Question #42529

A cylindrical rod of length $L$ and cross-sectional area $A$ is placed between two heat reservoirs maintained at temperatures $T_1 = 800 \text{ K}$ (at $x = 0$) and $T_2 = 200 \text{ K}$ (at $x = L$). The thermal conductivity of the rod varies with position $x$ as $k(x) = k_0 \left(1 + \frac{x}{L}\right)^2$, where $k_0$ is a positive constant. The lateral surface of the rod is perfectly insulated. In the steady state, a small hole is made at the midpoint of the rod ($x = L/2$) to observe the emitted blackbody radiation. If Wien's displacement constant is $b$, what is the wavelength $\lambda_{\text{max}}$ corresponding to the maximum spectral emissive power of the radiation emitted from the midpoint?