Question

The ends of a metre stick are maintained at 100°C and 0°C. One end of a rod is maintained at 25°C. Where should its other end be touched on the metre stick so that there is no heat current in the rod in steady state?

Answer 1

One end of the rod is at a temperature of 25°C. So, if no heat current flows through the rod in steady state, then the other end of the rod should also be at a temperature of 25°C.
Let the point at which the other end of the rod is touched be C.
No heat flows through the rod when the temperature at point C is also 25°C.

Heat current through AC = Heat current through CB

$$\begin{gathered} \Rightarrow \frac{{\left(\mathrm{\Delta }T\right)}_{\mathrm{AC}}}{{\displaystyle \frac{x}{\mathrm{kA}}}}=\frac{{\left(\mathrm{\Delta }T\right)}_{\mathrm{CB}}}{{\displaystyle \frac{100-x}{\mathrm{kA}}}} \\ \frac{\left(100-25\right)}{x}=\frac{25-0}{100-x} \\ \frac{3}{x}=\frac{1}{100-x} \\ 300-3x=x \\ 300=4x \\ x=75\mathrm{cm} \end{gathered}$$

Thus, it should be touched at 75 cm from 100°C end.