Question

Three identical rods of length $1m$ each, having cross-section area of $1c{m}^2$ each and made of Aluminium, copper and steel respectively are maintained at temperatures of ${12}^{\circ }C$, $4^{\circ }C$ and ${50}^{\circ }C$ respectively at their separate ends. Find the temperature of their common junction.
$[K_{cu}=400W/m-K,K_{Al}=200W/m-K,K_{steel}=50W/m-K]$

• A. $T={10}^o$
• B. $T={100}^o$
• C. $T={60}^o$
• D. $T={90}^o$

Answer 1

The correct option is A $T={10}^o$

$R_{Al}=\frac{L}{KA}=\frac{1}{{10}^{-4}\times 200}=\frac{{10}^4}{200}$
Similarly $R_{steel}=\frac{{10}^4}{50}$ and $R_{copper}=\frac{{10}^4}{400}$
Let temperature of common junction = T
Then from Kirchoff's current laws.
$i_{Al}+i_{steel}+i_{cu}=0$
$\Rightarrow$ $\frac{T-12}{R_{Al}}+\frac{T-50}{R_{steel}}+\frac{T-4}{R_{cu}}=0$
$\Rightarrow$ $(T-12)200+(T-50)50+(T-4)400$
$\Rightarrow$ 4(T - 12) + (T - 50) + 8 (T - 4) = 0
$\Rightarrow$ 13T = 48 + 50 + 32 = 130
$\Rightarrow$ T = ${10}^{\circ }$C