Two cylindrical resistors are made of the same material and have the same resistance.The resistors,R₁ and R₂,have different radii,r₁ and r₂,and different lengths,L₁ and L₂.Which of the following relative values for radii and lengths would result in equal resistances? A)r₁ = r₂ and L₁ = 2L₂ B)2r₁ = r₂ and L₁ = 2L₂ C)r₁ = r₂ and 4L₁ = L₂ D)2r₁ = r₂ and 4L₁ = L₂

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Quizplus · Accepted Answer

The resistance of a cylindrical resistor is given by $R = \rho \frac{L}{A}$, where $\rho$ is the resistivity of the material, $L$ is the length, and $A$ is the cross-sectional area. For a cylinder, $A = \pi r^2$. If two resistors have the same resistance, their $\frac{L}{A}$ ratios must be equal. Doubling the radius quadruples the area ($A \propto r^2$), so to keep the resistance the same, the length must also be quadrupled ($L \propto A$), maintaining the $\frac{L}{A}$ ratio constant.

Two Cylindrical Resistors Are Made of the Same Material and Have