Three rods made of the same material and having the same...

Three rods made of the same material and having the same cross-section have been joined as shown in Fig. $11.2$. Each rod is of the same length. The left and right ends are kept at $0^{\circ} \mathrm{C}$ and $90^{\circ} \mathrm{C}$, respectively. The temperature of the junction of the three rods will be (A) $45^{\circ} \mathrm{C}$ (B) $60^{\circ} \mathrm{C}$ (C) $30^{\circ} \mathrm{C}$ (D) $20^{\circ} \mathrm{C}$

Three rods made of the same material and having the same cross-section have been joined as shown in Fig. $11.2$. Each rod is of the same length. The left and right ends are kept at $0^{\circ} \mathrm{C}$ and $90^{\circ} \mathrm{C}$, respectively. The temperature of the junction of the three rods will be
(A) $45^{\circ} \mathrm{C}$
(B) $60^{\circ} \mathrm{C}$
(C) $30^{\circ} \mathrm{C}$
(D) $20^{\circ} \mathrm{C}$

Key Concepts

Energy Conservation at Junctions

At any junction where multiple conductive paths meet, the principle of conservation of energy (or the thermal equivalent of Kirchhoff’s current law) applies. The sum of all incoming heat fluxes must equal the sum of all outgoing fluxes. This balance is used to determine unknown temperatures at the junction by ensuring that the conductive heat flows from all connecting rods satisfy the equilibrium condition.

Thermal Resistance

Thermal resistance is a measure that quantifies a material's opposition to heat flow, analogous to electrical resistance in circuits. When multiple conductive paths exist, such as in parallel or series configurations, the thermal resistances can be combined to analyze how the overall heat transfer occurs within the system.

Fourier's Law of Heat Conduction

This law states that the rate at which heat is transferred through a material is proportional to the negative gradient of temperatures and the cross-sectional area, and inversely proportional to the length. It is the fundamental principle used to describe how heat flows in a solid medium, and it is essential for calculating temperature distributions along conductive paths.

Steady-State Heat Transfer

In steady-state conditions, the temperature distribution in a system no longer changes with time. This means that the amount of heat entering any portion of the system equals the amount of heat leaving it. This concept is crucial when analyzing networks of conductors, as it ensures that the net heat flow at any junction or interface is zero.

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