---
title: "Energy stored in an inductor"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:energy-density-inductor-magnetic-field"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/energy-density-inductor-magnetic-field
site: BCT Engineering
---

# Energy stored in an inductor

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2074 Chaitra, 5 marks; 2072 Chaitra, 5 marks; 2069 Chaitra, 5 marks; 2067 Asoj, 5 marks (asked 4 times)
- Video: [Energy Stored in an Inductor](https://www.youtube.com/watch?v=_91Zv3CZ4qc) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/energy-density-inductor-magnetic-field
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Determine the energy stored in an inductor (derive the expression). Hence show that the magnetic energy density is directly proportional to the square of the magnetic field (flux density), and compare it with the electric energy density.

## Short answer

Energy stored in an inductor: U = ½ L I². Magnetic energy density u_B = B²/(2μ₀), proportional to B², analogous to electric energy density u_E = ½ ε₀ E².

## Answer

#### Energy stored in an inductor
When current i flows in an inductor of inductance L, the induced emf opposes the rise: ε = −L di/dt. The source must do work against it.

1. Power supplied against the back emf: P = ε_back · i = L i (di/dt)
2. Work in time dt: dW = L i di
3. Total work to raise current from 0 to I:

U = ∫₀ᴵ L i di = ½ L I²

**U = ½ L I²**, stored in the magnetic field.

#### Magnetic energy density
Take a long solenoid of length l, area A, n turns per unit length. B = μ0 n I and L = μ0 n² A l.

1. U = ½ (μ0 n² A l) I² = (1/2μ0)(μ0 n I)² A l = B²/(2μ0) · (A l)
2. Volume = A l, so energy per unit volume:

**u_B = B²/(2μ0)**

Thus u_B ∝ B², the square of the flux density. This holds for any magnetic field in vacuum.

#### Comparison with electric energy density
| Property | Magnetic | Electric |
|---|---|---|
| Energy density | u_B = B²/(2μ0) | u_E = ½ ε0 E² |
| Depends on | square of B | square of E |
| Stored in device | inductor, U = ½LI² | capacitor, U = ½CV² |
| Constant | μ0 (in denominator) | ε0 (in numerator) |

Both are proportional to the square of the field and exist throughout space where the field is present.

## Check yourself

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

1. The energy stored in an inductor carrying I is
   A. L I^2
   B. ½ L I^2
   C. ½ L I
   D. L I^2/2μ0
   Answer: B. U = ∫L i di from 0 to I = ½ L I^2. (Key point: Stored energy)

2. The work done in raising current is dW =
   A. L i^2 dt
   B. L di
   C. L i di
   D. i^2 R dt
   Answer: C. Power against back emf is L i di/dt, so dW = L i di. (Key point: Work element)

3. The magnetic energy density is
   A. B^2/(2μ0)
   B. μ0 B^2/2
   C. B/(2μ0)
   D. ½ ε0 B^2
   Answer: A. Dividing U = B^2 Al/(2μ0) by volume Al gives B^2/(2μ0), so u ∝ B^2. (Key point: Energy density)

4. How does u_B compare with u_E = ½ε0E^2?
   A. u_E depends on E linearly
   B. Both are proportional to the square of the field
   C. Only the electric one is stored in space
   D. μ0 appears in the numerator for u_B
   Answer: B. Both are quadratic in the field; μ0 is in the denominator for u_B and ε0 in the numerator for u_E. (Key point: Comparison)

## Related in Engineering Physics

- [Electric and magnetic energy densities](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aenergy-density-electric-magnetic-proportional) — Engineering Physics, chapter 8
- [Cyclotron deuterons, flux density and energy](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acyclotron-deuteron-flux-density) — Engineering Physics, chapter 8
- [Magnetic field on axis of circular coil](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acircular-coil-axis-field) — Engineering Physics, chapter 8, asked 2×
- [Energy stored in coil, time constants](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acoil-energy-stored-time-constants) — Engineering Physics, chapter 8
- [Energy stored in parallel capacitors](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aparallel-capacitors-energy-stored) — Engineering Physics, chapter 7
- [Induced magnetic field, displacement current](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-induced-field-circular-plates) — Engineering Physics, chapter 9

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