Electrostatic Energy

6.1.Potential Energy of A Group of Point Charges

            By the electrostatic energy of a group of m point charges, we means the potential energy of the system relative to the state in which all point charges are infinitely separated from one other. This energy may be obtained rather easily by calculating the work to assemble the charges. The first charge q₁ may be placed in position without any work, W₁ = 0. Placing the second charge q₂ requires

            W₂ =

Where r₂₁ = . For the third charge q₃,

            W₃ = q₃

The work required to bring in the fourth charge, fifth charge, and so on, may be written in a similiar fashion. The total electrostatic energy of the assembled m-charge system is the sum of the W’s, namely,

            U =

Let us abbreviate this result for U as

            U =

           

Thus, the electrostatic energy of the system is

            U =

6.2.Electrostatic Energy of A Charge Distribution

           

But since all charges are at the same fraction, , of their final values, the potential  = (x,y,z), where  is the final value of the potential at(x,y,z). Making this subtitution, we find that the integration over  is readily done, and yields

           

This equation gives the desired result for the energy of charge distribution. It is important to note that the volume of integration V must be large enough to include all of the charge density in the problem, and that the potential  is just that due to the charge density (and ) itself. Normally the charge density vanishes outside some bounded region, in which case V may be taken to include all space. If all space is filled with a single dielectric medium except for certain conductors, the potential is given by

           

For the electrostatic energy of a charge distribution, which includes conductors, becomes

           

Where the last summation is over all conductors, and the surface integral is restricted to nonconducting surfaces.

6.3.ENERGY DENSITY OF AN ELECTROSTATIC FIELD

            We again consider an orbitary distribution of charge characterized by the densities  and . For convenience, it will be assumed that the charge system is bounded-that is, it is possible to construct a closed surface, S’, of finite dimensions that encloses all of the charge. In addition, all surface densities of charge  will be assumed to reside on conductor surfaces. The last statment is really no restriction at all, since a surface charge density on a dielectric-dielectric interface may be spread out slightly and then treated as a volume density . The densities  and  are related to the electric displacement:

           

Throughout the dielectric regions, and

           

On the conductor surfaces.

           

The volume integral here refers to the region where  is different from zero, and this is the region external to the conductors. The surface integral is over the conductors.

           

Of the two volume integrals resulting from this transformation, the first may be converted to a surface integral through the use of the divergence theorem.

           

If the charge distribution bears zero net charge, then the potential at large distance acts like some multipole and falls off more rapidly than r^-1. Again the contribution from S’ may be seen to vanish. Thus, for the electrostatic energy, we have

           

We are led to the concept of energy density in an electrostatic field:

           

           

6.4.Energy of A System of Charged conductors: Coefisients of Potential

            The problem is to find the potential of an uncharged spherical conductor in the presence of a point charge q at distance r from the center of the sphere, where r > R, and R is the radius of the spherical conductor.

Solution: the point charge and sphere are taken to be a system of two conductors, and use is made of the equality p₁₂ = p₂₁. If the sphere is charged(Q) and the “point” un charged, then the potential of the “point” is Q/4 . Thus,

            P₁₂ = p₂₁ =  

Evidently, when the “point” has charge q and the sphere is uncharged, the potential of the latter is q/4

6.5.Coefficients of Capacitance and Induction

            Q_i =

Where c_ij is called a coefficient of capacitance and c_ij ( ) is coefficient of induction. Properties of the c’s follow from those of the p’s, which we have already discussed. Thus: (1) c_ji, (2) c_jj > 0, (3) the coefficients of induction are negative of zero.

           

6.6.Capacitors

            Two conductors that can store equal and opposite charges , with a potential difference between them that is independent of whether other conductors in the system are charged, from what is called a capacitors.

           

Where C = (p₁₁ + p₂₂ + p₁₂)^-1 is called the capacitance of the capacitor

           

Where A is the area of one plate. The potential difference . Therefore,

           

Is capacitance of this capacitors.

           

If two uncharged capacitors are connected in series and subsequently charged, conservation of charge requires that each capacitor acquire the same charge. Thus, the equivalent capacitance C of the combination is related to C₁ and C₂ by the expression

           

6.7.Force and Torques

            Let us suppose we are dealing with an isolated system composed of a number of parts (conductors, point charges, dielectrics), and we allow one of these parts to make a small displacement dr under the influence of the electrical forces F acting upon it. The work performed by the electrical force on the system in these circumstances is

            dW = F.dr = F_x dx + F_y dy + F_z dz

because the system is isolated, this work is at the expense of the electrostatic energy U.

            dW = -dU

            -dU = F_x dx + F_y dy + F_z dz

And

            F_x =

With similiar experience for F_x and F_z. That is, in this case F is a concervative force, and F = - U.

Posted in: Foundations of Electromagnetic Theory