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Thermal Properties Of Matter

Question

A slab consists of portions of different materials of same thickness and having the conductivities K₁ and K₂. The equivalent thermal conductivity of the slab is

K₁ + K₂

$\sqrt{K_{1} + K_{2}}$

$\frac{2 K_{1} K_{2}}{K_{1} + K_{2}}$

$\sqrt{\frac{K_{1} K_{2}}{K_{1} + K_{2}}}$

Solution

$\frac{2 K_{1} K_{2}}{K_{1} + K_{2}}$

As both slabs are of same thickness made of different materials i.e d₁ = d₂ = d

As two portions of the slab are connected in series,

∴ K = $\frac{Σ x_{i}}{Σ \frac{x_{i}}{K_{i}}}$

K = $\frac{d + d}{\frac{d}{K_{1}} + \frac{d}{K_{2}}}$

= $\frac{2 d}{d (\frac{1}{K_{1}} + \frac{1}{K_{2}})}$

K = $\frac{2 K_{1} K_{2}}{K_{1} + K_{2}}$