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Dimensional Analysis

Two rod of sa...

Question

Two rod of same length and same diameter are drawn from equal masses and same quantity of heat I is supplied to the rods. If the are rods joined together to $100^{o} C$ and $40^{o} C$ at the ends, find ratio of their conductivities.

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Solution

The two rods have same length $L$ and same cross-sectional $A (= \frac{π D^{2}}{4})$ but different conductivities $K_{1}$ & $K_{2}$ , joined in series:- [Ref. above image]
The current $(I = \frac{d Q}{d t}, Q : Heat supplied)$ is constant in series convection.
The conductivity in series (equivalent) would be:
$K^{'} = \frac{K_{1} K_{2}}{K_{1} + K_{2}}$ $[A, L$ are same $]$
We can write,
$I = K^{'} (Δ T)$ , where $Δ T$ : Temp. difference
$\Rightarrow K^{'} = \frac{I}{Δ T} = \frac{I}{100 - 40} = \frac{I}{60}$ .
Thus, $\frac{K_{1} K_{2}}{K_{1} + K_{2}} = \frac{I}{60} \Rightarrow 60 K_{1} K_{2} = I K_{1} + I K_{2}$
which given ratio:
$\frac{K_{1}}{K_{2}} = \frac{60 - I}{60}$

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