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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
20060202