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 differenceis built up between the plates by transferring a chargefrom 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 energyis needed to transfer a further charge

(1)

If a dielectric is placed between the plates thenwill be changed since the dielectric changes the value ofbut onlythe 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 areaand plate separationthe energy stored when it is filled with dielectric isThe volume of the capacitor isand therefore the energy density is

We can regard the energy densityas 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 tois done in assembling the charge distribution whereis the potential at the position ofIf the free charges are distributed with surface charge densityon a number of conducting surfaces, and with volume charge densityin the regionbounded by the conductors, the sum is replaced by an integral and(2)

On the conducting surfaces the outward normal tois the inward normal to the conductor soand together withwe 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

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