Three conductions of same length having thermal conductivity k 1 , k 2 and k 3 are connected as shown in figure. Area of cross sections of 1 st and 2 nd conductor are same and for 3 rd conductor it is double of the 1 st conductor. The temperatures are given in the figure. In steady state condition, the value of θ is ° C . (Given : k 1 =60 Js -1 m -1 K -1 , k 2 =120 Js -1 m -1 K -1 , k 3 = 135 Js -1 m -1 K -1 )

Physics · Heat Transfer

JEE Main 2025 — 22 January, Morning Shift — Question 63

Three conductions of same length having thermal conductivity $\mathrm{k}_{1}, \mathrm{k}_{2}$ and $\mathrm{k}_{3}$ are connected as shown in figure. Area of cross sections of $1^{\text {st }}$ and $2^{\text {nd }}$ conductor are same and for $3^{\text {rd }}$ conductor it is double of the $1^{\text {st }}$ conductor. The temperatures are given in the figure. In steady state condition, the value of $\theta$ is $\qquad$ ${ }^{\circ} \mathrm{C}$ . (Given : $\mathrm{k}_{1}=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_{2}=120 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_{3}=$ $135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}$ )

Answer: 40

Numerical answer — enter this value.

Step-by-step solution

Let each conductor length be $L$ . For conductors $1$ and $2$ in parallel, each has area $A$ , so $R_1 = \frac{L}{k_1 A}$ and $R_2 = \frac{L}{k_2 A}$ . For conductor $3$ , area is $2A$ , so $R_3 = \frac{L}{k_3 \cdot 2A}$ . In steady state, heat current through the parallel combination equals heat current through conductor $3$ :

\frac{100 - \theta}{R_{\text{parallel}}} = \frac{\theta - 0}{R_3}

where $\frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{A}{L}(k_1 + k_2)$ . Thus:

(100 - \theta) \frac{A}{L}(k_1 + k_2) = \theta \frac{k_3 \cdot 2A}{L}

Cancel $\frac{A}{L}$ :

(100 - \theta)(60 + 120) = \theta \cdot 270

180(100 - \theta) = 270\theta

18000 = 450\theta \implies \theta = 40^\circ\text{C}

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 22 January, Morning Shift
Subject: Physics
Chapter: Heat Transfer
Topic: Introductory Heat transfer by Conduction

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