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1 results

  • Appendix B - Forces, Torques, and Responses of Localized Charge Densities

  • Jene A. Golovchenko, Harvard University, Massachusetts, Edlyn V. Levine, Harvard University, Massachusetts
  • Book:
    Lectures on Classical Electromagnetism
    Published online:
    16 October 2026
    Print publication:
    19 November 2026, pp 252-259

  • Summary

    This appendix derives the Lorentz force and torque on localized charge and current distributions in terms of their multipole moments. Starting from the general Lorentz force density, it introduces a comoving reference point X(t), rewrites the charge and current in a particle frame, and expands the external fields about that point. This yields a multipole expansion of the total force in terms of the net charge, electric dipole moment, charge quadrupole tensor, and magnetic dipole moment, and identifies the leading contributions, q0 E(X,t), (p⋅∇)E, and ∇(μ⋅B). The appendix defines the electric and magnetic dipole moments and discusses when higher-order multipoles can be neglected for slowly varying fields. It then interprets the force law as an effective equation of motion for a localized system carrying internal structure. A final section derives the torques on electric and magnetic dipoles, τE = p × E and τB = μ × B, summarizing how localized systems respond translationally and rotationally to external electromagnetic fields.