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(A) The thermoelectric power $S$ is defined as the rate of change of thermo electromotive force $E$ with respect to temperature $T$,given by $S = \fra

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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 electromotive force $E$ with respect to temperature $T$,given by $S = \frac{dE}{dT}$.Given the expression $E = K(T - T_r) \left[ T_0 - \frac{1}{2}(T + T_r) \right]$.Let $x = (T - T_r)$,then $T = x + T_r$. The expression becomes $E = Kx \left[ T_0 - \frac{1}{2}(x + 2T_r) \right] = K \left[ T_0 x - \frac{1}{2}x^2 - T_r x \right]$.Differentiating with respect to $T$ (noting $\frac{dx}{dT} = 1$):$S = \frac{dE}{dT} = K \left[ T_0 - x - T_r \right] = K \left[ T_0 - (T - T_r) - T_r \right] = K(T_0 - T)$.Substituting $T = \frac{1}{2}T_0$ into the expression for $S$:$S = K \left( T_0 - \frac{1}{2}T_0 \right) = \frac{1}{2} K T_0$.

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