= Solution
A generator couple has a p-type and an n-type thermoelement joined electrically at the hot face, with their other ends connected through a load at the cold face. The legs are \b[electrically in series and thermally in parallel]. Electrically insulating, thermally conducting plates couple them to the hot source and cold sink; metal interconnects carry current. The <Seebeck effect> provides an open-circuit electromotive force $\mathcal E=\alpha_{np}\Delta T$, where the magnitude of $\alpha_{np}$ is the difference between the two leg Seebeck coefficients.
For fixed junction temperatures, constant properties and internal resistance $R$,
$$
P_L=I^2R_L=\frac{(\alpha_{np}\Delta T)^2R_L}{(R+R_L)^2}.
$$
Differentiate with respect to load resistance:
$$
\frac{dP_L}{dR_L}=(\alpha_{np}\Delta T)^2\frac{R-R_L}{(R+R_L)^3}.
$$
The derivative changes from positive to negative at $R_L=R$, so
$$
\boxed{R_L=R,\qquad I_{P_{\max}}=\frac{\alpha_{np}\Delta T}{2R},\qquad P_{\max}=\frac{(\alpha_{np}\Delta T)^2}{4R}.}
$$
The load voltage is then half the open-circuit voltage. The heat source and sink must sustain the temperature difference; an isolated hot object cools as energy is extracted. \b[Maximum power and maximum conversion efficiency are different operating points]: the efficiency-optimal load is generally larger than $R$.
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