One end of a thermally insulated rod is kept at a temperature $T_{1}$ and the other at $T_{2}$. The rod is composed of two sections of lengths $\ell_{1}$ and $\ell_{2}$ and thermal conductivities $k_{1}$ and $k_{2}$ respectively. The temperature at the interface of the two sections is

(C) $(k_{1} \ell_{2} T_{1}+k_{2} \ell_{1} T_{2}) /(k_{1} \ell_{2}+k_{2} \ell_{1})$

(A) $(k_{2} \ell_{2} T_{1}+k_{1} \ell_{1} T_{2}) /(k_{1} \ell_{1}+k_{2} \ell_{2})$

(B) $(k_{2} \ell_{1} T_{1}+k_{1} \ell_{1} T_{2}) /(k_{2} \ell_{1}+k_{1} \ell_{2})$

(D) $(k_{1} \ell_{1} T_{1}+k_{2} \ell_{2} T_{2}) /(k_{1} \ell_{1}+k_{2} \ell_{2})$

JEE Main · 2007 · 29 Apr 2007 · Offline · Heat & Thermodynamics

One end of a thermally insulated rod is kept at a temperature

T_{1}

and the other at

T_{2}

. The rod is composed of two sections of lengths

ℓ_{1}

and

ℓ_{2}

and thermal conductivities

k_{1}

and

k_{2}

respectively. The temperature at the interface of the two sections is

(A) $(k_{2} ℓ_{2} T_{1} + k_{1} ℓ_{1} T_{2}) / (k_{1} ℓ_{1} + k_{2} ℓ_{2})$
(B) $(k_{2} ℓ_{1} T_{1} + k_{1} ℓ_{1} T_{2}) / (k_{2} ℓ_{1} + k_{1} ℓ_{2})$
(C) $(k_{1} ℓ_{2} T_{1} + k_{2} ℓ_{1} T_{2}) / (k_{1} ℓ_{2} + k_{2} ℓ_{1})$ ✓ Correct
(D) $(k_{1} ℓ_{1} T_{1} + k_{2} ℓ_{2} T_{2}) / (k_{1} ℓ_{1} + k_{2} ℓ_{2})$

Answer: (C)

In the steady state the same heat current flows through both sections, so at the junction temperature

T

we have

\frac{k _{1} ( T _{1} - T )}{ℓ _{1}} = \frac{k _{2} ( T - T _{2} )}{ℓ _{2}}

. Cross-multiplying and collecting the terms in

T

gives

T = \frac{k _{1} ℓ _{2} T _{1} + k _{2} ℓ _{1} T _{2}}{k _{1} ℓ _{2} + k _{2} ℓ _{1}}

, which is a conductance-weighted average of the two end temperatures.

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option