Thermal Conduction

Question

A metal rod of length 0.5 m and uniform cross-sectional area $4	imes10^{-4}\,	ext{m}^2$ has its ends maintained at $100^\circ	ext{C}$ and $0^\circ	ext{C}$. If the thermal conductivity of the metal is $200\,	ext{W m}^{-1}\,^\circ	ext{C}^{-1}$, what is the steady rate of heat conduction along the rod?

RankGuru · Accepted Answer

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 $k = 200$, $A = 4	imes10^{-4}\,	ext{m}^2$, $(T_1 - T_2) = 100^\circ	ext{C}$ and $L = 0.5\,	ext{m}$ gives $\frac{Q}{t} = \frac{200	imes4	imes10^{-4}	imes100}{0.5} = \frac{8}{0.5} = 16\,	ext{W}$. 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.

Thermal Conduction

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