---
title: "Self induction; solenoid and toroid inductance"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:solenoid-toroid-inductance"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/solenoid-toroid-inductance
site: BCT Engineering
---

# Self induction; solenoid and toroid inductance

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2074 Asoj, 5 marks; 2073 Shrawan, 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/solenoid-toroid-inductance
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Explain the meaning of self induction. Calculate the inductance of a solenoid and a toroid.

## Short answer

Self-induction is the property where a changing current induces an opposing emf in the same coil, ε = -L di/dt. For a solenoid, L = μ0 N^2 A / l. For a toroid, L = μ0 N^2 A / (2πr). Both depend on geometry and core material.

## Answer

**Self-induction** is the property of a coil by which a changing current in it induces an emf in the same coil, opposing the change (Lenz's law). The induced emf is ε = −L di/dt, where the **self-inductance** L = NΦ/i has unit henry (H).

#### Inductance of a solenoid
Length l, cross-sectional area A, N turns, n = N/l turns per metre, current i.

1. Field inside (Ampere's law): B = μ0 n i
2. Flux through each turn: Φ = BA = μ0 n i A
3. Flux linkage: NΦ = (nl)(μ0 n i A) = μ0 n² A l i
4. L = NΦ/i

**L = μ0 n² A l = μ0 N² A / l**

With a core of relative permeability μr, multiply by μr.

#### Inductance of a toroid
Toroid of N turns, mean radius r, cross-section area A (small compared with r), current i.

```text
       .------------.
     /    ( toroid )   \
    |     (  core  )    |
     \    (        )   /
       '------------'
         mean radius r
```

1. Amperian loop of radius r inside the core: B (2πr) = μ0 N i
2. B = μ0 N i / (2πr)
3. Flux per turn: Φ = BA = μ0 N i A / (2πr)
4. L = NΦ/i

**L = μ0 N² A / (2πr)**

Both depend only on geometry (and core material), not on the current.

## Check yourself

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

1. Self-induction is
   A. An emf induced in a nearby coil
   B. An emf in the same coil opposing change in its current
   C. Flux due to a magnet
   D. Resistance of a coil
   Answer: B. Changing current induces emf ε = −L di/dt in the same coil. (Key point: Self-induction)

2. The inductance of a solenoid is
   A. μ0 n A l
   B. μ0 n^2 A l
   C. μ0 N A/l^2
   D. μ0 n^2 A/l
   Answer: B. NΦ = (nl)(μ0 n i A) = μ0 n^2 A l i, so L = μ0 n^2 A l = μ0 N^2 A/l. (Key point: Solenoid)

3. The toroid inductance is
   A. μ0 N^2 A/(2πr)
   B. μ0 N A/(2πr)
   C. μ0 N^2 A/(πr)
   D. μ0 N^2 A r/(2π)
   Answer: A. B = μ0Ni/(2πr), so L = NΦ/i = μ0 N^2 A/(2πr). (Key point: Toroid)

4. L of these coils depends on
   A. The current i
   B. Only geometry and core material
   C. The rate of change di/dt
   D. The emf of the source
   Answer: B. L is set by geometry and core permeability, not by the current. (Key point: Dependence)

## Related in Engineering Physics

- [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-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×
- [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×
- [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