UY1: Capacitance Of A Cylindrical Capacitor

A coaxial capacitor is formed by two long concentric conductors, common in cables and transmission systems.

1) At a glance

• Prerequisites: cylindrical Gauss symmetry (Common Charge Distributions) and log integrals (Integration Techniques)
• Outcomes: derive the capacitance of a long coaxial capacitor and understand why the answer depends on ln(b/a)
• Key result: C = 2πε₀L/(ln(b/a)), C'≡C/L = 2πε₀/(ln(b/a))
• Common trap: sign confusion between Vₐ-V_b and V_b-Vₐ

Motivation / intuition

In cylindrical symmetry, the field falls like 1/r, so the potential difference involves ∫ dr/r, which is why logarithms appear. The per-unit-length form C' is especially useful because real coaxial capacitors are often “long” compared to their radii.

2) Setup

Inner cylinder radius a has linear charge density + λ, outer cylinder inner radius b has -λ, with a < b.

Assume long geometry so edge effects are negligible.

3) Core derivation/explanation

For a < r < b, use cylindrical Gaussian surface (radius r, length L): E(2π rL) = (λ L)/ε₀ E(r) = λ/2πε₀r

Potential difference magnitude: V_ab = Vₐ-V_b = ∫ₐ^b E dr V_ab = λ/2πε₀∫ₐ^bdr/r = λ/2πε₀ ln(b/a)

Total charge on length L is Q = λ L, so C = Q/V_ab = 2πε₀L/(ln(b/a))

Per unit length: C' = C/L = 2πε₀/(ln(b/a))

4) Worked example(s)

Given a = 1.0 mm, b = 5.0 mm, L = 2.0 m.

C' = 2πε₀/(ln(5)) ≈ 3.45 × 10⁻¹¹ F/m C = C'L ≈ 6.90 × 10⁻¹¹ F = 69.0 pF

5) Practice set (with hints + answers)

1. If b/a increases, what happens to C'?
2. If both radii scale up by same factor, does C' change?
3. Why do we report both C and C' for cylindrical capacitors?

Hints

1. Look at denominator ln(b/a).
2. Ratio b/a is unchanged.
3. Practical devices often differ mainly in length.

Answers

1. C' decreases.
2. No, ideal C' stays the same.
3. C' is geometry-intrinsic for long coaxial structures; C then scales with L.

6) Summary + next steps

• Gauss’s law gives the 1/r field in coaxial geometry.
• Integrating field gives logarithmic voltage dependence.
• Capacitance depends strongly on radius ratio through ln(b/a).

Next: Transferring Charge And Energy Between Capacitors Previous: Capacitance Of Spherical Capacitor Back To Electromagnetism (UY1)