Two rods one made of copper and other made of steel of the same...

Two rods one made of copper and other made of steel of the same length and same cross sectional area are joined together. The thermal conductivity of copper and steel are

385 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~K}^{-1} \mathrm{~m}^{-1}

and

50 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1}

respectively. The free ends of copper and steel are held at

100^{\circ} \mathrm{C}

and

0^{\circ} \mathrm{C}

respectively. The temperature at the junction is, nearly:

Options

A $12^{\circ} \mathrm{C}$
B $50^{\circ} \mathrm{C}$
C $73^{\circ} \mathrm{C}$
D $88.5^{\circ} \mathrm{C}$

Correct answer

D. $88.5^{\circ} \mathrm{C}$

Step-by-step solution

As per question, we have;

\begin{aligned} & \theta=\text{ ? } \\ & \text{As } \\ & \mathrm{K}=\frac{Q d}{\mathrm{ A} \Delta t}, \\ & \text{where, } \mathrm{K}=\text{ thermal conductivity } \\ & Q=\text{ amount\text{ of } heat transferred } \\ & d=\text{ distance between two planes } \\ & \mathrm{A}=\text{ area of surface, and } \\ & \Delta \mathrm{T}=\text{ difference in Temp. } \\ & \frac{\Delta \mathrm{Q}}{\Delta t}=\frac{385 \times \mathrm{A} \times\left(100^{\circ} \mathrm{C}-\theta\right)}{l} \\ & =\frac{50 \times \mathrm{A} \times\left(\theta-0^{\circ} \mathrm{C}\right)}{l} \\ & \Rightarrow 385\left(100^{\circ} \mathrm{C}-\theta\right)=50\left(\theta-0^{\circ} \mathrm{C}\right) \\ & \Rightarrow \theta=88.5^{\circ} \mathrm{C} \end{aligned}

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