Question

The thermo-emf of a thermocouple varies with the temperature $\theta$ of the hot junction as $E=a\theta +b{{\theta }^{2}}$ in volts where the ratio $\dfrac{a}{b}$ is $700{}^\circ C$. If the cold junction is kept at $0{}^\circ C$, then the neutral temperature is:
A. $700{}^\circ C$
B. $350{}^\circ C$
C. $1400{}^\circ C$
D. no neutral temperature is possible for this thermocouple

Answer 1

Firstly, you could recall the definition of the neutral temperature of a thermocouple which is the temperature for which emf attains maximum value. This could be attained by differentiating the given expression of variation of emf with temperature and then equating it to zero. The temperature so attained is the temperature difference between hot junction and that of the cold junction.

Complete answer:
In the question, we are given the variation of thermo-emf of a thermocouple with temperature $\theta$. We are also given the ratio of the constants in it. We are asked to find the neutral temperature when the cold junction is kept at $0{}^\circ C$ using the given information.
We know that, by definition, neutral temperature is the temperature of the hot junction of a thermocouple at which its emf attains a maximum value for the constant temperature of $0{}^\circ C$ for cold junction is maintained.
So we are given the variation of emf with temperature as,
$E=a\theta +b{{\theta }^{2}}$
In order to find the maximum value of emf for a $\theta$, we need to maximize the given equation. Maximization can be done by differentiating the given equation with respect to$\theta$ and then equating to zero. So, differentiating we get,
$\dfrac{dE}{d\theta }=a+2b\theta$
Now we could maximize this by equating to zero as,
$a+2b\theta =0$
$\Rightarrow \theta =\dfrac{-a}{2b}$
But we are given the ratio of a and b as,
$\dfrac{a}{b}=700{}^\circ C$
Substituting this value,
$\theta =\dfrac{-700}{2}$
$\therefore \theta =-350{}^\circ C$
Therefore, we found that the maximum value possible for $\theta$ as per the given conditions in the question is $-350{}^\circ C$. But, this maximum value of $\theta$ cannot be considered as neutral temperature as it should be always positive as it is by definition the temperature difference between hot junction and that of the cold junction.
Therefore, we could say that there is possible neutral temperature for this thermocouple.

Hence, option D is the correct answer.

Note:
A thermocouple is that device that produces a temperature dependent voltage which results from thermoelectric effect and is a widely used type of temperature sensor. Precision is the major limitation of this device. It is really very difficult to obtain system errors that are less than one degree Celsius.