Question

The two rods shown in figure have identical geometrical dimensions and are made of different materials. They are in contact with two heat baths at temperatures ${100}^{\circ }\text{C}$ and $0^{\circ }\text{C}$. The temperature of the junction is ${70}^{\circ }\text{C}$. Find the temperature of the junction if the rods are interchanged.

• A. ${30}^{\circ }\text{C}$
• B. ${50}^{\circ }\text{C}$
• C. ${70}^{\circ }\text{C}$
• D. ${60}^{\circ }\text{C}$

Answer 1

The correct option is A ${30}^{\circ }\text{C}$

Case 1:
Case 2:
As the rods are connected in series, the rate of flow of heat will be same in both the cases.
Case 1 :
Rate of flow of heat is given by
$\frac{dq}{dt}=\frac{kA\Delta T}{l}$

Rate of heat in rod $P$ will be same as that in rod $Q$,
$\therefore =\frac{K_P\times A\times (100-70)}{l}=\frac{kQ\times A\times (70-0)}{l}$

$\Rightarrow 30K_P=70K_Q$

$\Rightarrow K_Q=\frac{3}{7}{k}_p......\left(i\right)$

Case 2:
Again, the rate of flow of heat will be same in rod $P$ and $Q$

$\therefore \frac{k_q\times A\times (100-T)}{l}=\frac{k_p\times A\times (T-0)}{l}$

$100k_q-k_{q}T=k_{p}T$

$100k_q-k_{q}T=\frac{70}{30}{k}_{q}T\left(\text{using}\right(i\left)\right)$

$100-T=\frac{7}{3}T$

$100=\frac{10}{3}T$

$\Rightarrow T={30}^{\circ }\text{C}$

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