---
title: "Self-inductance of a toroid"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:toroid-self-inductance-turns-radius"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/toroid-self-inductance-turns-radius
site: BCT Engineering
---

# Self-inductance of a toroid

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2071 Shrawan, 5 marks; 2069 Asar, 5 marks (asked 2 times)
- Video: [Self Inductance of Inductors & Coils - Solenoids & Toroids - Physics](https://www.youtube.com/watch?v=gMn0MDl4vkc) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/toroid-self-inductance-turns-radius
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Explain the phenomenon of self-induction. Find an expression for the self-induction (inductance) of a toroid having N turns, radius r and carrying current i.

## Short answer

Self-induction: changing current induces an opposing emf in the same coil, ε = -L di/dt. For a toroid, Ampere's law gives B = μ0 N i/(2πr), flux Φ = BA, so L = NΦ/i = μ0 N^2 A/(2πr). It depends only on geometry and core.

## Answer

**Self-induction.** When the current in a coil changes, the magnetic flux linked with the coil itself changes, and an emf is induced in the same coil which opposes the change in current (Lenz's law). ε = −L di/dt, where L = NΦ/i is the self-inductance (henry).

#### Toroid
A toroid is a solenoid bent into a ring. Let it have N turns, mean radius r, core cross-sectional area A, and current i.

```text
        _____________
      /  ___________  \
     |  /   (turns)   \ |   <- mean radius r
     |  \_____________/ |
      \________________/
```

#### Derivation
1. Choose a circular Amperian loop of radius r inside the core. By symmetry B is tangential and constant along it.
2. Ampere's law: B (2πr) = μ0 N i  (the loop encloses N turns, each carrying i)
3. B = μ0 N i / (2πr)
4. Flux through one turn: Φ = B A = μ0 N i A / (2πr)
5. Flux linkage: NΦ = μ0 N² i A / (2πr)
6. Inductance: L = NΦ/i

**L = μ0 N² A / (2πr)**

(Using the mean radius r, with A ≪ r².) With a core of relative permeability μr, L = μ0μr N² A/(2πr). Outside the toroid the field is zero, since the loop encloses zero net current.

L depends only on the geometry (N, A, r) and the core material, not on i.

## Check yourself

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

1. The Amperian loop of radius r in the core encloses
   A. Zero net current
   B. One turn
   C. N turns each carrying i
   D. The whole toroid twice
   Answer: C. Hence B(2πr) = μ0 N i. (Key point: Amperian loop)

2. Flux through one turn is
   A. μ0 N i A/(2πr)
   B. μ0 N^2 i A/(2πr)
   C. μ0 i A/(2πr)
   D. μ0 N i/(2πr)
   Answer: A. Φ = BA with B = μ0Ni/(2πr). (Key point: Flux)

3. L of the toroid is
   A. μ0 N A/(2πr)
   B. μ0 N^2 A/(2πr)
   C. μ0 N^2 A/r
   D. 2πr μ0 N^2/A
   Answer: B. L = NΦ/i = μ0 N^2 A/(2πr). (Key point: Inductance)

4. Outside the toroid the field is zero because
   A. The core shields it
   B. Current is zero
   C. The loop encloses zero net current
   D. B is infinite
   Answer: C. The loop encloses turns going in and out, so I_enc = 0. (Key point: Outside field)

## Close variants set in other exams

- [Self-inductance of a toroid](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ainduction-vs-self-induction-toroid) — Engineering Physics, chapter 8, asked 3×

## Related in Engineering Physics

- [Self induction; solenoid and toroid inductance](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asolenoid-toroid-inductance) — Engineering Physics, chapter 8, asked 2×
- [Self induction of solenoid and toroid](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aself-induction-solenoid-toroid-derive) — Engineering Physics, chapter 8
- [Self induction and inductance of a coil](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aself-induction-inductance-permeability) — Engineering Physics, chapter 8
- [Inductance of a toroid numerical](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atoroid-inductance-numerical) — Engineering Physics, chapter 8

## Related in other subjects

- [Inductance of twin-bundle line, 16 mm radius](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Atwin-bundle-150km-16mm) — Power System Analysis I, chapter 4, asked 3×
- [Inductance of transposed double circuit, 0.9 cm radius](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Adouble-circuit-inductance-0-9cm) — Power System Analysis I, chapter 4
- [Ring with sinusoidal current, inductance and emf](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aem-oscillating-ring-inductance) — Electrical Machines, chapter 1
- [Iron ring with air gap: current, inductance, energy](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Airon-ring-15cm-stored-energy) — Electrical Machines, chapter 1