---
title: "Capacitance of cylindrical capacitor"
subject: "Engineering Physics"
chapter: 7
key: "semester_1/engineering_physics:capacitance-general-method-cylindrical"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/capacitance-general-method-cylindrical
site: BCT Engineering
---

# Capacitance of cylindrical capacitor

Engineering Physics · Chapter 7: Electrostatics

- Asked in: 2072 Chaitra, 5 marks; 2071 Chaitra, 5 marks (asked 2 times)
- Video: [Derivation of Capacitance for a Cylindrical Capacitor](https://www.youtube.com/watch?v=kAXg1xMkR_4) — Patrick Kaplo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/capacitance-general-method-cylindrical
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-7

## Question

Write the general method to calculate the capacitance of a capacitor and hence determine the capacitance of a cylindrical capacitor of inner and outer radii 'a' and 'b' respectively.

## Short answer

General method: assume charges ±q, find E via Gauss's law, compute V = ∫E·dl, then C = q/V. For a cylindrical capacitor, E = q/(2πε₀Lr), V = (q/(2πε₀L)) ln(b/a), so C = 2πε₀L/ln(b/a).

## Answer

A capacitor's capacitance is C = q/V, where q is the magnitude of charge on either plate and V is the potential difference between the plates. It depends only on geometry and the medium.

#### General method
1. Place charge +q on one conductor and −q on the other.
2. Choose a Gaussian surface of suitable symmetry and use Gauss's law (∮E·dA = q/ε0) to find E between the conductors.
3. Find the potential difference V = V₊ − V₋ = ∫E·dl, integrated along a path from the positive to the negative conductor.
4. Substitute in C = q/V; q cancels, leaving C in terms of dimensions only.

#### Cylindrical capacitor
Two coaxial cylinders, inner radius *a* (charge +q), outer radius *b* (charge −q), length L ≫ (b − a), so edge effects are ignored.

```text
      outer cylinder (-q), radius b
     ______________________________
    |   ______________________     |
    |  |  inner (+q), radius a |    |
    |  |<-- Gaussian surface r -->  |
    |  |______________________|    |
    |______________________________|
                 length L
```

1. Take a coaxial Gaussian cylinder of radius r (a < r < b) and length L. Flux passes only through the curved surface: E(2πrL) = q/ε0.
2. E = q/(2πε0 L r), directed radially outward.
3. V = ∫ₐᵇ E dr = (q/(2πε0 L)) ∫ₐᵇ dr/r = (q/(2πε0 L)) ln(b/a)
4. C = q/V

**Answer: C = 2πε0 L / ln(b/a)**

Capacitance per unit length is C/L = 2πε0/ln(b/a). With a dielectric of constant k, replace ε0 by kε0. It increases when the gap (b − a) is made small.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which sequence gives capacitance by the general method?
   A. Choose V, find E from dipole formulas, then C = V/q
   B. Find V, then differentiate to get q, then C = dV/dq
   C. Find E, take C = E x area
   D. Put ±q on the conductors, find E by Gauss's law, find V = ∫E·dl, then C = q/V
   Answer: D. q cancels in C = q/V, so C depends only on geometry and medium. (Key point: General method)

2. What is E between the coaxial cylinders at radius r?
   A. E = q/(4πε0 r^2)
   B. E = q/(2ε0 A)
   C. E = q/(2πε0 L r)
   D. E = q r/(2πε0 L)
   Answer: C. Flux is through the curved surface only: E(2πrL) = q/ε0. (Key point: Field of cylinder)

3. What is the capacitance of the cylindrical capacitor?
   A. C = 2πε0 L ln(b/a)
   B. C = 2πε0 L / ln(b/a)
   C. C = 4πε0 ab/(b - a)
   D. C = 2πε0 L/(b - a)
   Answer: B. V = (q/2πε0 L) ln(b/a), so C = q/V. (Key point: Capacitance)

4. How does C change with the gap and with a dielectric of constant k?
   A. It rises as the gap narrows, and ε0 is replaced by kε0
   B. It falls as the gap narrows; ε0 is replaced by ε0/k
   C. It is independent of the gap; ε0 is replaced by kε0
   D. It rises as the gap widens; ε0 is unchanged
   Answer: A. ln(b/a) decreases for a narrow gap, increasing C, and a dielectric multiplies C by k. (Key point: Effects of geometry and medium)

## Related in Engineering Physics

- [General capacitance method, spherical capacitor](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acapacitance-general-method-spherical) — Engineering Physics, chapter 7
- [Capacitance of dielectric-filled spherical shells](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aspherical-shells-dielectric-capacitance) — Engineering Physics, chapter 7
- [Current in outer portion of wire](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acurrent-density-wire-outer-portion) — Engineering Physics, chapter 8
- [Parallel plate capacitor with mica](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amica-capacitor-field-free-charge) — Engineering Physics, chapter 7

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