Maxwell equations

Equations {Maxwell's equations} {Maxwell equations} can find all electric and magnetic properties. For stationary and moving charges, electric-field and magnetic-field relations are Gauss's law, Gauss's law for magnets, Faraday's law, and Ampere's law.

stationary

Partial derivative of electric field with distance equals negative of partial derivative of magnetic field with time. Partial second derivative of electric field with distance equals electric permittivity times magnetic permeability times partial second derivative of electric field with time.

tensors

Maxwell's equations are equivalent to two equations. For magnetostatics and magnetodynamics equations, exterior derivative of electromagnetic-field tensor F equals zero: dF = 0. Electromagnetic-field tensor is a linear operator on velocity vector. Electromagnetic-field tensor has covariant components. This tensor is equivalent to delta function. For electrostatics and electrodynamics equations, exterior derivative of electromagnetic-field-tensor dual F* equals four times pi times four-current dual J*: dF* = 4 * pi * J*. This tensor is equivalent to delta scalar product.

current

The four-current has one component for charge density and three components for current densities in three spatial directions.

duals

Rank-x antisymmetric tensors relate to rank 4 - x antisymmetric tensors {dual, tensor}. Dual of dual gives original tensor, if rank is greater than two.

invariant

Electromagnetism invariant is current squared minus light speed times charge density squared, which equals negative of momentum times light speed squared.

retarded and advanced

Electromagnetic-field changes follow charge accelerations {retarded solution}. However, field changes can happen before charge accelerations {advanced solution}, because equations are symmetric. Other solutions can be linear retarded-solution and advanced-solution combinations.

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5-Physics-Wave-Electromagnetic

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Date Modified: 2022.0224