Q: Two wires of equal diameters of resistivities ρ1 and ρ2 and length x1 and x2 respectively are joined in series . Find the equivalent resistivity of the combination.
Sol. Resistance, $\large R_1 = \rho_1 \frac{l_1}{A_1} $
$\large R_2 = \rho_2 \frac{l_2}{A_2} $
As the wires are of equal diameters A1 = A2 = A
R = R1 + R2
$\large \rho \frac{x}{A} = \rho_1 \frac{x_1}{A} + \rho_2 \frac{x_2}{A} $
$\large \rho x = \rho_1 x_1 + \rho_2 x_2 $
$\large \rho ( x_1 + x_2 ) = \rho_1 x_1 + \rho_2 x_2 $ (since , x = x1 + x2 )
$\large \rho = \frac{\rho_1 x_1 + \rho_2 x_2}{( x_1 + x_2 )} $