Two uniform rods A and B of the same…

Two uniform rods A and B of the same length L and cross-section S are joined end to end. The thermal conductivity of A is 3k and that of B is k. The…

One or more options correct

Two uniform rods

A

and

B

of the same length

L

and cross-section

S

are joined end to end. The thermal conductivity of

A

3k

and that of

B

k

. The free end of

A

is kept at

100^\circ\mathrm{C}

and the free end of

B

0^\circ\mathrm{C}

. The lateral surfaces are insulated and the steady state is reached. Which of the following statements is/are correct?

A. The temperature of the junction is $75^\circ\mathrm{C}$ .
B. The rate of heat flow along the rods is $75\,kS/L$ (in units where temperature is in $^\circ\mathrm{C}$ ).
C. If the same two rods were instead connected in parallel between the same temperatures, the rate of heat flow would be $\dfrac83$ times that in the series arrangement.
D. If the rods are interchanged (B at $100^\circ\mathrm{C}$ and A at $0^\circ\mathrm{C}$ ), the junction temperature becomes $25^\circ\mathrm{C}$ .

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Answer: A, B, D

In series the heat current is the same in both rods:

\frac{3kS(100-T)}{L}=\frac{kS(T-0)}{L}

, giving

T=75^\circ\mathrm{C}

, so (A) is true. The heat current is then

\frac{kS\cdot75}{L}=75\,kS/L

, so (B) is true. In parallel, each rod spans the full

100^\circ\mathrm{C}

difference, so

H=\frac{(3k+k)S\cdot100}{L}=400\,kS/L

, and the ratio to the series value is

\frac{400}{75}=\frac{16}{3}

, not

\frac83

, so (C) is false. When interchanged,

\frac{kS(100-T)}{L}=\frac{3kS\,T}{L}

, giving

T=25^\circ\mathrm{C}

, so (D) is true. A rod of higher conductivity has the smaller temperature drop across it. The reasoning is the thermal analogue of resistors: the conductance

kS/L

of each rod plays the role of

1/R

, with series conductances combining as reciprocals and parallel conductances adding directly.

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Original question written in the JEE Advanced 2020 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