Question

A temperature difference of $120{}^{\circ }\text{C}$ is maintained between two ends of a uniform rod $AB$ of length $2L$. Another bent rod $PQ$, of same cross-section as $AB$ and length $\frac{3L}{2}$, is connected across $AB$ (see figure). In steady state, temperature difference between $P$ and $Q$ will be close to:

• A. $60{}^{\circ }\text{C}$
• B. $35{}^{\circ }\text{C}$
• C. $45{}^{\circ }\text{C}$
• D. $75{}^{\circ }\text{C}$

Answer 1

The correct option is C $45{}^{\circ }\text{C}$

Comparing the given structure in terms of thermal resistance as shown below,

$R_{PQ}=\frac{\frac{3R}{2}.R}{\frac{3R}{2}+R}=\frac{3}{5}R$

$R_{AB}=\frac{R}{2}+\frac{3}{5}R+\frac{R}{2}=\frac{8}{5}R$

As thermal current through $AB$ and $PQ$ are same,

$\frac{\Delta T_{AB}}{R_{AB}}=\frac{\Delta T_{PQ}}{R_{PQ}}$

$\Rightarrow \Delta T_{PQ}=\frac{120}{\frac{8}{5}R}\times \frac{3}{5}R=\frac{360}{8}$

$\Rightarrow \Delta T_{PQ}={45}^{\circ }\text{C}$

Hence, option $\left(A\right)$ is correct.