Cylindrical capacitors: The capacitance per unit length of a very long coaxial cable, made of two concentric cylinders, is 50 pF/m. What is the radius of the outer cylinder if the radius of the inner one is 1.0 mm? (k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²)

Question

Quizplus · Accepted Answer

The capacitance per unit length $C'$ of a coaxial cable is given by $C' = \frac{2\pi\epsilon_0}{\ln(b/a)}$, where $a$ is the radius of the inner cylinder, $b$ is the radius of the outer cylinder, and $\epsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} F/m$). Given $C' = 50 pF/m = 50 \times 10^{-12} F/m$ and $a = 1.0 mm = 1.0 \times 10^{-3} m$, we can solve for $b$. Rearranging the formula and substituting the given values, we find that $b$ corresponds to a radius that makes the equation true for the given capacitance per unit length, leading to the correct answer of 3.0 mm after calculation.

Cylindrical Capacitors: the Capacitance Per Unit Length of a Very