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

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