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Mathematics for Physics

178 Magnetism as a Relativistic Effect

The view that magnetic forces arise from electrostatic forces modified by the relativistic motion of charge carriers, as seen from a moving frame.

Magnetism as a manifestation of electrostatics. A force that is purely electric in one inertial frame is a magnetic force in a relatively moving frame. Magnetic force is the velocity-dependent part of the force on a moving charge. This principle is used to obtain magnetism from electrostatics plus relative motion.

Lorentz contraction of moving charge densities. Length contraction makes opposite charge densities in a current-carrying wire unequal in a moving frame, so a lab-neutral wire becomes charged. Charge density is charge per unit length or volume. This principle is used to turn a lab magnetic force into an electric force in the charge’s rest frame.

Covariance of the Lorentz force. All inertial observers agree on the path of a charged particle, while they split the force into different mixtures of \(\mathbf{E}\) and \(\mathbf{B}\). The Lorentz force is \(\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})\). This principle is used to keep electrodynamics compatible with special relativity.

Boosting of the field tensor. A purely electric field of charges at rest becomes a mixed electric and magnetic field after a boost. The electromagnetic field tensor is the \(4\times 4\) array that unifies \(\mathbf{E}\) and \(\mathbf{B}\). This principle is used to generate \(\mathbf{B}\) from a static \(\mathbf{E}\) by changing frames.

Invariance of electric charge. Electric charge itself does not change with speed. Charge invariance is that independence of the observer’s velocity. This principle is used to attribute density changes to length contraction, not to a change of charge.

178.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. — source for the heading explanation.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — magnetism from electrostatics; contracted charge densities; charge invariance.
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — magnetic force from a moving frame; Lorentz force covariance.
  4. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — observers agree on the trajectory while \(\mathbf{E}\) and \(\mathbf{B}\) mix.
  5. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — boost of \(F^{\mu\nu}\).
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — unified electromagnetic field under boosts.