---
title: "Capacitance of dielectric-filled spherical shells"
subject: "Engineering Physics"
chapter: 7
key: "semester_1/engineering_physics:spherical-shells-dielectric-capacitance"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/spherical-shells-dielectric-capacitance
site: BCT Engineering
---

# Capacitance of dielectric-filled spherical shells

Engineering Physics · Chapter 7: Electrostatics

- Asked in: 2073 Shrawan, 5 marks
- Video: [5. Capacitance of a spherical capacitor with two different dielectrics](https://www.youtube.com/watch?v=ogmwET7J1V0) — Electrical Power For Dummies
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/spherical-shells-dielectric-capacitance
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-7

## Question

The space between two concentric conducting spherical shells of radii b = 1.70 cm and a = 1.20 cm is filled with a substance of dielectric constant k = 23.5. A potential difference V = 73 V is applied across the inner and outer shells. Determine (a) the capacitance of the device (b) the free charge q on the inner shell.

## Short answer

Capacitance C = 4πε₀ k ab/(b−a) = 1.113×10⁻¹⁰ × 23.5 × 4.08×10⁻² ≈ 1.07×10⁻¹⁰ F (107 pF). Free charge q = CV = 1.07×10⁻¹⁰ × 73 ≈ 7.8×10⁻⁹ C (7.8 nC).

## Answer

Given: inner radius a = 1.20 cm = 1.20 × 10⁻² m, outer radius b = 1.70 cm = 1.70 × 10⁻² m, k = 23.5, V = 73 V.

#### (a) Capacitance
For a spherical capacitor with vacuum, C₀ = 4πε0 ab/(b − a). With a dielectric filling the gap, C = k C₀.

1. C = 4πε0 k ab/(b − a)
2. ab = (1.20 × 10⁻²)(1.70 × 10⁻²) = 2.04 × 10⁻⁴ m²
3. b − a = 0.50 × 10⁻² = 5.0 × 10⁻³ m
4. ab/(b − a) = 2.04 × 10⁻⁴ / 5.0 × 10⁻³ = 4.08 × 10⁻² m
5. 4πε0 = 1.113 × 10⁻¹⁰ F/m
6. C = 1.113 × 10⁻¹⁰ × 23.5 × 4.08 × 10⁻² = 1.07 × 10⁻¹⁰ F

**Answer (a): C ≈ 1.07 × 10⁻¹⁰ F = 107 pF**

#### (b) Free charge on the inner shell
q = CV = 1.067 × 10⁻¹⁰ × 73 = 7.79 × 10⁻⁹ C

**Answer (b): q ≈ 7.8 × 10⁻⁹ C = 7.8 nC** (the free charge is on the inner shell; the outer shell carries −7.8 nC).

## Check yourself

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

1. Which formula gives the capacitance of a dielectric-filled spherical capacitor?
   A. C = 4πε0 k ab/(b - a)
   B. C = 4πε0 k ab/(b + a)
   C. C = 4πε0 ab/[k (b - a)]
   D. C = 2πε0 k L/ln(b/a)
   Answer: A. Vacuum C0 = 4πε0 ab/(b - a); the dielectric multiplies it by k. (Key point: Capacitance formula)

2. What is the capacitance for a = 1.20 cm, b = 1.70 cm, k = 23.5?
   A. About 4.5 pF (forgetting k)
   B. About 1.07 nF
   C. About 25 pF
   D. About 107 pF
   Answer: D. C = 1.113 x 10^-10 x 23.5 x 4.08 x 10^-2 ≈ 1.07 x 10^-10 F. (Key point: Value)

3. How is the free charge on the inner shell found, and what is the outer shell's charge?
   A. q = C/V ≈ 1.5 pC; the outer carries the same sign
   B. q = CV ≈ 7.8 nC; the outer shell is uncharged
   C. q = CV ≈ 7.8 nC; the outer shell carries -7.8 nC
   D. q = CV/k ≈ 0.33 nC; the outer carries -0.33 nC
   Answer: C. The free charge q = CV = 1.067 x 10^-10 x 73, and the outer shell has equal and opposite charge. (Key point: Free charge)

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