Home Back

Energy Density Formula Calculator

\[ u = \frac{1}{2} \varepsilon_0 E^2 + \frac{1}{2} \frac{B^2}{\mu_0} \]

F/m
H/m

1. What is the Energy Density Formula Calculator?

Definition: This calculator computes the energy density (\(u\)) of an electromagnetic field in a vacuum using the formula \( u = \frac{1}{2} \varepsilon_0 E^2 + \frac{1}{2} \frac{B^2}{\mu_0} \), where \(\varepsilon_0\) is the permittivity of free space, \(E\) is the electric field strength, \(\mu_0\) is the permeability of free space, and \(B\) is the magnetic field strength.

Purpose: It is used in electromagnetism to quantify the energy stored per unit volume in an electromagnetic field, applicable in physics, optics, and electromagnetic wave studies.

2. How Does the Calculator Work?

The calculator uses the energy density formula for an electromagnetic field:

Formula: \[ u = \frac{1}{2} \varepsilon_0 E^2 + \frac{1}{2} \frac{B^2}{\mu_0} \] where:

  • \(u\): Energy density (J/m³, erg/cm³)
  • \(\varepsilon_0\): Permittivity of free space (F/m)
  • \(E\): Electric field (V/m, kV/m)
  • \(\mu_0\): Permeability of free space (H/m)
  • \(B\): Magnetic field (T, mT, G)

Unit Conversions:

  • Electric Field:
    • 1 V/m = 1 V/m
    • 1 kV/m = 1000 V/m
  • Magnetic Field:
    • 1 T = 1 T
    • 1 mT = 0.001 T
    • 1 G = \( 10^{-4} \) T
  • Energy Density (Output):
    • 1 J/m³ = 1 J/m³
    • 1 erg/cm³ = 0.1 J/m³
The energy density is calculated in J/m³ and can be converted to the selected output unit (J/m³, erg/cm³). Results greater than 10,000 or less than 0.001 are displayed in scientific notation; otherwise, they are shown with 4 decimal places.

Steps:

  • Enter the permittivity (\(\varepsilon_0\)), electric field (\(E\)), permeability (\(\mu_0\)), and magnetic field (\(B\)) with their units (default: \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(E = 1000 \, \text{V/m}\), \(\mu_0 = 1.25663706212 \times 10^{-6} \, \text{H/m}\), \(B = 0.001 \, \text{T}\)).
  • Convert electric and magnetic fields to SI units (V/m, T).
  • Validate that permittivity and permeability are greater than 0, and electric and magnetic fields are non-negative.
  • Calculate the energy density in J/m³ using the formula.
  • Convert the energy density to the selected output unit.
  • Display the result, using scientific notation if the value is greater than 10,000 or less than 0.001, otherwise rounded to 4 decimal places.

3. Importance of Energy Density Calculation

Calculating energy density is crucial for:

  • Electromagnetism: Quantifying the energy stored in electromagnetic fields, such as in capacitors, inductors, and electromagnetic waves (e.g., light).
  • Physics: Understanding the propagation of electromagnetic waves, where energy density relates to the intensity of the wave (via the Poynting vector).
  • Engineering: Designing electromagnetic devices like antennas, lasers, and microwave systems, where energy density influences performance and efficiency.

4. Using the Calculator

Examples:

  • Example 1: Calculate the energy density for \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(E = 1000 \, \text{V/m}\), \(\mu_0 = 1.25663706212 \times 10^{-6} \, \text{H/m}\), \(B = 0.001 \, \text{T}\), output in J/m³:
    • Enter \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(E = 1000 \, \text{V/m}\), \(\mu_0 = 1.25663706212 \times 10^{-6} \, \text{H/m}\), \(B = 0.001 \, \text{T}\).
    • Electric field term: \(\frac{1}{2} \varepsilon_0 E^2 = 0.5 \times 8.8541878128 \times 10^{-12} \times (1000)^2 \approx 4.427 \times 10^{-6} \, \text{J/m}^3\).
    • Magnetic field term: \(\frac{1}{2} \frac{B^2}{\mu_0} = 0.5 \times \frac{(0.001)^2}{1.25663706212 \times 10^{-6}} \approx 397.887 \, \text{J/m}^3\).
    • Total energy density: \(u = 4.427 \times 10^{-6} + 397.887 \approx 397.887 \, \text{J/m}^3\).
    • Output unit: J/m³ (no conversion needed).
    • Result: \( \text{Energy Density} = 397.8916 \, \text{J/m}^3 \).
  • Example 2: Calculate the energy density for \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(E = 1 \, \text{kV/m}\), \(\mu_0 = 1.25663706212 \times 10^{-6} \, \text{H/m}\), \(B = 10000 \, \text{G}\), output in erg/cm³:
    • Enter \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(E = 1 \, \text{kV/m}\), \(\mu_0 = 1.25663706212 \times 10^{-6} \, \text{H/m}\), \(B = 10000 \, \text{G}\).
    • Convert: \(E = 1 \times 1000 = 1000 \, \text{V/m}\), \(B = 10000 \times 0.0001 = 1 \, \text{T}\).
    • Electric field term: \(\frac{1}{2} \varepsilon_0 E^2 = 0.5 \times 8.8541878128 \times 10^{-12} \times (1000)^2 \approx 4.427 \times 10^{-6} \, \text{J/m}^3\).
    • Magnetic field term: \(\frac{1}{2} \frac{B^2}{\mu_0} = 0.5 \times \frac{(1)^2}{1.25663706212 \times 10^{-6}} \approx 397887 \, \text{J/m}^3\).
    • Total energy density in J/m³: \(u = 4.427 \times 10^{-6} + 397887 \approx 397887 \, \text{J/m}^3\).
    • Convert to output unit (erg/cm³): \(397887 \times 0.1 = 39788.7 \, \text{erg/cm}^3\).
    • Result: \( \text{Energy Density} = 3.9789 \times 10^4 \, \text{erg/cm}^3 \).

5. Frequently Asked Questions (FAQ)

Q: What is energy density in an electromagnetic field?
A: Energy density (\(u\)) is the energy stored per unit volume in an electromagnetic field, consisting of contributions from both the electric field (\(\frac{1}{2} \varepsilon_0 E^2\)) and the magnetic field (\(\frac{1}{2} \frac{B^2}{\mu_0}\)). It is typically measured in J/m³.

Q: Why must permittivity and permeability be greater than zero?
A: Permittivity (\(\varepsilon_0\)) and permeability (\(\mu_0\)) are physical constants that define the properties of the medium (in this case, free space). They must be greater than zero to represent physical properties and avoid division by zero in the formula.

Q: Does this formula apply in media other than a vacuum?
A: The formula \( u = \frac{1}{2} \varepsilon_0 E^2 + \frac{1}{2} \frac{B^2}{\mu_0} \) is specific to a vacuum, where \(\varepsilon_0\) and \(\mu_0\) are the permittivity and permeability of free space. In other media, you would use the permittivity (\(\varepsilon\)) and permeability (\(\mu\)) of the medium: \( u = \frac{1}{2} \varepsilon E^2 + \frac{1}{2} \frac{B^2}{\mu} \). This calculator assumes a vacuum.

Energy Density Formula Calculator© - All Rights Reserved 2025