First slide

Heat transfer

Question

Three rods of identical area of cross-section and made from the same metal form the sides of an isosceles triangle ABC, right angled at B. The points A and B are maintained at temperatures T and $\sqrt{2} T$ , respectively. In the steady state, the temperature of the point C is T_C. Assuming that only heat conduction takes place, T_C/T is equal to

Moderate

A

$\frac{1}{(\sqrt{2} + 1)}$
B

$\frac{3}{(\sqrt{2} + 1)}$
C

$\frac{1}{2 (\sqrt{2} - 1)}$
D

$\frac{1}{\sqrt{3} (\sqrt{2} - 1)}$

Solution

$∵ T_{B} > T_{A} \Rightarrow$ Heat will flow B to A via two paths
(i) B to A (ii) and along BCA as shown.
Rate of flow of heat in path BCA will be same
$\begin{array}{l} i . e . {(\frac{Q}{t})}_{BC} = {(\frac{Q}{t})}_{CA} \\ \Rightarrow \frac{k (\sqrt{2} T - T_{C}) A}{a} = \frac{k (T_{C} - T) A}{\sqrt{2} a} \\ \Rightarrow \frac{T_{C}}{T} = \frac{3}{1 + \sqrt{2}} \end{array}$

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