Q: Two wires of equal diameters of resistivities ρ_{1} and ρ_{2} and length x_{1} and x_{2} 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 A_{1} = A_{2} = A

R = R_{1} + R_{2}

$\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 = x_{1} + x_{2} )

$\large \rho = \frac{\rho_1 x_1 + \rho_2 x_2}{( x_1 + x_2 )} $