Energy Density of Electric and Magnetic Fields
Energy density quantifies how much energy is stored per unit volume in electric and magnetic fields. Electric field energy density uE = ½ε₀E² and magnetic field energy density uB = B²/(2μ₀). For electromagnetic waves in vacuum, uE = uB.
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Electric and magnetic energy densities are equal in EM waves in vacuum Capacitor energy U = ½CV² equals uE × volume for uniform fields Inductor energy U = ½LI² equals uB × volume for uniform fields Dielectric εᵣ and magnetic μᵣ materials increase energy density
Ready to run the numbers?
Why: Energy density determines capacitor and inductor storage capacity, RF power handling, and electromagnetic wave intensity. It underpins power electronics, MRI design, and wireless transmission.
How: Enter electric field E, magnetic field B, or capacitor/inductor parameters. The calculator computes energy densities using uE = ½ε₀E², uB = B²/(2μ₀), and the Poynting vector for power flow.
Run the calculator when you are ready.
⚡ Parallel Plate Capacitor
Capacitor: 100 μF, 12V, Volume 1 cm³
🔋 Solenoid Inductor
Inductor: 10 mH, 5A current, Volume 50 cm³
📡 Electromagnetic Wave
EM wave: E = 1000 V/m, B = 3.33 μT
🔄 Transformer Core
Transformer: E = 500 V/m, B = 1.5 T (iron core)
❄️ Superconductor Field
Superconductor: B = 0 T (Meissner effect), E = 100 V/m
📊 Poynting Vector
EM wave power flow: S = 1000 W/m²
Input Parameters
Electric Field Parameters
Material Properties
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Electric energy density uE = ½ε₀E² scales with E²; doubling field quadruples energy density.
— Griffiths Electrodynamics
MRI machines use magnetic field energy densities of ~10⁵ J/m³.
— NIST
Sunlight delivers ~1.4 kW/m² to Earth via the Poynting vector.
— HyperPhysics
Supercapacitors achieve high energy density through high-permittivity dielectrics.
— IEEE
📋 Key Takeaways
- • Electric field energy density: uE = ½ε₀E² — energy stored per unit volume in electric fields
- • Magnetic field energy density: uB = B²/(2μ₀) — energy stored per unit volume in magnetic fields
- • Total energy density: u = uE + uB — sum of electric and magnetic contributions
- • Capacitor energy: U = ½CV² — energy stored in capacitors is proportional to capacitance and voltage squared
- • Inductor energy: U = ½LI² — energy stored in inductors is proportional to inductance and current squared
💡 Did You Know?
📖 How Energy Density Calculation Works
Energy density calculations use fundamental electromagnetic theory from Maxwell's equations. The calculator employs energy conservation principles to compute energy densities for electric fields, magnetic fields, and their combinations.
Electric Field Energy Density
Energy stored in electric fields: uE = ½ε₀E²
Where ε₀ is vacuum permittivity (8.854×10⁻¹² F/m) and E is electric field strength in V/m. For materials with relative permittivity εᵣ, multiply by εᵣ.
Magnetic Field Energy Density
Energy stored in magnetic fields: uB = B²/(2μ₀)
Where μ₀ is vacuum permeability (4π×10⁻⁷ H/m) and B is magnetic field strength in Tesla. For magnetic materials with relative permeability μᵣ, divide by μᵣ.
Total Electromagnetic Energy Density
The total energy density is the sum: u = uE + uB
For electromagnetic waves in vacuum, electric and magnetic energy densities are equal, so u = ε₀E² = B²/μ₀.
🎯 Expert Tips for Energy Density Calculations
💡 Maximize Capacitor Energy Density
Use high-permittivity dielectrics (εᵣ > 100) and minimize plate separation. Energy density scales with E², so high-voltage designs store more energy per volume.
💡 Optimize Inductor Design
High-permeability cores (μᵣ > 1000) dramatically increase magnetic energy density. Ferrite cores can achieve energy densities 1000× higher than air-core inductors.
