Thermal Conduction

In steady-state conduction heat flows from the hot end to the cold end through the lattice and free electrons, and the rate is given by Fourier's law $\frac{Q}{t}=\frac{kA(T_1-T_2)}{L}$, where $k$ is the thermal conductivity, $A$ the area, $L$ the length and $(T_1-T_2)$ the temperature difference across the rod. Substituting $k=200$, $A=4\times 10^{-4}{\text{m}}^2$, $(T_1-T_2)=100^{\circ }\text{C}$ and $L=0.5\text{m}$ gives $\frac{Q}{t}=\frac{200\times 4\times 10^{-4}\times 100}{0.5}=\frac{8}{0.5}=16\text{W}$. The value 8 W omits a factor by using twice the length. The value 32 W halves the length incorrectly. The value 40 W uses an inconsistent temperature difference. This direct application of Fourier's law mirrors the NCERT treatment of conduction in solids, where in the steady state every cross-section carries the same heat current and the temperature falls linearly along the rod. The quantity $kA/L$ is the thermal conductance, the direct analogue of electrical conductance, so heat flow through a rod is mathematically identical to charge flow through a resistor. As a plausibility check, the answer carries units of watts because conductivity times area times temperature gradient yields power, and a 16 W flow through a short, good-conducting metal rod under a 100-degree difference is physically reasonable.