Question

Two metallic wires of identical dimensions are connected in series. If σ1 and σ2 are the conductivities of 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)

$\frac{2\sigma_1 \sigma_2}{\sigma_1 + \sigma_2}$

Answer 2

Equivalent resistance,
$R = R_1 + R_2$
$⇒$ $\frac{I_1 + I_2}{\sigma_A} = \frac{I_1}{\sigma_{1A}} + \frac{I_2}{\sigma_{2A}}$
$⇒$ $\frac{2}{\sigma} = \frac{1}{\sigma_1} + \frac{1}{\sigma_2}$
The effective conductivity of the combination
$\sigma = \frac{2\sigma_1 \sigma_2}{\sigma_1 + \sigma_2}$
So, the correct option is (B): $\frac{2\sigma_1 \sigma_2}{\sigma_1 + \sigma_2}$