A uniform metal ring with centre C have two points A and B,...

A uniform metal ring with centre $C$ have two points $A$ and $B$, such that angle $A C B$ is $\theta . A$ and $B$ are maintainedat two different constant temperatures. If the angle between $A$ and $B$, i.e. $\theta=180^{\circ}$, the rate of heat flow from $A$ to $B$ is $1.2 \mathrm{~W}$, then what will be the rate, when $\theta=90^{\circ}$ ? (a) $0.6 \mathrm{~W}$ (b) $0.9 \mathrm{~W}$ (c) $1.6 \mathrm{~W}$ (d) $1.8 \mathrm{~W}$

A uniform metal ring with centre $C$ have two points $A$ and $B$, such that angle $A C B$ is $\theta . A$ and $B$ are maintainedat two different constant temperatures. If the angle between $A$ and $B$, i.e. $\theta=180^{\circ}$, the rate of heat flow from $A$ to $B$ is $1.2 \mathrm{~W}$, then what will be the rate, when $\theta=90^{\circ}$ ?
(a) $0.6 \mathrm{~W}$
(b) $0.9 \mathrm{~W}$
(c) $1.6 \mathrm{~W}$
(d) $1.8 \mathrm{~W}$

Key Concepts

Heat Conduction

Heat conduction refers to the process by which thermal energy is transmitted through a material due to a temperature gradient. In the context of a metal ring or any continuous medium, the rate at which heat flows between two points depends on the material’s conductivity, the area through which the heat is being transferred, and the distance between the points (or the effective length of the path).

Thermal Resistance

Thermal resistance is a property of materials that quantifies how strongly a material opposes the flow of heat. In conduction problems, it is analogous to electrical resistance in circuits, where the resistance is proportional to the length of the material the heat must travel through and inversely proportional to its thermal conductivity and cross-sectional area. This concept simplifies the analysis of heat flow by allowing the use of resistance networks.

Parallel Combination of Thermal Resistances

When there are multiple paths for heat to flow between two points, such as along different segments of a ring, the thermal resistances of these paths act in parallel. The effective thermal resistance is reduced by combining these individual resistances using the formula for parallel resistors. This means that heat flow will be greater when there are multiple conductive paths, each with its own resistance, compared to a single path.

Geometrical Influence on Heat Flow

The geometry of the system, such as the angular separation between points on a ring, directly affects the lengths of the paths available for heat conduction and, consequently, the thermal resistance along each path. As the total path length varies with the angle, the division of the ring into different arcs leads to different resistances along each arc, influencing the overall effective resistance and thereby the rate of heat transfer between the points.

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