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(D) Let $R_1 = \frac{l}{K_1 A}$ be the thermal resistance of a copper wire and $R_2 = \frac{l}{K_2 A}$ be the thermal resistance of an iron wire.The a

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$\frac{2l}{(K_1 + K_2)A}$

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(D) Let $R_1 = \frac{l}{K_1 A}$ be the thermal resistance of a copper wire and $R_2 = \frac{l}{K_2 A}$ be the thermal resistance of an iron wire.The arrangement forms a Wheatstone bridge network. The arms $AB$ and $BC$ have resistance $R_1$,while arms $AD$ and $DC$ have resistance $R_2$. The central arm $BD$ consists of two wires (one copper and one iron) in parallel,but due to the symmetry of the bridge,the potential (temperature) at $B$ and $D$ is equal.Since the temperatures at $B$ and $D$ are equal,no heat flows through the central arm $BD$. Thus,the bridge is balanced and the central arm is ineffective.The circuit simplifies to two parallel branches,each containing two resistors in series:Branch $ABC$: $R_{ABC} = R_1 + R_1 = 2R_1$Branch $ADC$: $R_{ADC} = R_2 + R_2 = 2R_2$The equivalent thermal resistance $R_{eq}$ between $A$ and $C$ is given by the parallel combination of these two branches:$R_{eq} = \frac{(2R_1)(2R_2)}{2R_1 + 2R_2} = \frac{4R_1 R_2}{2(R_1 + R_2)} = \frac{2R_1 R_2}{R_1 + R_2}$Substituting $R_1 = \frac{l}{K_1 A}$ and $R_2 = \frac{l}{K_2 A}$:$R_{eq} = \frac{2 \left( \frac{l}{K_1 A} \right) \left( \frac{l}{K_2 A} \right)}{\frac{l}{K_1 A} + \frac{l}{K_2 A}} = \frac{2 \frac{l^2}{K_1 K_2 A^2}}{\frac{l(K_1 + K_2)}{K_1 K_2 A}} = \frac{2l}{A(K_1 + K_2)}$

Six wires each of cross-sectional area $A$ and length $l$ are combined as shown in the figure. The thermal conductivities of copper and iron are $K_1$ and $K_2$ respectively. The equivalent thermal resistance between points $A$ and $C$ is

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