A large cylindrical rod of length $L$ is made by joining two identical rods of copper and steel of length $(\frac {L}{2})$ each . The rods are completely insulated from the surroundings. If the free end of copper rod is maintained at $100\,^oC$ and that of steel at $0\,^oC$ then the temperature of junction is........$^oC$ (Thermal conductivity of copper is $9\,times$ that of steel)
AIEEE 2012, Medium
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Let conductivity of steel ${K_{steel}{ = }}k\,then$
From question
Conductivity of copper ${K_{copper}} = 9k$
${\theta _{copper}} = {100^ \circ }C$
${\theta _{steel}} = {0^ \circ }C$
${l_{steel}} = {l_{copper}} = \frac{L}{2}$
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 \times 0 \times \frac{L}{2}}}{{9k \times \frac{L}{2} + k \times \frac{L}{2}}}$
$ = \frac{{\frac{{900}}{2}}}{{\frac{{10kL}}{2}}} = {90^ \circ }C$
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