Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity $\mathrm{K}_{1}$, $\mathrm{K}_{2}$ and $\mathrm{K}_{3}$, respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at $100^{\circ} \mathrm{C}$ and the other at $0^{\circ} \mathrm{C}$ (see figure). If the joints of the rod are at $70^{\circ} \mathrm{C}$ and $20^{\circ} \mathrm{C}$ in steady state and there is no loss of energy from the surface of the rod, the correct relationship between $\mathrm{K}_{1}$, $\mathrm{K}_{2}$ and $\mathrm{K}_{3}$ is:

(A) $\mathrm{K}_{1}: \mathrm{K}_{3}=2: 3, \mathrm{~K}_{2}: \mathrm{K}_{3}=2: 5$

(B) $\mathrm{K}_{1}<\mathrm{K}_{2}<\mathrm{K}_{3}$

(C) $\mathrm{K}_{1}: \mathrm{K}_{2}=5: 2, \mathrm{~K}_{1}: \mathrm{K}_{3}=3: 5$

(D) $\mathrm{K}_{1}>\mathrm{K}_{2}>\mathrm{K}_{3}$

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Heat & Thermodynamics

Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity

K_{1}

K_{2}

and

K_{3}

, respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at

10 0^{\circ} C

and the other at

0^{\circ} C

(see figure). If the joints of the rod are at

7 0^{\circ} C

and

2 0^{\circ} C

in steady state and there is no loss of energy from the surface of the rod, the correct relationship between

K_{1}

K_{2}

and

K_{3}

is:

Answer: (A)

In the steady state the same heat current flows through the three rods in series, so

\frac{K _{1} A ( 100 - 70 )}{ℓ} = \frac{K _{2} A ( 70 - 20 )}{ℓ} = \frac{K _{3} A ( 20 - 0 )}{ℓ}

, which reduces to

30 K_{1} = 50 K_{2} = 20 K_{3}

. Comparing the first and third parts,

K_{1} : K_{3} = 20 : 30 = 2 : 3

; comparing the second and third,

K_{2} : K_{3} = 20 : 50 = 2 : 5

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Single correct option