Energy density within a dielectric medium

Consider a system of free charges embedded in a dielectric medium. The increase in the total energy when a small amount of free charge $\delta\rho_f$ is added to the system is given by

$$\delta W = \int \phi \delta\rho_f d^3{\bf r},$$ (847)

where the integral is taken over all space, and $\phi({\bf r})$ is the electrostatic potential. Here, it is assumed that the original charges and the dielectric are held fixed, so that no mechanical work is performed. It follows from Eq. (807) that

$$\delta W = \int \phi \nabla\!\cdot\!\delta {\bf D} d^3{\bf r},$$ (848)

where $\delta{\bf D}$ is the change in the electric displacement associated with the charge increment. Now the above equation can also be written

$$\delta W = \int \nabla\!\cdot\!(\phi \delta{\bf D}) d^3{\bf r} - \int \nabla\phi\!\cdot\!\delta{\bf D} d^3{\bf r},$$ (849)

giving

$$\delta W = \int \phi \delta{\bf D}\!\cdot\!d{\bf S} - \int \nabla\phi\!\cdot\!\delta{\bf D} d^3{\bf r},$$ (850)

where use has been made of Gauss' theorem. If the dielectric medium is of finite spatial extent then we can neglect the surface term to give

$$\delta W = -\int\nabla\phi \!\cdot\!\delta{\bf D} d^3{\bf r} = \int {\bf E}\!\cdot\! \delta {\bf D} d^3{\bf r}.$$ (851)

This energy increment cannot be integrated unless ${\bf E}$ is a known function of ${\bf D}$. Let us adopt the conventional approach, and assume that ${\bf D} =\epsilon_0 \epsilon {\bf E}$, where the dielectric constant $\epsilon$ is independent of the electric field. The change in energy associated with taking the displacement field from zero to ${\bf D}({\bf r})$ at all points in space is given by

$$W = \int_{\bf0}^{\bf D} \delta W = \int_{\bf0}^{\bf D }\int {\bf E}\!\cdot\! \delta{\bf D} d^3{\bf r},$$ (852)

or

$$W = \int \int_0^E \frac{\epsilon_0 \epsilon \delta (E^2)}... ... = \frac{1}{2} \int \epsilon_0 \epsilon E^2 d^{3} {\bf r},$$ (853)

which reduces to

$$W = \frac{1}{2} \int {\bf E}\!\cdot\!{\bf D} d^3{\bf r}.$$ (854)

Thus, the electrostatic energy density inside a dielectric medium is given by

$$U = \frac{1}{2} {\bf E}\!\cdot\!{\bf D}.$$ (855)

This is a standard result, and is often quoted in textbooks. Nevertheless, it is important to realize that the above formula is only valid in dielectric media in which the electric displacement ${\bf D}$ varies linearly with the electric field ${\bf E}$. Note that Eq. (855) is consistent with the expression (595) which we obtained earlier.