Three rods of same dimensions are arranged as shown in the figure. They have thermal conductivities $K_{1}, K_{1}$ and $K_{3}$ . The points $A$ and $B$ are maintained at different temperatures.For the heat to flow at the same along $A C B$ and $A B$

K_{3} = \frac{K_{1} K_{2}}{K_{1} + K_{2}}

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K_{3} = 2 (K_{1} + K_{2})

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K_{3} = \frac{K_{1} + K_{2}}{2}

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K_{3} = K_{1} + K_{2}

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Solution

The correct option is A $K_{3} = \frac{K_{1} K_{2}}{K_{1} + K_{2}}$
For the heat to flow at the same along ACB and AB,we have to do following operation,
$Q_{C} = Q_{A B}$
and here $A B$ are connected end to end so
$\frac{Q}{t} = \frac{K_{3} A (T_{1} - T_{2})}{L}$
$\frac{Q}{t} = \frac{A (T_{1} - T_{2})}{\frac{L}{K_{1}} + \frac{L}{K_{2}}}$
$\frac{K_{3} A (T_{1} - T_{2})}{L} = \frac{Q}{t} = \frac{A (T_{1} - T_{2})}{\frac{L}{K_{1}} + \frac{L}{K_{2}}}$
$K_{3} = \frac{K_{1} K_{2}}{K_{1} + K_{2}}$

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Three rods of same dimensions are arranged as shown in the figure. They have thermal conductivities K1,K1 and K3 . The points A and B are maintained at different temperatures.For the heat to flow at the same along ACB and AB

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