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(A) The thermoelectric power $S$ is defined as the rate of change of thermo-emf with respect to temperature,given by $S = \frac{dE}{dT}$.Given the exp

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$\frac{1}{2} K T_0$

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(A) The thermoelectric power $S$ is defined as the rate of change of thermo-emf with respect to temperature,given by $S = \frac{dE}{dT}$.Given the expression for emf: $E = K(T - T_r) [T_0 - \frac{1}{2}(T + T_r)]$.Let $x = (T - T_r)$. Then $T = x + T_r$. Substituting this into the expression:$E = Kx [T_0 - \frac{1}{2}(x + T_r + T_r)] = Kx [T_0 - \frac{1}{2}(x + 2T_r)] = Kx [T_0 - \frac{1}{2}x - T_r] = K [x(T_0 - T_r) - \frac{1}{2}x^2]$.Differentiating with respect to $T$ (noting $\frac{dx}{dT} = 1$):$\frac{dE}{dT} = K [(T_0 - T_r) - x] = K [T_0 - T_r - (T - T_r)] = K(T_0 - T)$.At $T = \frac{1}{2}T_0$:$S = K(T_0 - \frac{1}{2}T_0) = K(\frac{1}{2}T_0) = \frac{1}{2} K T_0$.

One junction of a thermocouple is kept at a constant temperature $T_r$ and the other junction is kept at a temperature $T$. If the thermo-emf is given by $E = K(T - T_r) [T_0 - \frac{1}{2}(T + T_r)]$,what is the thermoelectric power at $T = \frac{1}{2}T_0$?

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