Q. A large cylindrical rod of length L is made by joining two identical rods of copper and steel of length $\left(\frac{L}{2}\right)$ each. The rods are completely insulated from the surroundings. If the free end of copper rod is maintained at $100^°C$ and that of steel at $0^°C$ then the temperature of junction is (Thermal conductivity of copper is 9 times that of steel)
AIEEEAIEEE 2012Thermal Properties of Matter
Solution:
Let conductivity of steel $K_{steel} = k$ then from question
Conductivity of copper $K_{copper} = 9k$
$\theta_{copper}=100^{°}C$
$\theta_{steel}=0^{°}C$
$l_{steel}=l_{copper}=\frac{L}{2}$
From formula temperature of junction;
$\theta=\frac{K_{copper}\,\theta_{copper}\,l_{steel}+k_{steel}\,\theta_{steel}\,l_{copper}}{K_{copper}\,l_{steel}+\,K_{steel}\,l_{copper}}$
$=\frac{9k\times 100\times\frac{L}{2}+k\times0\times\frac{L}{2}}{9k\times\frac{L}{2}+k\times\frac{L}{2}}$
$=\frac{\frac{900}{2}kL}{\frac{10kL}{2}}=90^{°}C$
