Two rods P and Q have equal lengths. Their thermal...

Two rods $P$ and $Q$ have equal lengths. Their thermal conductivities are $K_{1}$ and $K_{2}$ and cross-sectional areas are $A_{1}$ and $A_{2}$. When the temperature at ends of each rod are $T_{1}$ and $T_{2}$ respectively, the rate of flow of heat through $P$ and $Q$ will be equal, if (a) $\frac{A_{1}}{A_{2}}=\frac{K_{2}}{K_{1}}$ (b) $\frac{A_{1}}{A_{2}}=\frac{K_{2}}{K_{1}} \times \frac{T_{2}}{T_{1}}$ (c) $\frac{A_{1}}{A_{2}}=\sqrt{\frac{K_{1}}{K_{2}}}$ (d) $\frac{A_{1}}{A_{2}}=\left(\frac{K_{1}}{K_{1}}\right)^{2}$

Two rods $P$ and $Q$ have equal lengths. Their thermal conductivities are $K_{1}$ and $K_{2}$ and cross-sectional areas are $A_{1}$ and $A_{2}$. When the temperature at ends of each rod are $T_{1}$ and $T_{2}$ respectively, the rate of flow of heat through $P$ and $Q$ will be equal, if
(a) $\frac{A_{1}}{A_{2}}=\frac{K_{2}}{K_{1}}$
(b) $\frac{A_{1}}{A_{2}}=\frac{K_{2}}{K_{1}} \times \frac{T_{2}}{T_{1}}$
(c) $\frac{A_{1}}{A_{2}}=\sqrt{\frac{K_{1}}{K_{2}}}$
(d) $\frac{A_{1}}{A_{2}}=\left(\frac{K_{1}}{K_{1}}\right)^{2}$

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Key Concepts

Thermal Conductivity

Thermal conductivity is an intrinsic property of a material that measures its ability to conduct heat. A material with a high thermal conductivity will allow heat to pass through it more efficiently compared to one with a low thermal conductivity. This concept is essential when comparing the heat transfer rates in different materials under similar conditions.

Cross-sectional Area in Heat Conduction

The cross-sectional area of a material directly affects its ability to conduct heat. A larger cross-sectional area provides a greater pathway for the transfer of thermal energy, thereby increasing the rate at which heat flows through the material. This relationship is critical when adjusting dimensions to achieve equal heat flow in different conductive paths.

Fourier's Law of Heat Conduction

Fourier's Law states that the rate of heat transfer through a material is directly proportional to the material's thermal conductivity, the cross-sectional area, and the temperature difference across it, and inversely proportional to the length of the material. This fundamental principle explains how various factors such as material properties and geometric dimensions influence the conduction of heat.

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