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 $$a/b$$ is $$700^\circ$$C. If the cold junction is kept at $$0^\circ$$C, then the neutral temperature is

The thermo emf ($$E$$) of the thermocouple varies with the temperature ($$\theta$$) of the hot junction according to the given equation:

$$ E = a\theta + b\theta^2 $$

The neutral temperature ($$\theta_n$$) is defined as the temperature of the hot junction at which the thermo emf reaches its maximum value. At this maximum point, the derivative of the emf with respect to temperature is zero:

$$ \frac{dE}{d\theta} = 0 $$

Differentiating the given equation with respect to $$\theta$$:

$$ \frac{dE}{d\theta} = a + 2b\theta $$

Equating this derivative to zero to find the neutral temperature $$\theta_n$$:

$$ a + 2b\theta_n = 0 $$

$$ 2b\theta_n = -a $$

$$ \theta_n = -\frac{a}{2b} $$

We are given that the ratio $$\frac{a}{b} = 700^\circ\text{C}$$. Substituting this value into our expression for $$\theta_n$$:

$$ \theta_n = -\frac{1}{2} \left( \frac{a}{b} \right) $$

$$ \theta_n = -\frac{1}{2} (700^\circ\text{C}) $$

$$ \theta_n = -350^\circ\text{C} $$

The calculated neutral temperature is $$-350^\circ\text{C}$$. However, according to the laws of thermodynamics, the lowest possible temperature achievable (absolute zero) is $$0\text{ K}$$, which corresponds to approximately $$-273.15^\circ\text{C}$$.

Since $$-350^\circ\text{C}$$ falls below absolute zero, it is a physically impossible temperature.

Therefore, no neutral temperature is possible for this particular thermocouple.

Answer: Option (D)