- #1

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I take as granted that electric energy within a volume [itex]\Omega[/itex] is defined by:

[tex]W = \int_\Omega \phi \cdot \rho \cdot d^3r[/tex]

where [itex]\phi = \phi(\textbf{r})[/itex] is the eletric potential, [itex]\rho = \rho(\textbf{r})[/itex] is the charge density and [itex]d^3r \stackrel{\Delta}{=} [/itex] volume element. Now that the energy density is defined by

[tex]U = \phi \cdot \rho[/tex]

To my understanding, the tutorial is trying to show that

[tex] U = \frac{1}{2} \textbf{E}\cdot \textbf{D} [/tex]

where [itex]\textbf{E} = \textbf{E}(\textbf{r})[/itex] is eletric field strength and [itex]\textbf{D} = \epsilon_0 \textbf{E} + \textbf{P}[/itex] is the electric displacement (FYI: definition of electric displacement if needed).

Now that the tutorial begines with introducing a change of free charge density [itex](\delta\rho_f)[/itex] and yielding a change of total energy (within the volume I SUPPOSE):

[itex]\delta W = \int_\Omega \phi \cdot (\delta\rho_f) \cdot d^3r \; -- \; (1)[/itex]

then by [itex]\nabla \textbf{D} = \rho_f[/itex] equation [itex](1)[/itex] reduces to

[itex]\int_\Omega \phi \cdot \nabla (\delta \textbf{D}) \cdot d^3r [/itex]

[itex]= \int_\Omega \nabla (\phi \cdot (\delta \textbf{D})) \cdot d^3r - \int_\Omega \nabla \phi \cdot (\delta \textbf{D}) \cdot d^3r [/itex]

[itex]= \int_{\partial\Omega} \phi \cdot (\delta \textbf{D}) \cdot d\textbf{S} - \int_\Omega \nabla \phi \cdot (\delta \textbf{D}) \cdot d^3r \; -- \; (2)[/itex]

where use has been made of Integration by Parts and Divergence Theorem. I'm fine with the derivation by far.

Here comes the part that I don't understand. The tutorial says "If the dielectric medium is of finite spatial extent then we can neglect the surface term to give [itex]\delta W = - \int_\Omega \nabla \phi \cdot (\delta \textbf{D}) \cdot d^3r[/itex]" which implies that [itex] \int_{\partial\Omega} \phi \cdot (\delta \textbf{D}) \cdot d\textbf{S} = 0[/itex].

This doesn't seem trivial to me. I consulted some of my friends majored in Physics but most of them just took [itex] U = \frac{1}{2} \textbf{E}\cdot \textbf{D} [/itex] as granted when using it and some are still trying to help.

Hope I can get luck in this forum, any help will be appreciated :)