Additional Problems

We will collect here suggestions for additional problems, keyed to the Chapter and Section to which they pertain.

Chap. 2 (Sections 2.1 and 2.6). Investigate the motion of a charged particle in the field of an electric, or magnetic, dipole: see McGuire, Am.J.Phys. 71, 809 (2003) and Alonzo, Am.J.Phys. 72, 10 (2004).

Chap. 2 (Section 2.4). Consider a localized charge distribution that is placed in an externally applied electric field described by the potential function Phi(r). The potential function can be expanded as a (3-dimensional) Taylor series about an origin in the vicinity of the charges. The charge distribution can be described by a multipole expansion about the same origin. Investigate the potential energy of the system to show that it can be expressed as a series involving the multipole moments of the distribution (the first two terms are the familiar energies of a monopole and dipole). Work out the form of the third term involving the quadrupole moment of the distribution (show that it depends upon the product of elements of the quadrupole tensor and a kind of gradient of the field). [Cribbed from Jackson Sec.4.2 and Eq.4.24.]

Chap. 3 (Introduction). The method of images is not discussed in this text. Nontrivial applications of the image concept are explored in Engel and Friedrichs, Am.J.Phys. 70, 428 (2002).

Chap. 3 (Section 3.3 or 3.6?). Investigate the distribution of surface charge on a long thin wire. See Jackson, Am.J.Phys. 70, 409 (2002) and references therein; Keller, Am.J.Phys. 71, 282 (2003).

Chap. 4 (Section 4.3). A small bar magnet is dropped through a compact circular coil (the axes of magnet and coil are aligned with the vertical). What is the shape of the EMF pulse induced in the coil? See Singh et al., Am.J.Phys. 70, 424 (2002); Kingman et al., Am.J.Phys. 70, 595 (2002); Wood, Rottmann, and Barrera, Am.J.Phys. 72, 376 (2004).

Chap. 4 (Section 4.7). A well-known elementary problem is to consider two capacitors, one initially charged (hence storing electrostatic energy) and the other uncharged. When they are connected together and allowed to equilibrate, some energy appears to be lost. Where has it gone? See Boykin et al., Am.J.Phys. 70, 415 (2002); Choy, Am.J.Phys. 72, 662 (2004).

Chap. 6 (Section 6.3). Under conditions of total reflection, the phase of the reflected wave is different for the two polarizations, parallel and perpendicular to the plane of incidence. Show that the maximum difference of phase occurs at the incident angle A given by sin^2(A) = (2n^2)/(1+n^2), where n is the ratio of the indexes of the two media (n<1). Show that this maximum phase difference D is given by tan(D/2) = (1-n^2)/(2n). For a glass-air interface with n = 1/1.5, D is very nearly 45deg. How might you use this effect to convert linear to circular polarization? (Hint: use two reflections.) [Cribbed from Born and Wolf Sec.1.5.]

Chap. 10 (Section 10.3). The polarizability of the H2O molecule is 4.07x10^-26 cm^3, and the molecule has an intrinsic moment of 1.85x10^-18 esu-cm. The measured dielectric constant at 293K is 80.4. Are these data consistent with the formula of Prob. 10-19(c)?

Chap. 10 (Section 10.3). Consider moderately dense matter consisting of molecules for which only one resonance dominates the sum in Eqs. (10.24) and (10.53). Take the molecular field as given in terms of the coefficient eta of Eq.(10.47), without restricting it to the Lorentz value of Eq.(10.48). Show that the dielectric constant can now be written in the 'Sellmeier' form of Eqs. (10.25) and (10.50) (i.e., as if there were no local-field correction), with the resonance frequency shifted by a correction dependent on eta. [Cribbed from Reitz-Milford-Christy, 4th ed., pp.497-8.]

Chap. 13 (Section 13.1). A homework problem, or a mini research project, can be based on the computational method of Dauger, Computers in Physics 10, 591-604 (1996).

The author will be grateful for further suggestions: mheald@frontiernet.net

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