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

# Self-inductance of a toroid

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2074 Chaitra, 5 marks
- 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/induction-vs-self-induction-toroid
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Differentiate between electromagnetic induction and self-induction. Develop an expression for self-inductance of a toroid having N turns, radius r and carrying current i.

## Short answer

Electromagnetic induction is emf from any changing flux; self-induction is emf from a coil’s own changing current. For a toroid, Ampere’s law gives B = μ₀Ni/(2πr), so flux linkage NΦ = μ₀N²Ai/(2πr), yielding self-inductance L = μ₀N²A/(2πr).

## Answer

#### Electromagnetic induction versus self-induction
| Electromagnetic induction | Self-induction |
|---|---|
| An emf is induced in a circuit when the flux linked with it changes, from any cause | An emf is induced in a coil due to the change of current in that same coil |
| The changing flux may come from another circuit, a moving magnet or a change in area or orientation | The changing flux is produced by the coil's own current |
| Mutual induction involves two coils | Involves one coil |
| Law: ε = −dΦ/dt | ε = −L di/dt |
| Depends on the rate of flux change | Depends on the inductance L and di/dt |
| Example: generator, transformer | Example: choke coil, solenoid |

Self-induction is a special case of electromagnetic induction.

#### Self-inductance of a toroid
Toroid with N turns, mean radius r, core area A, carrying current i.

```text
       .------------.
     /   N turns, i    \
    |  .------------.   |   mean radius r
     \ '------------' /
       '------------'
```

1. Take a circular Amperian loop of radius r inside the core: B (2πr) = μ0 N i
2. B = μ0 N i / (2πr)
3. Flux through one turn: Φ = B A = μ0 N i A / (2πr)
4. Total flux linkage: NΦ = μ0 N² A i / (2πr)
5. Inductance L = NΦ/i:

**L = μ0 N² A / (2πr)**

It depends only on the geometry and core material, increasing with N² and A and decreasing with r. With an iron core of relative permeability μr, L is multiplied by μr.

## Check yourself

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

1. Self-induction differs because the flux change is
   A. From another coil
   B. From the coil's own current
   C. From a moving magnet
   D. From temperature
   Answer: B. EM induction covers any flux change; self-induction is the own-current case. (Key point: Difference)

2. The self-induction law is
   A. ε = −dΦ/dt only
   B. ε = −L di/dt
   C. ε = −Li
   D. ε = L dΦ
   Answer: B. It depends on L and di/dt. (Key point: Laws)

3. L of a toroid is
   A. μ0 N^2 A/(2πr)
   B. μ0 N A/(2πr)
   C. μ0 N^2/(2πr A)
   D. μ0 N^2 A r
   Answer: A. B = μ0Ni/(2πr) and L = NΦ/i. (Key point: Toroid result)

4. L increases with
   A. r
   B. N^2 and A
   C. i
   D. Resistance
   Answer: B. It increases with N^2 and A and decreases with r. (Key point: Dependence)

## Close variants set in other exams

- [Self-inductance of a toroid](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atoroid-self-inductance-turns-radius) — 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

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