Two metal rods of same length and same malerial are...

Two metal rods of same length and same malerial are connecled betwcen two heat rescrvoirs so that a total of $Q$ (cal/s) heat is transferred from hot reservoir to cold reservoir through the rods If the respective cross-scctional area of rods be $A_{1}$ and $\Lambda_{2}$, then the respective heal flow rates are (a) $\frac{Q}{2}, \frac{Q}{2}$ (b) $\frac{Q A_{1}}{\Lambda_{1} \mid \Lambda_{2}}, \frac{Q \Lambda_{2}}{\Lambda_{1} \mid \Lambda_{2}}$ (c) $\frac{Q \Lambda_{2}}{\Lambda_{1} \mid \Lambda_{2}}, \frac{Q_{1}}{A_{1} \mid \Lambda_{2}}$ (d) $\frac{Q_{1}}{A_{2}}, \frac{Q A_{2}}{A_{1}}$

Two metal rods of same length and same malerial are connecled betwcen two heat rescrvoirs so that a total of $Q$ (cal/s) heat is transferred from hot reservoir to cold reservoir through the rods If the respective cross-scctional area of rods be $A_{1}$ and $\Lambda_{2}$, then the respective heal flow rates are
(a) $\frac{Q}{2}, \frac{Q}{2}$
(b) $\frac{Q A_{1}}{\Lambda_{1} \mid \Lambda_{2}}, \frac{Q \Lambda_{2}}{\Lambda_{1} \mid \Lambda_{2}}$
(c) $\frac{Q \Lambda_{2}}{\Lambda_{1} \mid \Lambda_{2}}, \frac{Q_{1}}{A_{1} \mid \Lambda_{2}}$
(d) $\frac{Q_{1}}{A_{2}}, \frac{Q A_{2}}{A_{1}}$

Key Concepts

Thermal Resistance

Thermal resistance is a measure of a material's opposition to heat flow and is given by R = L/(kA) for conduction problems. In cases where multiple conductive paths are present, such as rods in parallel, the effective thermal resistance of the system is determined by the combination of the individual resistances. This concept helps in analyzing how changes in geometry (like cross-sectional area) influence the rate of heat transmission.

Fourier's Law of Heat Conduction

Fourier's law states that the rate at which heat is conducted through a material is proportional to the negative gradient of temperature and the cross-sectional area through which the heat flows. This law is typically expressed as Q/t = -kA(?T/L), where k is the thermal conductivity, A is the cross-sectional area, ?T is the temperature difference, and L is the length of the material. This principle allows us to predict heat transfer through conductive media such as rods.

Thermal Conduction in Parallel

When multiple conductive paths, such as metal rods, are placed in parallel between two reservoirs with the same temperature difference, each path carries a fraction of the total heat according to its own conductive properties. Since the rods have the same material and length, differences in cross-sectional area directly influence the individual heat flow rates. This concept is essential for understanding how heat is distributed among parallel conductive elements.

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