Electromagnetic Field Energy Density Calculator

Compute electric (uₑ) and magnetic (uₘ) energy densities stored in an electromagnetic field. Supports vacuum or linear dielectric/magnetic media. Visualize energy distribution and explore Maxwell's energy theorem.

Magnitude of electric field (non-negative).
Magnetic field strength in Tesla.
Dielectric constant (vacuum = 1). For water ≈ 80, FR4 ≈ 4.5.
Magnetic constant (vacuum = 1). Ferrites can be > 1000.
Quick examples:
? EM wave (E=10 V/m, B≈3.33e-8 T)
⚡ Capacitor (E=1e6 V/m, εr=2.2)
? Inductor (B=0.5 T, μr=800)
? Strong RF (E=5000 V/m, B=0.02 T)
? Vacuum reference (E=1kV/m, B=1mT)
Client-side processing: All calculations run locally in your browser using double-precision arithmetic. No data is transmitted to any server.

Fundamentals of Field Energy Density

In classical electrodynamics, energy can be stored in electric and magnetic fields. The instantaneous energy density (energy per unit volume) for an electric field in a linear isotropic medium is given by uₑ = ½ ε E² where ε = ε₀ εᵣ. Similarly, magnetic energy density is uₘ = ½ (B² / μ) with μ = μ₀ μᵣ. These expressions originate from Poynting's theorem and are fundamental for capacitors, inductors, transformers, and electromagnetic waves.

utotal = ½ ε₀ εᵣ E² + ½ (B² / (μ₀ μᵣ))

[J/m³] – valid for linear isotropic media, no hysteresis.

Historical & Scientific Context

James Clerk Maxwell unified electricity and magnetism, showing that energy resides in the fields themselves. The concept of electromagnetic energy density was refined by John Henry Poynting (1884) and forms the backbone of modern electromagnetics. This calculator uses the exact vacuum permittivity and permeability recommended by CODATA (2018), ensuring traceability to international measurement standards.

Applications Across Disciplines

  • Capacitor design: Maximizing energy density for pulse power systems (u ∝ εᵣ E²).
  • Inductor & transformer core selection: High μᵣ materials store greater magnetic energy per volume.
  • Wireless power transfer: Near-field energy density determines coupling efficiency.
  • Electromagnetic wave propagation: For plane waves in vacuum uₑ = uₘ, total u = ε₀ E² = B²/μ₀.
  • MRI & RF safety: SAR (Specific Absorption Rate) is derived from electric energy density.
Case Study: High-Energy Capacitor Banks

In a typical pulsed-power capacitor (εᵣ ≈ 3, dielectric strength 20 MV/m), the maximum electric energy density reaches ~½·(3·8.85e-12)·(2e7)² ≈ 5310 J/m³. Our calculator reveals how material selection and field strength directly impact volumetric efficiency. Compare with magnetic energy density in superconducting magnets (B ~ 10 T, μᵣ≈1) → uₘ ≈ 3.98 × 10⁷ J/m³. This explains why fusion experiments prefer magnetic confinement – orders of magnitude higher energy density.

Step-by-Step Calculation & Derivations

The tool implements the exact formulas: First, absolute permittivity ε = ε₀ * εᵣ, absolute permeability μ = μ₀ * μᵣ. For electric component: uₑ = 0.5 * ε * E². For magnetic component: uₘ = 0.5 * B² / μ. All quantities in SI units. The algorithm also checks for non-positive εᵣ or μᵣ and issues warnings.

Medium / Scenario Typical E (V/m) or B (T) Energy Density (J/m³) Application
Air breakdown threshold E ≈ 3×10⁶ V/m ~39.8 J/m³ Lightning, spark gaps
Electrostatic loudspeaker E = 5000 V/m 0.11 J/m³ Audio transducers
Neodymium magnet surface B ≈ 1.2 T ~5.73×10⁵ J/m³ Permanent magnet motors
Solar wind (Earth orbit) E ≈ 0.01 V/m, B≈6 nT ~1.4×10⁻¹⁴ J/m³ Space plasma physics

The Electromagnetic Wave Connection

For a plane electromagnetic wave in vacuum, the electric and magnetic energy densities are equal: ½ ε₀ E² = ½ B²/μ₀. The total energy density u = ε₀ E² = B²/μ₀. Our calculator confirms this relationship automatically (notice when E/B = c ≈ 299792458 m/s). This consistency is a direct consequence of Maxwell's equations. Test it yourself: enter E = 3×10⁶ V/m and compute the corresponding B = E/c (≈ 0.01 T) – you will see uₑ = uₘ.

Common Misconceptions Clarified

  • Myth: "Energy density depends on field direction." → Reality: Energy density is scalar, depends on magnitude squared.
  • Myth: "Materials don't affect magnetic energy density." → Reality: μᵣ directly scales uₘ for a given B (or scales B for fixed H).
  • Myth: "High εᵣ always increases stored energy." → True for fixed E, but dielectrics have breakdown limits.

Frequently Asked Questions (FAQ)

SI unit: Joules per cubic meter (J/m³). Equivalent to Pa (Pascal) or N/m².

The calculator assumes isotropic linear media. For anisotropic crystals, energy density would require tensor permittivity/permeability, which is beyond this tool's scope.

Yes, instantaneous energy density holds for time-varying fields as long as medium is non-dispersive. For dispersive media, stored energy definition is more subtle.

The ½ factor arises from the work done to establish the field from zero, linear relationship between field and source (D = εE, B = μH).

Yes – multiply the energy density by the volume (in m³) after integrating. Our tool provides density; you can manually compute total energy.
References & Further Reading: CODATA recommended values (ε₀, μ₀), Griffiths, D.J. "Introduction to Electrodynamics" (4th ed), Feynman Lectures Vol.II Ch.27.
All calculations are derived directly from these standard references. The tool's output matches theoretical predictions (e.g., uₑ = uₘ for plane waves in vacuum). Last verification based on CODATA 2018 constants.
Disclaimer: This tool implements standard electromagnetic formulas. No proprietary or confidential data is used. All results are for educational and reference purposes.