Two rods of the same cross-section S…

Two rods of the same cross-section S are joined end to end. Rod A has length L and thermal conductivity 2K; rod B has length 2L and thermal…

One or more options correct

Two rods of the same cross-section

S

are joined end to end. Rod

A

has length

L

and thermal conductivity

2K

; rod

B

has length

2L

and thermal conductivity

K

. The free end of

A

is held at

100^\circ\mathrm{C}

, the free end of

B

0^\circ\mathrm{C}

, and the sides are perfectly insulated. Steady state is reached. Which of the following statements is/are correct?

A. The temperature of the junction is $80^\circ\mathrm{C}$ .
B. The combination behaves like a single rod of length $3L$ with conductivity $1.2K$ .
C. If instead the two rods were placed side by side between the same two temperatures (each end-to-end across the full temperature difference, with their own lengths), the heat currents in $A$ and $B$ would be in the ratio $4:1$ .
D. If the rods are interchanged, i.e. $B$ is placed at the $100^\circ\mathrm{C}$ end and $A$ at the $0^\circ\mathrm{C}$ end, the junction temperature remains $80^\circ\mathrm{C}$ .

Show answer and solution

Answer: A, B, C

Thermal resistance is

R_{th}=\ell/(\kappa S)

. Rod

A

R_A=\dfrac{L}{2KS}

; rod

B

R_B=\dfrac{2L}{KS}=4R_A

. In series the same heat current flows, so the temperature drop is proportional to resistance: the drop across

A

\tfrac15\times100=20^\circ\mathrm{C}

and the junction sits at

80^\circ\mathrm{C}

(A true). The equivalent conductivity is

\kappa_{eq}=\dfrac{3L}{S(R_A+R_B)}=\dfrac{3L}{2.5L/K}=1.2K

(B true). For two rods each spanning the same temperature difference, the heat current is inversely proportional to

R_{th}

, so

H_A:H_B=R_B:R_A=4:1

(C true). Interchanging the rods places the large resistance

B

at the hot end, so

B

drops

80^\circ\mathrm{C}

and the junction is at

100-80=20^\circ\mathrm{C}

, not

80^\circ\mathrm{C}

(D false).

Original question written in the JEE Advanced 2016 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake