One end of a solid cylindrical copper rod 0.200 m long is maintained at a temperature of 20.00 K

step 1 The rate at which heat is absorbed at the blackened end of the rod is the heat current in the ro

One end of a solid cylindrical copper rod 0.200 m long is...

One end of a solid cylindrical copper rod 0.200 $\mathrm{m}$ long is maintained at a temperature of 20.00 $\mathrm{K}$ . The other end is blackencl and exposed to thermal radiation from surrounding walls at 500.0 $\mathrm{K}$ . The sides of the rod are insulated, so no energy is lost or gained except at the ends of the rod. When equilibrium is reached, what is the temperature of the blackened end? (Hint: Since copper is a very good conductor of heat at low temperature, with $k=1670 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ at 20 $\mathrm{K}$ , the temperature of the blackened end is only slightly higher than $20.00 \mathrm{K} .$ )

Key Concepts

Fourier's Law of Heat Conduction

This law quantifies the rate of heat transfer through a material in response to a temperature gradient. It relates the heat transfer rate to the material's thermal conductivity, the cross-sectional area, and the temperature difference over a given distance. In the context of the problem, Fourier's law is used to assess how effectively the copper rod conducts heat from the cold to the warmer end, which is crucial for establishing the equilibrium temperature at the blackened end.

Stefan-Boltzmann Law

This law governs the power radiated from a black body in terms of its temperature. It states that the emitted energy per unit area is proportional to the fourth power of the absolute temperature. In the problem, this law is used to determine the radiative heat loss from the blackened end of the rod to its surroundings, thereby linking the temperature at that end to the energy balance with conduction.

Thermal Equilibrium and Energy Balance

Thermal equilibrium occurs when the rate of heat conduction into a region equals the rate of heat loss due to radiation, resulting in a steady temperature distribution. In this problem, the energy conducted to the blackened end of the rod is balanced by the radiation loss from that end, leading to a near-steady state condition where the temperature at the radiating end is only marginally higher than the fixed cold temperature.

Material Thermal Conductivity

Thermal conductivity is a material property that measures its ability to conduct heat. Copper, known for its high thermal conductivity, allows efficient heat transfer, minimizing temperature gradients within it. This high conductivity implies that even when one end is exposed to much higher temperatures, the temperature rise at that end is limited when in equilibrium with radiative losses.

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