---
title: "Electric and magnetic energy densities"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:energy-density-electric-magnetic-proportional"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/energy-density-electric-magnetic-proportional
site: BCT Engineering
---

# Electric and magnetic energy densities

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2069 Asar, 5 marks
- Video: [Energy density in Magnetic fields | Inductance | Physics | Khan Academy](https://www.youtube.com/watch?v=BTa5l6giwto) — Khan Academy India - English
- 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-electric-magnetic-proportional
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Show that the energy per unit volume in electric field and magnetic field are proportional to the square of their fields.

## Short answer

From a parallel-plate capacitor, u_E = U/volume = ½ ε₀ E². From a long solenoid, u_B = B²/(2μ₀). Both energy densities are proportional to the square of the respective field; in a medium, ε₀→ε, μ₀→μ.

## Answer

Energy density u is the energy stored per unit volume.

#### Electric field
Take a parallel plate capacitor of plate area A and separation d, so the field is E = V/d, and C = ε₀A/d.

```text
U = ½ C V² = ½ (ε₀A/d)(E d)² = ½ ε₀ E² (A d)
u_E = U/(A d) = ½ ε₀ E²
```

#### Magnetic field
Take a long solenoid, length ℓ, area A, n turns per metre. B = μ₀nI, L = μ₀n²Aℓ.

```text
U = ½ L I² = ½ (μ₀ n² A ℓ) I²
I = B/(μ₀ n)  →  U = ½ μ₀ n² A ℓ · B²/(μ₀² n²) = B² A ℓ /(2μ₀)
u_B = U/(A ℓ) = B²/(2μ₀)
```

**Result:**
- u_E = ½ ε₀E² ∝ E²
- u_B = B²/(2μ₀) ∝ B²

Both are proportional to the square of the field. In a medium, replace ε₀ with ε and μ₀ with μ. In an EM wave E = cB, so u_E = u_B.

## Check yourself

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

1. Energy per unit volume in an electric field is:
   A. u_E = ε₀E
   B. u_E = ½ε₀E²
   C. u_E = ½ε₀E
   D. u_E = ε₀E²
   Answer: B. U = ½CV² with C = ε₀A/d and V = Ed gives ½ε₀E² × Ad. (Key point: Electric energy density)

2. Energy per unit volume in a magnetic field is:
   A. u_B = B²/(2μ₀)
   B. u_B = μ₀B²/2
   C. u_B = B/(2μ₀)
   D. u_B = B²μ₀
   Answer: A. U = ½LI² with L = μ₀n²Aℓ and I = B/(μ₀n). (Key point: Magnetic energy density)

3. In the magnetic derivation, which expression is substituted for I?
   A. I = μ₀ n B
   B. I = B n/μ₀
   C. I = B/(μ₀ n)
   D. I = B/(μ₀ n²)
   Answer: C. From B = μ₀nI inside a long solenoid. (Key point: Method for B)

4. What do the results show?
   A. Both are proportional to the field itself
   B. Only electric energy density depends on the field squared
   C. Energy density is independent of field strength
   D. Both energy densities are proportional to the square of their fields
   Answer: D. u_E ∝ E² and u_B ∝ B². (Key point: Conclusion)

## Related in Engineering Physics

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