Question

Two metal rods with identical length and area of cross-section are connected in series. If ${\sigma }_1\text{and}{\sigma }_2$ are the thermal conductance of material of two rods then equivalent conductance of combination is equal to

Answer 1

Thermal resistance for both the rods are:
$R_1=\frac{L}{{\sigma }_{1}A}\text{and}{R}_2=\frac{L}{{\sigma }_{2}A}$
Since, rods are connected in series:
$R_{\text{net}}=R_1+R_2=\frac{L}{A}(\frac{1}{{\sigma }_1}+\frac{1}{{\sigma }_2})$

It must be equivalent to, $R_{\text{net}}=\frac{2L}{\sigma A}$

So, $\frac{2L}{\sigma A}=\frac{L}{A}(\frac{1}{{\sigma }_1}+\frac{1}{{\sigma }_2})$

$\sigma =\frac{2{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$