Question

Two metallic wires of identical dimensions are connected is series. If ${\sigma }_1$ and ${\sigma }_2$ are the conductivities of the these wires respectively, the effective conductivity of the combination is :

• A. $\frac{{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$
• B. $\frac{2{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$
• C. $\frac{{\sigma }_1+{\sigma }_2}{2{\sigma }_1{\sigma }_2}$
• D. $\frac{{\sigma }_1+{\sigma }_2}{{\sigma }_1{\sigma }_2}$

Answer 1

Answer: (B)

Let length of wire be ' $\ell$ '
Area of wire as ' $A$ '
For equivalent wire length $=2\ell \&$ area will be A
Thermal resistance
$R_{eq}=R_1+R_2$
$\frac{2\ell }{{\sigma }_{eq}A}=\frac{\ell }{{\sigma }_{1}A}+\frac{\ell }{{\sigma }_{1}A}$
$\frac{2\ell }{{\sigma }_{eq}}=\frac{\ell }{{\sigma }_1}+\frac{\ell }{{\sigma }_2}$
$\Rightarrow {\sigma }_{eq}=\frac{2{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$