thermometer reads R = 20.0 $Ω$ , 27.5 $Ω$ , and 50.0 $Ω$ at the ice point ( $0^{\circ} C$ ), the steam point ( $100^{\circ} C$ ) and the zinc point ( $420^{\circ} C$ ) respectively. Assuming that the resistance varies with temperature as $R_{θ} = R_{0} (1 + α θ + β θ^{2})$ , find the values of $R_{0}, α$ and $β$ . Here $θ$ represents the temperature on Celsius scale.