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 the figure. Fach 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 the figure. Fach 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

Steady-State Heat Conduction

This concept refers to a condition where the temperature distribution within a system remains constant over time because the rate of heat entering any part of the system is equal to the rate of heat leaving. It underpins the analysis of systems where conduction pathways are in balance and is governed by Fourier’s law of heat conduction.

Thermal Resistance

Thermal resistance quantifies the opposition to heat flow through a material, analogous to electrical resistance in circuits. It is defined based on the material's conductivity, cross-sectional area, and length, and it facilitates the calculation of temperature differences across components in a system.

Parallel Heat Conduction Paths

When conduction elements (such as rods) are arranged in parallel between two fixed temperatures, they share the heat flow. This arrangement reduces the effective thermal resistance of the system, thereby increasing the overall rate of heat transfer, similar to how parallel electrical resistances lower total resistance.

Energy Balance in Conduction

This principle, analogous to Kirchhoff’s current law in electrical circuits, states that in a steady state the sum of all heat flows entering a junction must equal the sum of all heat flows leaving it. This energy conservation approach is crucial for determining unknown temperatures at points where multiple conduction paths meet.

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