*Notes to a video lecture on http://www.unizor.com*

__Electric Field Energy__

Let's calculate the

**potential energy density**

*of an electric field inside a capacitor as a function of the field's intensity*

**P**_{E}*, which we assume to be uniform between the plates of a capacitor.*

**E**While the methodology will depend on the fact that this electric field is between the plates of a capacitor, the final formula will depend only on the field's intensity and, as a field's local characteristic, will be the same, no matter what is the outside source of an electric field, whether it's plates of a capacitor or a few point charges, or the result of electromagnetic oscillations far from the source of these oscillations.

Consider a battery used to charge a capacitor from initial no electric charge between plates with the difference of potential between the plates being equal to zero to some charge

*with corresponding difference of potential between the plates (voltage) being equal*

**Q**_{max}*. This process of charging requires some work performed by the battery.*

**V**_{max}The total energy accumulated inside a capacitor as a result of charging a capacitor should be equal to this amount of work.

This work performed by a battery to charge a capacitor from

*voltage to*

**0***can be considered a function of accumulated charge*

**V**_{max}*and it grows from zero to some maximum*

**W=W(Q)***- the energy accumulated by a capacitor as it accumulated a charge of*

**W**_{max}*.*

**Q**_{max}During the charging process the voltage between the plates of a capacitor can also be considered as a function of accumulated charge

*, that is*

**Q***, and it grows from zero to*

**V=V(Q)***.*

**V**_{max}Recall that the difference in electric potential between two points of an electric field (voltage) is the amount of work needed to transfer a unit of electric charge (1 coulomb) from one point to another.

Therefore, to transfer an additional infinitesimal amount of electric charge

*d*from one plate of a capacitor to another, when there is already transferred amount of electric charge

**Q***that creates a voltage*

**Q***between the plates, the battery has to spend an additional amount of work*

**V(Q)***d*

**W(Q) = V(Q)·**d**Q**Further recall that the electric charge on each plate of a capacitor (positive on one plate and negative on another)

*and the voltage between the plates*

**Q***are proportional to each other with a capacitor's*

**V***capacitance*

*being a factor:*

**C***or*

**Q = V·C**

**V = Q/C**Therefore, the above expression for an additional amount of work can be written as

*d*[

**W(Q) =***]*

**Q/C**

**·**d**Q**To calculate a total amount of energy

*needed to charge a capacitor from zero to*

**W**_{max}*, we have to integrate*

**Q**_{max}*d*from

**W***to*

**0***:*

**Q**_{max}

**W**_{max}= ∫_{[0,Qmax]}[

*]*

**Q/C***=*

**·**d**Q**

**= ½·Q²**_{max}/COf course, the amount of charge

*can be any from zero to some practical maximum, so we can drop an index*

**Q**_{max}*max*, and the work spent by a battery will be

**W = ½·Q²/C = ½V²·C**Let's approach the same problem from a different viewpoint that involves the intensity

*of an electric field between the plates of a capacitor charged with*

**E***amount of electricity to a voltage*

**Q***between its plates.*

**V**The capacitance

*of a capacitor was discussed in a lecture "Electromagnetism" - "Electric Field" - "Capacitors" of this course and, as was shown there, depends on the area of each plate*

**C***, distance between plates*

**A***and electric*

**d***permittivity*of a medium between the plates

*:*

**ε**

**C = ε·A/d**with the vacuum having the permittivity

*and any other medium having it as*

**ε**_{0}

**ε = ε**_{r}·ε_{0}where

*is*

**ε**_{r}*relative permittivity*of a medium.

Using this expression for a capacitance

*, the total work to charge a capacitor with*

**C***amount of electricity to a voltage level*

**Q***can be written as*

**V**

**W = ½·Q²·d/(ε·A) = ½V²·ε·A/d**The definition of the electric field's intensity

*is the force with which a field acts on a unit charge, while voltage*

**E***between plates is an amount of work needed to move a unit of charge from one plate to another.*

**V**Therefore, using the principal

Work = Force ⨯ Distance,

we can write

**V = E·d**Substituting this into a formula for the total work yields

**W = ½E²·d²·ε·A/d = ½ε·E²·(A·d)**Notice that

*is a volume of the space between the plates of a capacitor, occupied by an electric field.*

**A·d**Therefore, the expression

*characterizes the*

**W/(A·d)***density*

*of a potential energy of an electric field between the plates of a capacitor.*

**P**_{E}Hence, the

**potential energy density of an electric field**is a local characteristic that depends on the field's intensity and the permittivity of a medium where the field propagates

**P**_{E}= ½ε·E²Final comment is related to the fact that we used a simple kind of an electric field to derive the formula above - the static uniform finite field between two plates of a capacitor. The formula contains only the local property of the field - its intensity at any point

*.*

**E**Even if the field is of a more complex kind (variable, non-uniform, infinite etc.), since only its intensity participates in the above expression for a potential energy density, the formula should be valid.

In particular, it's valid for an electrical component of an oscillating electromagnetic field.