Question

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

• A. $\frac{1}{2(\sqrt{2}-1)}$
• B. $\frac{3}{\sqrt{2}+1}$
• C. $\frac{1}{\sqrt{3}(\sqrt{2}-1)}$
• D. $\frac{1}{\sqrt{2}+1}$

Answer 1

The correct option is B $\frac{3}{\sqrt{2}+1}$

The transfer of thermal energy from one single body to another and from one particular temperature to other - high to low temperature is possible only under steady or unsteady conditions. In this question, there is steady-state which means temperature does not change with time, at point c the temperature would remain the same which is $Tc.$

${\left(\frac{△Q}{△t}\right)}_{BC}={\left(\frac{△Q}{△t}\right)}_{CA}$
$\Rightarrow \frac{kA(\sqrt{2}T-T_C)}{a}=\frac{kA(T_C-T)}{\sqrt{2}a}$
Solve to get
$\frac{T_C}{T}=\frac{3T}{\sqrt{2}+1}$.

Therefore, heat conduction takes place is equal to $\frac{3T}{\sqrt{2}+1}.$