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(B) Let the temperature at junction $B$ be $T_B$. Since the rods are of the same material and cross-section, their thermal resistance $R$ is proportio

Vedclass · Accepted Answer

$3.5$

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(B) Let the temperature at junction $B$ be $T_B$. Since the rods are of the same material and cross-section, their thermal resistance $R$ is proportional to their length $L$ $(R = \frac{L}{kA})$. Given that there is no heat flow in $AB$, the temperature at $B$ must be equal to the temperature at $A$. Therefore, $T_B = T_A = 20 ^oC$. Since there is no heat flow in $AB$, all the heat flowing from $D$ to $B$ must flow from $B$ to $C$. Thus, the rate of heat flow $H_{DB} = H_{BC}$. Using the formula $H = \frac{\Delta T}{R}$, we have $\frac{T_D - T_B}{R_{BD}} = \frac{T_B - T_C}{R_{BC}}$. Substituting the values: $\frac{90 - 20}{R_{BD}} = \frac{20 - 0}{R_{BC}}$. $\frac{70}{R_{BD}} = \frac{20}{R_{BC}} \implies \frac{R_{BD}}{R_{BC}} = \frac{70}{20} = 3.5$. Since $R \propto L$, the ratio of lengths $\frac{L_{BD}}{L_{BC}} = \frac{R_{BD}}{R_{BC}} = 3.5$.

Three conducting rods of the same material and cross-section are shown in the figure. The temperatures at $A$, $D$, and $C$ are maintained at $20 ^oC$, $90 ^oC$, and $0 ^oC$ respectively. If there is no heat flow in $AB$, find the ratio of the lengths of $BD$ and $BC$.

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