Electric Potential Energy - Energy Stored in An Electrostatic Field Distribution

Energy Stored in An Electrostatic Field Distribution

The energy density, or energy per unit volume, of the electrostatic field of a continuous charge distribution is:

Outline of proof

One may take the equation for the electrostatic potential energy of a continuous charge distribution and put it in terms of the electrostatic field.

Since Gauss' law for electrostatic field in differential form states

where

  • is the electric field vector
  • is the total charge density including dipole charges bound in a material
  • is the permittivity of free space,

then,

\begin{align} U & = \frac{1}{2}\int \limits_{\text{all space}} \rho(r) \Phi(r) \, dV \\ & = \frac{1}{2}\int \limits_{\text{all space}} \varepsilon_0(\mathbf{\nabla}\cdot{\mathbf{E}})\Phi \, dV \end{align}

so, now using the following divergence vector identity

we have

using the divergence theorem and taking the area to be at infinity where

\begin{align} U & = \overbrace{\frac{\varepsilon_0}{2}\int\limits_{{}^\text{boundary}_\text{ of space}} \Phi\mathbf{E}\cdot d\mathbf A}^{0} - \frac{\varepsilon_0}{2}\int \limits_{\text{all space}} (-\mathbf{E})\cdot\mathbf{E} \, dV \\ & = \int \limits_{\text{all space}} \frac{1}{2}\varepsilon_0\left|{\mathbf{E}}\right|^2 \, dV. \end{align}

So, the energy density, or energy per unit volume of the electrostatic field is:

Read more about this topic:  Electric Potential Energy

Famous quotes containing the words energy, stored, field and/or distribution:

    “The chief function of the city is to convert power into form, energy into culture, dead matter into the living symbols of art, biological reproduction into social creativity.”
    —Lewis Mumford (1895–1990)

    “The idea that information can be stored in a changing world without an overwhelming depreciation of its value is false. It is scarcely less false than the more plausible claim that after a war we may take our existing weapons, fill their barrels with cylinder oil, and coat their outsides with sprayed rubber film, and let them statically await the next emergency.”
    —Norbert Wiener (1894–1964)

    “And through the field the road runs by
    To many-towered Camelot;”
    —Alfred Tennyson (1809–1892)

    “There is the illusion of time, which is very deep; who has disposed of it? Mor come to the conviction that what seems the succession of thought is only the distribution of wholes into causal series.”
    —Ralph Waldo Emerson (1803–1882)