Question

The thermo E.M.F E in volts of a certain thermocouple is said to vary with temperature difference $\theta$ in $^{\circ }C$ between the two junctions according to the relation
$E=30\theta -\dfrac{{{\theta }^{2}}}{15}$
The neutral temperature of the thermos couple will be?
A.)${{450}^{\circ }}C$
B.)${{400}^{\circ }}C$
C.)${{225}^{\circ }}C$
D.)${{30}^{\circ }}C$

Answer 1

Hint: A thermocouple produces a temperature-dependent voltage as a result of the thermoelectric effect. The temperature of the hot junction, at which the thermo emf produced is maximum, is called neutral temperature. So, we know that at a point of maximum the first-order differential of the E.M.F induced wrt temperature $\left( \theta \right)$ will be zero. So, differentiating the given equation with respect to temperature $\left( \theta \right)$ and equating it with zero, we can find the neutral temperature.

Complete step by step answer:
The neutral temperature is the temperature at which the e.m.f produced by the hot junction is maximum. So we are given an equation for thermos e.m.f as a function of $\theta$. So the neutral temperature will occur at a point where E is a maximum. So a maxima of a function will have its first order derivative of the function as zero. So the given equation is

$E=30\theta -\dfrac{{{\theta }^{2}}}{15}$

We will differentiate this equation and equate it to zero to find out the neutral temperature.
$\dfrac{dE}{d\theta }=30-\dfrac{2\theta }{15}$

If we equate the above derivative to zero, we get

$\begin{aligned} & 30-\dfrac{2\theta }{15}=0 \\\ & \Rightarrow \theta =\dfrac{30\times 15}{2} \\\ \end{aligned}$
$\therefore \theta ={{225}^{\circ }}C$

So the neutral temperature for the given thermocouple occurs at a temperature of ${{225}^{\circ }}C$.

So the answer to the question is option (C).

Additional Information:
A thermocouple is an electrical device consisting of two dissimilar electrical conductors forming an electrical junction. A thermocouple produces a temperature-dependent voltage as a result of the thermoelectric effect, and this voltage can be interpreted to measure temperature.

Note: At neutral temperature, the temperature of the cold junction is kept at ${{0}^{\circ }}C$.
The basic working principle of a thermocouple is that the electrons from metal with a greater electron density diffuse into the other which has a state of lower electron density, this process establishes a small potential gradient.
If one of the junctions in a thermocouple is heated, it will result in electron diffusion as a result produces a higher potential at the hot junction, which is greater than the potential at the cold junction. So, a thermal EMF is always produced, whenever there is a temperature variation between the junctions. So, when there is no temperature variation between the junctions, no thermal EMF is produced.