Two resistance wires are of same material having length, \( l \) and \( 2l \) and area of cross-section, \( 4A \) and \( A \) respectively. When they are connected in parallel, their equivalent resistance can be expressed as
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{2}{9}\,\frac{\rho l}{A} \)
The resistance of a wire is \( R = \rho l/A \). Find the two resistances, then combine them in parallel.
\( \displaystyle R_1 = \frac{\rho\,l}{4A} \qquad R_2 = \frac{\rho\,(2l)}{A} = \frac{2\rho l}{A} \)
Write \( x = \rho l/A \), so that \( R_1 = x/4 \) and \( R_2 = 2x \). In parallel the product over the sum gives
\( \displaystyle R_{\mathrm{eq}} = \frac{R_1R_2}{R_1 + R_2} = \frac{(x/4)(2x)}{x/4 + 2x} = \frac{x^{2}/2}{9x/4} = \frac{2}{9}\,x \)
\( \displaystyle R_{\mathrm{eq}} = \frac{2}{9}\,\frac{\rho l}{A} \)