---
title: "Inductance in terms of physical dimensions"
subject: "Basic Electrical Engineering"
chapter: 4
key: "semester_1/basic_electrical_engineering:inductance-physical-dimensions"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/inductance-physical-dimensions
site: BCT Engineering
---

# Inductance in terms of physical dimensions

Basic Electrical Engineering · Chapter 4: Inductance & Capacitance in Electric Circuits

- Asked in: 2072 Chaitra, 4 marks; 2071 Chaitra, 4 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/inductance-physical-dimensions
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-4

## Question

Derive the expression for the inductance of inductor in terms of its physical dimensions.

## Short answer

From Ampere’s law, H = NI/l, B = μH, flux Φ = BA = μ N I A/l. Inductance L = NΦ/I = μ N² A/l, where μ = μ0 μr, N turns, A cross-sectional area, l mean length. L increases with N² and core area, and decreases with length.

## Answer

Consider a coil (solenoid or toroid) of N turns wound on a core of mean length *l*, cross-sectional area *A* and relative permeability μ_r.

```text
   +-----------------------+
   | ( ( ( ( ( ( ( ( ( ( ( |   N turns, current I
   +-----------------------+
   <---------- l --------->    core area A, permeability μ
```

#### Derivation
**Step 1: magnetic field strength.** By Ampere's circuital law, the MMF of the coil is N I, so

H = N I/l (A/m)

**Step 2: flux density.** B = μ H = μ₀ μ_r N I/l (tesla)

**Step 3: flux.** Φ = B A = μ₀ μ_r N I A/l (weber)

**Step 4: inductance.** Inductance is the flux linkage per ampere:

L = N Φ/I = N × μ₀ μ_r N A/l

**L = μ₀ μ_r N² A/l = μ N² A/l henry**

where μ₀ = 4π × 10⁻⁷ H/m.

(Equivalent form: L = N²/S, where S = l/(μ A) is the reluctance of the magnetic circuit.)

#### Conclusions
- L is proportional to N²: doubling the turns makes the inductance four times larger.
- L is proportional to the core area A and permeability μ (iron core gives a much larger L than air).
- L is inversely proportional to the length *l*.

Example: N = 500, A = 4 cm², l = 20 cm, μ_r = 1000:
L = 4π × 10⁻⁷ × 1000 × 500² × 4 × 10⁻⁴/0.2 = 0.628 H.

## Check yourself

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

1. What is the MMF and field strength in the coil?
   A. MMF = N/I
   B. MMF = I/N, H = Il
   C. MMF = NI, H = NIl
   D. MMF = NI, H = NI/l
   Answer: D. By Ampere's law H = NI/l. (Key point: Step 1)

2. Which gives flux?
   A. Phi = mu N I l / A
   B. Phi = B/A
   C. Phi = BA = mu0 mur N I A / l
   D. Phi = H/A
   Answer: C. B = mu H, multiply by area A. (Key point: Flux)

3. What is inductance in terms of dimensions?
   A. L = N A / (mu l)
   B. L = mu N l / A
   C. L = mu N A / l^2
   D. L = mu N^2 A / l
   Answer: D. L = N Phi / I = mu N^2 A/l. (Key point: Final formula)

4. If turns are doubled, what happens to L?
   A. Becomes four times
   B. Doubles
   C. Halves
   D. Unchanged
   Answer: A. L is proportional to N^2. (Key point: Dependence)

## Related in Basic Electrical Engineering

- [Inductance from flux linkages](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Ainductance-flux-linkages-derivation) — Basic Electrical Engineering, chapter 4
- [Number of turns of iron-core inductor](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Ainductor-turns-iron-core-500mh) — Basic Electrical Engineering, chapter 4
- [Inductance, series mutual flux aiding](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Ainductance-series-aiding) — Basic Electrical Engineering, chapter 4
- [Inductance and inductors in parallel](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Ainductors-in-parallel-derivation) — Basic Electrical Engineering, chapter 4, asked 2×
- [Average power in pure inductor](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Aaverage-power-pure-inductor) — Basic Electrical Engineering, chapter 6
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