Question

The resistance of pure platinum increases linearly with temperature (over a small range). We can use this property to measure temperature of different bodies, relative to a scale.

If we scale the thermometer such that a resistance of 80 Ω denotes $0^{o}C$ and 90 Ω denotes ${100}^{o}C$, what temperature will a resistance of 86 Ω denote?

• A. 40⁰C
• B. 50⁰C
• C. 60⁰C
• D. 80⁰C

Answer 1

The correct option is C

60⁰C

A "linear” relation between two quantities, say, Q₁ and Q₂, means the following
$Q_1=m{Q}_2+c$
where m and c are constants. This follows from the fact that for a straight line on a plane, y and x are related as,
$y=mx+c$

$m$ being the slope and $c$ the y-intercept respectively.

In our case, this should be true between the temperature and the resistance at that temperature,
$R_{\theta }=\alpha \theta +\beta$
for some constants α & β.
Let's find out what these constants are from the information given. We can write
$80\Omega =\alpha \times 0^{o}C+\beta$
$\Rightarrow \beta =80\Omega$
$90\Omega =\alpha \times {100}^{o}C+\beta$
$\Rightarrow \alpha =\frac{90-80}{100}=0.1\Omega {/}^{o}C$

Using these values, the temperature($\theta$) can be written as
$86=0.1\times \theta +80$
$\Rightarrow \theta ={60}^{o}C$