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Conduction

Three rods A,...

Question

Three rods A,B and C have the same dimensions.Their conductivities are $K_{A}, K_{B}$ and $K_{C}$ respectively. A and B are placed end to end, with their free ends kept at certain temperature difference. C is placed separately with its ends kept at same temperature difference. The two arrangements conduct heat at the same rate $K_{c}$ must be equal to

A

K_{A} + K_{B}

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B

\frac{K_{A} + K_{B}}{K_{A} K_{B}}

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C

\frac{1}{2} (K_{A} + K_{B})

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D

\frac{2 K_{A} K_{B}}{K_{A} + K_{B}}

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Solution

The correct option is D $\frac{2 K_{A} K_{B}}{K_{A} + K_{B}}$
We know that $\frac{Q}{t} = \frac{K A (Δ T)}{L}$
so all rods having same dimensions,therefor for rod $A a n d B$
equivalent thermal conductivity is given as $K_{e q u i v a l e n t} = \frac{K_{A} + K_{B}}{K_{A} K_{B}}$
$\frac{Q}{t} = \frac{2 k_{a} k_{b}}{(k_{a} + k_{b})} \times \frac{A (T_{A} - T_{B})}{2 L}$
now for rod $C$ we say $\frac{Q}{t} = \frac{K_{c} A (T_{A} - T_{B})}{L}$ ,here it is mention that both are at same temperature difference.
so equating both of them, we find $K_{C} = \frac{2 K_{a} K_{b}}{(K_{a} + K_{b})}$

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