Capacitance per unit length of a coaxial cable Two coaxial metallic cylinders have inner radius R1 and outer radius R2. The space between them is uniformly filled with a dielectric of relative permittivity εr. What is the capacitance per metre length?

Introduction / Context:
The capacitance per unit length of a coaxial line is a staple result in electromagnetics and transmission line theory. It follows from Gauss’s law under cylindrical symmetry with a homogeneous dielectric between the conductors.

Given Data / Assumptions:

• Inner conductor radius R1, outer conductor inner radius R2 (R2 > R1).
• Dielectric is linear, homogeneous, isotropic with permittivity ε = ε0 * εr.
• Length considered is 1 metre (per-unit-length capacitance C′).

Concept / Approach:

Use Gauss’s law in integral form over a cylindrical surface to find the electric field E(r). Integrate the potential difference V between R1 and R2. The per-unit-length capacitance is C′ = Q / V for a 1 m length. The result contains a logarithm due to the 1/r dependence of E in cylindrical coordinates.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional analysis: ε has units F/m; logarithm is dimensionless; result has units F/m as required. For εr = 1, this reduces to the vacuum formula.

Why Other Options Are Wrong:

• Halving the numerator (π instead of 2π) underestimates C′ by 2.
• Multiplying by ln instead of dividing reverses the dependence.
• Using (R2 − R1) ignores cylindrical field variation (not a parallel-plate capacitor).

Common Pitfalls:

Confusing natural log with log base 10; forgetting ε = ε0 * εr; using diameter rather than radius.

Final Answer:

C′ = 2 * π * ε0 * εr / ln(R2 / R1)