Question 94 of 126

If ${K}_{1}$ and ${K}_{2}$ are the thermal conductivities ${L}_{1}$ and ${L}_{2}$ are the lengths and ${A}_{1}$ and ${A}_{2}$ are the cross sectional areas of steel and copper rods respectively such that $\dfrac{{K}_{2}}{{K}_{1}}=9,\dfrac{{A}_{1}}{{A}_{2}}=2,\dfrac{{L}_{1}}{{L}_{2}}=2$. Then, for the arrangement as shown in the figure. The value of temperature $T$ of the steel - copper junction in the steady state will be

A$18^{\circ}C$B$14^{\circ}C$C$45^{\circ}C$D$150^{\circ}C$

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Answer: (C) $45^{\circ}C$

At steady state, the heat conducted by both the rods will be the same.
The rate of heat conduction is given by, $\dfrac{dQ}{dt}=\dfrac{KA}{l}∆T$. Therefore, if the temperature of the junction is $T$,
$\dfrac{{K}_{1}{A}_{1}}{{l}_{1}}\left({T}_{1}-T\right)=\dfrac{{K}_{2}{A}_{2}}{{l}_{2}}\left(T-{T}_{2}\right)$
$\Rightarrow \dfrac{450-T}{T-0}=\dfrac{{K}_{2}{A}_{2}{l}_{1}}{{K}_{1}{A}_{1}{l}_{2}}=9\times \dfrac{1}{2}\times 2$
$\Rightarrow 450-T=9T \Rightarrow T=45^{\circ}C$