Chapter Questions
Show that, if the net charge $Q$ is zero, then the dipole moment of a charge distribution is independent of the choice of origin.
Two line charges $+Q$ and $-Q$ extend, respectively, from $(-a, 0, c)$ to $(a, 0, c)$ and from $(-a, 0,-c)$ to $(a, 0,-c)$. Calculate their dipole moment.
Calculate the dipole moment of a spherical shell of radius $R$ bearing a surface charge density $\sigma=\sigma_{0} \cos \theta$.
(a) Calculate the dipole moment of a spherical shell of radius $R$ whose surface charge density is $\sigma_{0}(1+\cos \theta)$.(b) What is the dipole moment if the center of the sphere is at $Z \hat{z} ?$(c) What is the dipole moment if the center of the sphere is at $X \hat{x}+Y \hat{y}+Z \hat{z} ?$
An alternate expression for the potential in the field of an electric dipole We found that, in the field of an electric dipole,$$V=\frac{Q}{4 \pi \epsilon_{0}}\left(\frac{1}{r_{b}}-\frac{1}{r_{a}}\right)$$Refer to Fig. 5-7. Show that, if the length of the dipole is small, then$$V=\frac{Q s}{4 \pi \epsilon_{0}}\left[\frac{d}{d z^{\prime}}\left(\frac{1}{r^{\prime}}\right)\right]_{z^{\prime}-0}$$where $z^{\prime}$ is the position of a point on the $z$-axis and $r^{\prime}=x \hat{x}+y \hat{y}+(z-$ $\left.z^{\prime}\right) \hat{z}$.
An alternate expression for the potential in the field of an electric quadrupoleSee Prob. 5-5 and refer to Fig. 5-8. Show that the potential in the field of a linear electric quadrupole is$$V=\frac{p s}{4 \pi \epsilon_{0}}\left[\frac{d}{d z^{\prime}}\left(\frac{\cos \theta}{r^{\prime 2}}\right)\right]_{z^{\prime}-0}$$
Multipolar expansion of the field of a single point charge A single point charge $Q$ is situated at $P^{\prime}(0,0, s)$ as in Fig. 5-7.First expand its potential at point $P$ in terms of multipoles. The vector $r$ that defines the position of $P$ forms an angle $\theta$ with the $z$-axis, and $r \gg s$. The distance from $Q$ to $P$ is $r^{\prime}$. Disregard terms of the order of $(s / r)^{4}$ and higher. Then write out the values of $V_{1}, V_{2}, V_{3}$.
Calculate $V$ for a dipole exactly, and identify the quadrupole and octupole terms. The octupole term varies as $(s / r)^{4}$. You can therefore disregard terms in $(s / r)^{5},(s / r)^{6}$, etc.
A cube of side $2 a$ carries a uniform volume charge density $\rho$. The origin of coordinates is at the center. Calculate $V_{1}, V_{2}, V_{3}$.
A line charge $Q$ extends from $z=-a / 2$ to $z=a / 2$.(a) Calculate the monopole, dipole, and quadrupole terms in the expansion for $V$.(b) For what value of the distance $r$ to the center of the charge is the quadrupole term less than $1 \%$ of the monopole term, if $3 n^{2}-1$ is of the order of unity?
In Fig. 5-6, let all the charges be $Q$. Calculate $V_{4}$ and $V_{5}$.