---
title: "Inductance from flux linkages"
subject: "Basic Electrical Engineering"
chapter: 4
key: "semester_1/basic_electrical_engineering:inductance-flux-linkages-derivation"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/inductance-flux-linkages-derivation
site: BCT Engineering
---

# Inductance from flux linkages

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

- Asked in: 2073 Shrawan, 4 marks
- 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-flux-linkages-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-4

## Question

Derive an equation for inductance L in terms of flux linkages and current change.

## Short answer

Inductance L equals flux linkage NΦ divided by current i. From Faraday’s law, induced emf e = -L di/dt; thus L = N dΦ/di = e/(di/dt). 1 H corresponds to 1 Wb-turn/A or 1 V per A/s.

## Answer

**Inductance** L of a coil is the flux linkage produced per unit current. Consider a coil of N turns carrying current i. Let φ be the flux linking each turn.

#### Derivation

1. The flux produced is proportional to the current (for a linear magnetic path), so the flux linkage is proportional to i:

   Nφ = L i  ⟹  L = Nφ / i

2. When the current changes, the flux linkage changes. By Faraday's law the induced emf is

   e = −d(Nφ)/dt = −N dφ/dt

3. Since Nφ = L i and L is constant,

   e = −L di/dt

   The negative sign shows (Lenz's law) that the induced emf opposes the change of current.

4. Taking magnitudes, the inductance in terms of flux linkage and current change is

**L = N dφ / di = e / (di/dt)**

For a finite change: L = N Δφ / Δi = Δ(flux linkage) / Δi.

So 1 henry means 1 weber-turn of flux linkage per ampere, or 1 V induced when current changes at 1 A/s.

## Check yourself

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

1. What is L?
   A. N phi / i
   B. i / N phi
   C. N i / phi
   D. phi / N
   Answer: A. Flux linkage per ampere. (Key point: Definition)

2. What is induced emf?
   A. e = -N i
   B. e = phi/t
   C. e = -d(N phi)/dt
   D. e = N phi
   Answer: C. Faraday's law. (Key point: Faraday)

3. What does the minus sign in e = -L di/dt show?
   A. L is negative
   B. current reverses
   C. Induced emf opposes the change of current
   D. emf is always negative
   Answer: C. Lenz's law. (Key point: Lenz)

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