Two identical metal of thermal conductivities $\mathrm{K}_{1}$ and $\mathrm{K}_{2}$ respectively are connected in series. The effective thermal conductivity of the combination is:

(C) $\frac{2 \mathrm{~K}_{1} \mathrm{~K}_{2}}{\mathrm{~K}_{1}+\mathrm{~K}_{2}}$

(A) $\frac{\mathrm{K}_{1}+\mathrm{K}_{2}}{\mathrm{~K}_{1} \mathrm{~K}_{2}}$

(B) $\frac{\mathrm{K}_{1}+\mathrm{K}_{2}}{2 \mathrm{~K}_{1} \mathrm{~K}_{2}}$

(D) $\frac{\mathrm{K}_{1} \mathrm{~K}_{2}}{\mathrm{~K}_{1}+\mathrm{~K}_{2}}$

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Heat & Thermodynamics

Two identical metal of thermal conductivities

K_{1}

and

K_{2}

respectively are connected in series. The effective thermal conductivity of the combination is:

Answer: (C)

Two identical pieces in series carry the same heat current, so their thermal resistances add. With each of length

L

and cross-section

A

, the combination of length

2 L

satisfies

\frac{2 L}{K _{eff} A} = \frac{L}{K _{1} A} + \frac{L}{K _{2} A}

\frac{2}{K _{eff}} = \frac{1}{K _{1}} + \frac{1}{K _{2}} = \frac{K _{1} + K _{2}}{K _{1} K _{2}}

The effective thermal conductivity of the combination is

\frac{2 K _{1} K _{2}}{K _{1} + K _{2}}

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option