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THE INFINITESIMAL ELECTRIC DIPOLE
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Figure 5.21 The Relation Between dx and ds Along the Parabola
Figure 5.22 The Relation Between dy and ds Along the Parabola
Looking at Figure 5.22 shows what is going on here. There is not a one-to-one relationship between dy and ds. Of course, evaluating the integral in this way produces exactly the same result for p,(Q, as you will prove in one of the exercises of this chapter.
5.6 THE INFINITESIMAL ELECTRIC DIPOLE
The example of the infinitesimal electric dipole is one of the more interesting appli- cations of the Dirac S-function and makes use of many of its properties.
5.6.1 Moments of a Charge Distribution In electromagnetism, the distribution of charge density in space p,(F) can be ex- panded, in what is generally called a multipole expansion, into a sum of its moments. These moments are useful for approximating the potential fields associated with com- plicated charge distributions in the far field limit (that is, far away from the charges). Each moment is generated by calculating a different volume integral of the charge distribution over all space. Because our goal is not to derive the mathematics of multipole expansions, but rather to demonstrate the use of the Dirac 6-functions, the multipole expansion results are stated here without proof. Derivations can be found
122 THE DIRAC &FUNCTION
in most any intermediate or advanced book on electromagnetism, such as Jackson’s Classical Electrodynamics. The lowest term in the expansion is a scalar called the monopole moment. It is just the total charge of the distribution and is determined by calculating the volume integral of p c :
The next highest moment is a vector quantity called the dipole moment, which is generated from the volume integral of the charge density times the position vector:
The next moment, referred to as the quadrapole moment, is a tensor quantity generated by the integral
Q = J’ dT (3TF - lT12T)pc(T). (5.82) All space
In this equation, the quantity T T is a dyad, and T is the identity tensor. There are an infinite number of higher-order moments beyond these three, but they are used less frequently, usually only in cases where the first three moments are zero. Far away from the charges, the electric potential can be approximatedby summing the contributionsfrom each of the moments. The potential field Q, due to the first few moments is
It is quite useful to know what charge distributions generate just a single term in this expansion, and what potentials and electric fields are associated with them. For example, what charge distribgtion has just a dipole term (that is, #^ 0)^ while all other terms are zero (Q = 0, Q = 0, etc.). The Dirac &function turns out to be quite useful in describing these particular distributions.
5.6.2 The Electric Monopole
The distribution that generatesjust the Q / r term in Equation 5.83 is called the electric monopole. As you may have suspected, it is simply the distribution of a point charge at the origin:
