Two ends of a conducting rod of varying cross-section are...

Two ends of a conducting rod of varying cross-section are maintained at $200^{\circ} \mathrm{C}$ and $0^{\circ} \mathrm{C}$ respectively. In steady state (a) temperature differences across $A B$ and $C D$ are equal (b) temperature difference across $A B$ is greater than that across $C D$ (c) temperature difference across $A B$ is less than that across $C D$ (d) temperature difference may be equal or different depending upon thermal conductivity of the rod

Two ends of a conducting rod of varying cross-section are maintained at $200^{\circ} \mathrm{C}$ and $0^{\circ} \mathrm{C}$ respectively. In steady state
(a) temperature differences across $A B$ and $C D$ are equal
(b) temperature difference across $A B$ is greater than that across $C D$
(c) temperature difference across $A B$ is less than that across $C D$
(d) temperature difference may be equal or different depending upon thermal conductivity of the rod

Key Concepts

Temperature Gradient

The temperature gradient is the rate at which temperature changes with distance within a material. In the context of steady state conduction, it is directly influenced by the cross-sectional area and thermal conductivity of the material. Where the cross-sectional area is smaller (or thermal resistance higher), a steeper temperature gradient is required for the same amount of heat to be transferred.

Thermal Resistance

Thermal resistance is a measure of a material's opposition to heat flow, analogous to electrical resistance in a circuit. It depends on the geometry of the material (length and cross-sectional area) and its thermal conductivity. In a rod with varying cross-section, different segments will have different thermal resistances, affecting the local temperature gradients that must adjust to maintain a uniform heat flux.

Fourier's Law

Fourier's law of heat conduction states that the rate of heat transfer through a material is proportional to the negative of the temperature gradient across it and to the cross-sectional area through which the heat is flowing. This law provides the fundamental relationship between the microscopic properties of a material (such as thermal conductivity) and the macroscopic rate of energy transfer.

Steady State Heat Conduction

This concept refers to a condition in which the temperature at any given point in a medium does not change over time, meaning that the heat flowing into any section of the material is exactly balanced by the heat flowing out. In steady state, despite spatial variations in material properties or geometry, the heat flux remains constant along the conducting path.

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