Chapter 12.5, Problem 15P

Expand the potential $V=K/d$ in (5.11) in the following way in order to see how the terms depend on the tensors mentioned above. In Figure 5.1 let $R$ have the coordinates X, Y, Z and r have coordinates x, y, z . [Note: The coordinate $x$ here is not the $x$ in ( 5.14).] Then .

ConsiderX, Y, Z as constants and expand $V(x,y,z)$ in a three variable power series about the origin. (See Chapter 4, Section 2, for discussion of two-variable power series and generalize the method.) You should find $V=\frac{K}{R}+\frac{K}{R^2}\left(\frac{X}{R}x+\cdots \right)$ and similar terms in $y,z,y^2,xz,$ and so on. Now letting $r=r_i$ and $K=K^{\prime }{q}_i$ for a charge distribution as in (5.18), and summing (or integrating) over the charge distribution, show that: the first term is just $\left(K^{\prime }/R\right)\cdot$ total charge; the next group of terms (in x, y, z ) involve the three components of the electric dipole moment; the sum of these terms is $\left(K^{\prime }/R^2\right)\cdot$ component of the dipole moment in the $R$ direction; the next group (quadratic terms) involve six quantities of the form ${\displaystyle \iiint x^2}\rho d\tau$ and similar y, z integrals, ${\displaystyle \iiint 2}xy\rho d\tau$ and similar xz, yz integrals.

If we split the 2xy term into xy and yx (and similarly for the 2xz and 2yz terms), we have the nine components of a 2^nd-rank tensor called the quadrupole moment. Use the "direct product" method of Chapter 10, Section 2 to show that it is a 2^nd–rank tensor. (Remember from Chapter 10 that, by definition, x, y, z are the components of a vector, that is, a 1^st-rank tensor.) Just as two charges +q and -q form an electric dipole, so four charges like this form an electric quadrupole and the quadratic terms in the V series give the potential of such a charge configuration. Again using Chapter 10, Section 2, show that the third-order terms in x, y, z form a 3^rd-rank tensor; this is known as the octopole moment. It can be represented physically by two quadrupoles side by side just as the quadrupole above was formed by two dipoles side by side.