💡 EM Wave Energy Transport
The Poynting vector magnitude S = EB/μ₀ represents power flow density. For plane waves, S = c × u, where c is speed of light and u is energy density.
💡 Material Properties Matter
Relative permittivity εᵣ and permeability μᵣ can increase energy density by orders of magnitude. Choose materials based on your application's frequency and field strength requirements.
⚖️ Energy Density Comparison: Capacitors vs Inductors
| Energy Storage Type | Typical Energy Density | Advantages | Applications |
|---|---|---|---|
| Electric Field (Capacitor) | 10⁴ - 10⁶ J/m³ | Fast charge/discharge, high power | Power electronics, pulse systems |
| Magnetic Field (Inductor) | 10⁵ - 10⁷ J/m³ | High energy density, efficient | Transformers, motors, energy storage |
| Electromagnetic Wave | 10⁻⁶ - 10⁻³ J/m³ | Wireless transmission | RF systems, antennas |
| Supercapacitor | 10⁵ - 10⁷ J/m³ | Very high power density | Hybrid vehicles, grid storage |
⚖️ Energy Density Comparison: Capacitors vs Inductors
| Energy Storage Type | Typical Energy Density | Advantages | Applications |
|---|---|---|---|
| Electric Field (Capacitor) | 10⁴ - 10⁶ J/m³ | Fast charge/discharge, high power | Power electronics, pulse systems |
| Magnetic Field (Inductor) | 10⁵ - 10⁷ J/m³ | High energy density, efficient | Transformers, motors, energy storage |
| Electromagnetic Wave | 10⁻⁶ - 10⁻³ J/m³ | Wireless transmission | RF systems, antennas |
| Supercapacitor | 10⁵ - 10⁷ J/m³ | Very high power density | Hybrid vehicles, grid storage |
❓ Frequently Asked Questions
What is the difference between electric and magnetic energy density?
Electric energy density (uE = ½ε₀E²) stores energy in electric fields, while magnetic energy density (uB = B²/(2μ₀)) stores energy in magnetic fields. Both contribute to total electromagnetic energy density: u = uE + uB.
Why does energy density scale with field strength squared?
Energy density is proportional to E² or B² because work done to establish the field is proportional to the field strength squared. Doubling the field strength quadruples the energy density.
How do dielectric materials affect energy density?
Dielectric materials with relative permittivity εᵣ increase electric energy density by a factor of εᵣ. For example, barium titanate (εᵣ ≈ 1000) stores 1000× more energy than vacuum for the same electric field.
What is the Poynting vector and how is it related to energy density?
The Poynting vector S = E×B/μ₀ represents power flow density in electromagnetic waves. Its magnitude equals the energy density times the speed of light: |S| = c × u for plane waves in vacuum.
Can energy density be negative?
No, energy density is always positive or zero. Both uE = ½ε₀E² and uB = B²/(2μ₀) are always non-negative since they involve squared quantities.
How does energy density relate to capacitor and inductor energy?
Capacitor energy U = ½CV² equals energy density times volume: U = uE × V. Similarly, inductor energy U = ½LI² equals magnetic energy density times volume: U = uB × V (for uniform fields).
What are typical energy densities in practical applications?
Capacitors: 10⁴-10⁶ J/m³, inductors: 10⁵-10⁷ J/m³, supercapacitors: 10⁵-10⁷ J/m³, batteries: 10⁶-10⁸ J/m³. Magnetic energy storage typically has higher energy density than electric field storage.
How do I calculate energy density for non-uniform fields?
For non-uniform fields, energy density varies with position. Total energy is the volume integral: U = ∫ u dV. This calculator assumes uniform fields or provides average energy density for the specified volume.
📊 Energy Density by the Numbers
📚 Official Data Sources
⚠️ Disclaimer: This calculator provides theoretical energy density calculations based on fundamental electromagnetic theory. Actual energy storage systems may have lower energy densities due to material limitations, losses, and practical constraints. Always verify calculations for critical applications and consult engineering references for material-specific properties.
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