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

# General capacitance method, spherical capacitor

Engineering Physics · Chapter 7: Electrostatics

- Asked in: 2072 Kartik, 5 marks
- Video: [Physics 39   Capacitors (10 of 37) The Spherical Capacitor](https://www.youtube.com/watch?v=APXKwFrrrOM) — Michel van Biezen
- 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-spherical
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-7

## Question

Give the general method of calculating capacitance of a capacitor. Use the method to calculate the capacitance of a spherical capacitor.

## Short answer

General method: put ±q on conductors, use Gauss’s law to find E, obtain V = ∫E·dl, then C = q/V. For concentric spheres of radii a<b, E = q/(4πε₀r²), giving C = 4πε₀ab/(b−a).

## Answer

The capacitance of a capacitor is C = q/V, with q the magnitude of the charge on each plate and V the potential difference between them.

#### General method
1. Put +q on one conductor and −q on the other.
2. Use Gauss's law with a symmetric surface to find E between the conductors.
3. Compute V = V₊ − V₋ = ∫E·dl along a path from the positive to the negative conductor.
4. Find C = q/V; q cancels, leaving a purely geometric result.

#### Spherical capacitor
Concentric spheres: inner radius *a* carries +q, outer shell of inner radius *b* carries −q.

```text
        .-----------------.
      /     outer (-q)      \
     |     .-----------.     |
     |    |  inner (+q) |    |
     |    |    radius a |    |
     |     '-----------'     |
      \       radius b       /
        '-----------------'
```

1. Gaussian sphere of radius r (a < r < b): E (4πr²) = q/ε0.
2. E = q / (4πε0 r²), radially outward.
3. V = ∫ₐᵇ E dr = (q/4πε0) ∫ₐᵇ dr/r² = (q/4πε0)(1/a − 1/b) = (q/4πε0)(b − a)/(ab)
4. C = q/V

**C = 4πε0 ab / (b − a)**

Special case: for b → ∞ (an isolated sphere), C = 4πε0 a. With a dielectric of constant k, C = 4πε0 k ab/(b − a).

## Check yourself

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

1. What is the correct order in the general method?
   A. Find V first, then E = V/d, then C = V/q
   B. Choose C, then compute q and V
   C. Put ±q on the conductors, find E by Gauss's law, find V = ∫E·dl, then C = q/V
   D. Compute V from q, then differentiate to get E
   Answer: C. C is purely geometric because q cancels. (Key point: General method)

2. What is the field between the concentric spheres?
   A. E = q/(2πε0 r)
   B. E = q/(4πε0 r^2)
   C. E = q/ε0
   D. E = q r/(4πε0)
   Answer: B. Gauss's law for a sphere: E(4πr^2) = q/ε0. (Key point: Field)

3. What is C of the spherical capacitor?
   A. C = 4πε0 ab/(b - a)
   B. C = 4πε0 (b - a)/(ab)
   C. C = 4πε0 (a + b)
   D. C = 4πε0 ab/(b + a)
   Answer: A. V = (q/4πε0)(1/a - 1/b) = (q/4πε0)(b - a)/(ab). (Key point: Result)

4. What happens as b -> infinity?
   A. C -> infinity
   B. C -> 0
   C. C -> 4πε0 b
   D. C -> 4πε0 a, the capacitance of an isolated sphere
   Answer: D. ab/(b - a) -> a as b becomes large. (Key point: Special case)

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