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Index/Electromagnetism/EMG-11

EMG-11

Electric energy density

u = ½ ε0 E² in vacuum.

Reading speed
EnergyMaxwell

Governing equation

u=12ε0E2u=\tfrac12\varepsilon_0 E^2

E∈[1,1×107]E \in \left[1, 1\times 10^{7}\right]

where

E
Electric field (V/m)
u
Energy density (J/m³)

The figure, part by part

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EMG-11 · gauge
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①

Governing law — red

u=12ε0E2u=\tfrac12\varepsilon_0 E^2

E∈[1,1×107]E \in \left[1, 1\times 10^{7}\right]

E=1E = 1u=4.43×10−12u = 4.43\times 10^{-12}

②

Domain of the variable — blue

Emin⁡=1,Emax⁡=1×107E_{\min} = 1,\quad E_{\max} = 1\times 10^{7}

E∈[1,1×107]E \in \left[1, 1\times 10^{7}\right]

E=1E = 1E=1E = 1

③

Response curve — green

u=12ε0E2u=\tfrac12\varepsilon_0 E^2

E∈[1,1×107]E \in \left[1, 1\times 10^{7}\right]

E=1E = 1u=4.43×10−12u = 4.43\times 10^{-12}

④

Rate of change — amber

dudE\frac{\mathrm{d}u}{\mathrm{d}E}

E∈[1,1×107]E \in \left[1, 1\times 10^{7}\right]

E=62894.1E = 62894.1u′=5.57×10−7u^{\prime} = 5.57\times 10^{-7}

Lecture brief

Historical brief

Coulomb, Gauss, Ampère, Faraday and Maxwell (1861–65) unified charge, current and light. The lab computes fields, induction, Poynting flux and the electromagnetic wave in SI. This sheet (EMG-11 — Electric energy density) is the form associated with Maxwell. Working symbols: EE →\rightarrow uu. The work to assemble a field configuration is stored locally as ½ E·D, reducing to ½ ε0 E² in vacuum.

Purpose

Purpose: compute uu from EE in Electromagnetism via u=12ε0E2u=\tfrac12\varepsilon_0 E^2 u = ½ ε0 E² in vacuum. Use it when a real electromagnetism question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given E=10000.000 V/mE = 10000.000\,\mathrm{V/m}, the governing relation u=12ε0E2u=\tfrac12\varepsilon_0 E^2 yields u=4.427e−4 J/m3u = 4.427e-4\,\mathrm{J/m^{3}}. Linear vacuum, uniform E. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Energy density u0.0004 J/m³
Reading speed

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EMG-11 · gauge
00:0 / 00:08

Narration of this film

Linear vacuum, uniform E.

The work to assemble a field configuration is stored locally as ½ E·D, reducing to ½ ε0 E² in vacuum.

Reading speed

Watch on YouTube