We can illustrate how energy is stored in an electric field using the example of a parallel plate capacitor. The capacitor is initially uncharged, and a potential difference
is built up between the plates by transferring a charge
from one plate to the other. Energy – supplied for example by a battery – is needed to move the charge, and at the moment when the potential difference is
an additional amount of energy
is needed to transfer a further charge
(1)
If a dielectric is placed between the plates then
will be changed since the dielectric changes the value of
but only
the free charge on the capacitor plate, appears in (1) which contains no reference to polarization charges. This is because it is only the free charge which is moved across the potential difference by external forces.
In a parallel plate capacitor of area
and plate separation
the energy stored when it is filled with dielectric is
The volume of the capacitor is
and therefore the energy density is
We can regard the energy density
as residing in the field. We can imagine field is generated by a large number of parallel plate capacitors by thin conductors placed along closely placed equipotentials. The total electrostatic energy stored in a volumeis then
The same equation applies to the potential energy stored by an arbitrary distribution of charges. We start with an unpolarised dielectric with no free charges and assemble the charge distribution by bringing the charges from infinity. Work equal to
is done in assembling the charge distribution where
is the potential at the position of
If the free charges are distributed with surface charge density
on a number of conducting surfaces, and with volume charge density
in the regionbounded by the conductors, the sum is replaced by an integral and
(2)
On the conducting surfaces the outward normal to
is the inward normal to the conductor so
and together with
we have from (2)
We can use Gauss's Divergence Theorem on the second term to give
Now we can use the vector identity
and put
to obtain