Resistance Formula Basics

The resistance ($R$) of a conductor depends on its material's resistivity ($\rho$), its length ($L$), and its cross-sectional area ($A$). The relationship is given by the formula:

$R = \rho \frac{L}{A}$

Applying the Formula

We have two conductors made of the same material, meaning their resistivity ($\rho$) is the same. Let the properties of the first conductor be $L_1, A_1, R_1$ and the second be $L_2, A_2, R_2$.

• Conductor 1: Length $L_1 = L$. Its resistance is $R_1 = \rho \frac{L}{A_1}$.
• Conductor 2: Length $L_2 = 2L$. Its resistance is $R_2 = \rho \frac{2L}{A_2}$.

Calculating the Area Ratio

The problem states that the conductors have equal resistance, so $R_1 = R_2$. We can set the expressions for their resistances equal to each other:

$ \rho \frac{L}{A_1} = \rho \frac{2L}{A_2} $

Since $\rho$ and $L$ are non-zero, we can cancel them from both sides:

$ \frac{1}{A_1} = \frac{2}{A_2} $

To find the ratio $A_1 : A_2$, we rearrange the equation:

$ \frac{A_1}{A_2} = \frac{1}{2} $

Therefore, the ratio of their cross-sectional areas ($A_1 : A_2$) is 1:2.