Two rods A and B of same length and cross-sectional area are...

Two rods $A$ and $B$ of same length and cross-sectional area are connected in parallel and a temperature difference of $100^{\circ} \mathrm{C}$ is maintained across the combination as shown in the figure. If the thermal conductivity of the $\operatorname{rod} A$ is $3 K$ and that of $\operatorname{rod} B$ is $K$, then what is the equivalent thermal resistance?(a) $R=\frac{1}{4 K A}$ (b) $R=\frac{1}{2 K A}$ (c) $R=\frac{2 l}{K A}$ (d) $R=\frac{3 l}{2 K A}$

Two rods $A$ and $B$ of same length and cross-sectional area are connected in parallel and a temperature difference of $100^{\circ} \mathrm{C}$ is maintained across the combination as shown in the figure. If the thermal conductivity of the $\operatorname{rod} A$ is $3 K$ and that of $\operatorname{rod} B$ is $K$, then what is the equivalent thermal resistance?(a) $R=\frac{1}{4 K A}$
(b) $R=\frac{1}{2 K A}$
(c) $R=\frac{2 l}{K A}$
(d) $R=\frac{3 l}{2 K A}$

Key Concepts

Fourier's Law of Heat Conduction

Fourier's Law is the fundamental principle governing heat conduction. It states that the rate of heat flow through a material is proportional to the negative gradient of temperature and the cross-sectional area through which the heat is flowing, with the proportionality constant being the material's thermal conductivity. This law serves as the basis for deriving equations for thermal resistance and analyzing heat transfer in various systems.

Parallel Heat Conduction

When heat flows through multiple conductive paths in parallel, such as rods arranged side by side between two temperatures, each path provides an alternative route for the heat. The overall or equivalent thermal resistance of the system is found by combining the individual resistances in a parallel configuration, similar to combining resistances in parallel in an electric circuit.

Thermal Conductivity

Thermal conductivity is a material-specific property that measures the ability of a material to conduct heat. Higher thermal conductivity indicates that a material can transfer heat more efficiently than a material with lower thermal conductivity. This property is key to understanding how different materials behave when they are subject to a temperature gradient.

Thermal Resistance

Thermal resistance quantifies how strongly a material or component opposes the flow of heat. It is analogous to electrical resistance in circuits and is derived from the dimensions and the thermal conductivity of the material. The formula for thermal resistance in a rod is given by the length of the rod divided by the product of its cross-sectional area and thermal conductivity.

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