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## 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 is added to the system is given by (847)

where the integral is taken over all space, and 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 (848)

where is the change in the electric displacement associated with the charge increment. Now the above equation can also be written (849)

giving (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 (851)

This energy increment cannot be integrated unless is a known function of . Let us adopt the conventional approach, and assume that , where the dielectric constant is independent of the electric field. The change in energy associated with taking the displacement field from zero to at all points in space is given by (852)

or (853)

which reduces to (854)

Thus, the electrostatic energy density inside a dielectric medium is given by (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 varies linearly with the electric field . Note that Eq. (855) is consistent with the expression (595) which we obtained earlier.   Next: Magnetization Up: Dielectric and magnetic media Previous: Boundary value problems with
Richard Fitzpatrick 2006-02-02