publication . Preprint . Article . 2012

Magnetic dipole moment of a moving electric dipole

V. Hnizdo;
Open Access English
  • Published: 04 Jan 2012
The current density of a moving electric dipole is expressed as the sum of polarization and magnetization currents. The magnetic field due to the latter current is that of a magnetic dipole moment that is consistent with the relativistic transformations of the polarization and magnetization of macroscopic electrodynamics.
Persistent Identifiers
free text keywords: Physics - Classical Physics, General Physics and Astronomy, Condensed matter physics, Electron magnetic dipole moment, Physics, Magnetic dipole, Transition dipole moment, Electric dipole transition, Magnetization, Dipole, Electric dipole moment, Quantum electrodynamics, Polarization density
16 references, page 1 of 2

1 W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Dover, New York, 2005), Section 18-6.

2 We use Gaussian units since the relativistic aspects of relations among the dipole moments are displayed most clearly in these units.

3 Reference 1, Section 18-4.

4 D. Bedford and P. Krumm, “On the origin of magnetic dynamics,” Am. J. Phys. 54, 1036-1039 (1986).

5 D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, Upper Saddle River, 1999), Problem 12.62.

6 This situation is somewhat paradoxical since we shall see that, unlike the acquired electric dipole moment, the acquired magnetic dipole moment is not a relativistic effect.

7 A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980), Chapter II, Section 4.

8 G. E. Vekstein, “On the electromagnetic force on a moving dipole,” Eur. J. Phys. 18, 113-117 (1997). [OpenAIRE]

9 G. P. Fisher, “The electric dipole moment of a moving magnetic dipole,” Am. J. Phys. 39, 1528-1533 (1971).

10 J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), Problems 6.21, 6.22 and 11.27 (a).

11 When a time dependence of the electric dipole itself is allowed, the polarization current density (7) is supplemented by a term p˙ 0δ(r − r0(t)), where the dot denotes the time derivative. It can be shown easily that this term generates already in the dipole's rest frame a magnetic field (p˙ 0 × n)/c|r − r0(t)|2, where n is the unit vector (17); this magnetic field must be then added to the moving dipole's magnetic field (14).

12 The magnetic force on a moving electric dipole p0, which is the Lorentz force on the sum (11) of the polarization and magnetization currents, can be derived in the Lagrangian formalism using an interaction −m · B, where m = −v × p0/c. See V. Hnizdo, “Comment on 'Electromagnetic force on a moving dipole',” Eur. J. Phys. 33, L3-L6 (2012); arXiv:1108.4332.

13 Reference 10, Problem 6.22.

14 See, e.g., Reference 10, Sec. 5.6.

15 It can be shown easily by integration by parts that (1/2c) R d3r r × Jm, where Jm = c∇ × [m δ(r−r0)], equals the magnetic moment m in the curl expression for Jm.

16 references, page 1 of 2
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