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

# Self induction of solenoid and toroid

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2070 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/self-induction-solenoid-toroid-derive
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Explain the phenomenon of self induction. Derive the expressions for inductance of (a) a long solenoid and (b) a toroid. Then show that inductance is the property of the coil.

## Short answer

Self induction is the opposing emf induced in a coil by its own changing current. For a long solenoid L = μ₀N²A/ℓ = μ₀n²Aℓ; for a toroid L = (μ₀N²h/2π) ln(b/a). Both depend only on geometry and permeability, showing inductance is a property of the coil.

## Answer

#### Self induction
When the current in a coil changes, its own magnetic flux changes, and by Faraday's law an emf is induced in the same coil. This emf opposes the change of current (Lenz's law). The phenomenon is **self induction**. The flux linkage is NΦ = LI, so the emf is ε = −L dI/dt, where L is the **self inductance** (unit henry).

#### (a) Long solenoid
Length ℓ, N turns, cross-section area A, current I.
1. Field inside (Ampere's law): B = μ₀ N I/ℓ.
2. Flux through one turn: Φ = BA = μ₀NIA/ℓ.
3. Flux linkage: NΦ = μ₀N²AI/ℓ.
4. L = NΦ/I:

**L = μ₀ N² A / ℓ = μ₀ n² A ℓ**  (n = N/ℓ)

#### (b) Toroid
Toroid with N turns, inner radius a, outer radius b, height h.

```text
      ____________
     /   b        \
    |   ( a )      |   B is circular, inside the core
     \____________/
```

1. Ampere's law on a circle of radius r inside the core: B(2πr) = μ₀NI, so B = μ₀NI/(2πr).
2. Flux through the rectangular section: Φ = ∫ₐᵇ B h dr = (μ₀NIh/2π) ln(b/a).
3. L = NΦ/I:

**L = μ₀ N² h ln(b/a) / (2π)**

#### L is a property of the coil
In both results L contains only N, the dimensions (A, ℓ, a, b, h) and μ₀ (or μ for a core). There is no I or dI/dt in the expressions. So L depends only on the geometry of the coil and the medium, and not on the current flowing. Therefore inductance is a property of the coil itself, like resistance for a resistor.

## Check yourself

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

1. Self induction means:
   A. a changing current in one coil induces emf in another coil
   B. a changing current in a coil induces an emf in the same coil that opposes the change
   C. a steady current induces a steady emf
   D. the emf aids the change of current
   Answer: B. Lenz's law makes the self-induced emf oppose the change: ε = −L dI/dt. (Key point: Self induction)

2. The inductance of a long solenoid is:
   A. L = μ₀N²A/ℓ
   B. L = μ₀NA/ℓ
   C. L = μ₀N²ℓ/A
   D. L = μ₀ N A ℓ
   Answer: A. Φ = μ₀NIA/ℓ and L = NΦ/I. (Key point: Solenoid inductance)

3. In the toroid derivation, the flux through the rectangular section is found by:
   A. multiplying B at the inner radius by the area
   B. integrating B dr over 0 to ∞
   C. integrating B h dr from a to b, giving (μ₀NIh/2π)ln(b/a)
   D. using B = μ₀NI/ℓ with ℓ = h
   Answer: C. B varies as 1/r across the section, so integration gives a logarithm. (Key point: Toroid derivation)

4. The inductance of the toroid is:
   A. L = μ₀N h ln(b/a)/(2π)
   B. L = μ₀N² h (b−a)/(2π)
   C. L = μ₀ N² ln(b/a)/(2π h)
   D. L = μ₀N²h ln(b/a)/(2π)
   Answer: D. Multiply the flux by N and divide by I. (Key point: Toroid result)

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