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Mathematics for Physics

196 Relativistic Electrodynamics

The study of electric and magnetic fields in special relativity, including how those fields transform between inertial frames.

Magnetism as a relativistic effect. Magnetic forces are a relativistic consequence of electrostatic forces when charges are in relative motion. Relativistic magnetism is the extra force a moving test charge feels because moving charge densities are Lorentz contracted. This principle is used to obtain magnetic forces from Coulomb’s law plus relative motion.

Frame-dependent field unification. Electric and magnetic fields are frame-dependent parts of one electromagnetic field. Field mixing is the exchange of electric and magnetic components under a change of inertial frame. This principle is used to show that a pure electric field in one frame is mixed with a magnetic field in another.

The covariant field tensor. The six independent electric and magnetic components form one antisymmetric field tensor. The electromagnetic field tensor is a \(4\times 4\) array whose entries are those field components. This principle is used to write Maxwell’s equations and the Lorentz force in covariant form.

Direction-dependent field transformations. Field components parallel to the relative motion are unchanged. Transverse components mix and scale with the Lorentz factor. A longitudinal component is a field part along the boost. A transverse component is a field part perpendicular to the boost. This principle is used to compute the fields of a moving charge from the rest-frame fields.

196.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. — source for the heading explanation.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — field mixing; \(F^{\mu\nu}\); longitudinal and transverse transformations.
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — magnetism from electrostatics; field transformations.
  4. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — electric and magnetic fields mix between frames.
  5. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — electromagnetic field tensor; four-potential.
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — unified electromagnetic field; \(F_{\mu\nu}\).