Physics / advanced
Electromagnetism in spacetime
Express electromagnetic dynamics covariantly, relating field tensors, relativistic motion, conserved currents, and observer-dependent field decompositions.
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Module 01 advanced
Electromagnetism in spacetime
Express electromagnetic dynamics covariantly, relating field tensors, relativistic motion, conserved currents, and observer-dependent field decompositions.
- 01
Four-vectors and Minkowski contractions
Fix a spacetime convention, lower indices correctly, and use invariant contractions to distinguish physical measurements from frame components.
25 min - 02
Electromagnetic field tensor components
Construct the antisymmetric field tensor from potentials, check all index signs, and recover an electric and magnetic force from its components.
26 min - 03
Lorentz transformation of electric and magnetic fields
Transform field components between inertial observers and determine the direction and scale of electric–magnetic mixing without losing SI factors.
27 min - 04
Electromagnetic Lorentz invariants
Use the two electromagnetic invariants to classify fields, rule out impossible rest frames, and identify when a purely electric or magnetic description exists.
25 min - 05
The covariant Lorentz force
Recover relativistic momentum and energy equations from the tensor force law, and use antisymmetry to prove that electromagnetic forces preserve rest mass.
27 min - 06
Four-current and charge conservation
Transform charge and current density together, derive local continuity, and distinguish invariant total charge from observer-dependent density.
26 min - 07
The covariant Maxwell equations
Recover all four Maxwell equations from tensor equations, derive current conservation from antisymmetry, and distinguish dynamical equations from potential identities.
28 min - 08
The electromagnetic stress-energy tensor
Organize energy, energy flux, momentum density, and momentum flux with consistent c factors and relate field losses to the Lorentz force on matter.
29 min - 09
Relativistic charged-particle orbits
Solve ideal uniform magnetic and electric trajectories, distinguish laboratory and proper-time frequencies, and identify the relativistic momentum–velocity relation.
28 min - 10
Action for particles and electromagnetic fields
Combine proper-time particle motion, gauge coupling, and the Maxwell field action while distinguishing boundary gauge shifts, canonical momentum, and physical fields.
30 min