## Energy in Electric Fields in the Presence of Dielectrics

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 volume is 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 region bounded 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  