Math 2.0

Optics, electromagnetic theory Definition

Classical electromagnetism

Builds on Maxwell's equationsLorentz forceElectric fieldMagnetic fieldElectric chargeVector field

Definition

Classical electromagnetism is the physical theory with the following content.

  1. The state of the field at time tt is a pair of vector fields on ℝ3\R^3: the electric field EE and the magnetic field BB.
  2. The sources are a charge density ρ\rho and a current density JJ on ℝ3×ℝ\R^3 \times \R, made of electric charges and their motion.
  3. The fields obey Maxwell's equations with these sources.
  4. A point charge qq at position x(t)x(t) with velocity v=x˙v = \dot x feels the Lorentz force F=q(E(x,t)+v×B(x,t))F = q \big(E(x, t) + v \times B(x, t)\big). It moves by the relativistic law ddt(mγv)=F\frac{d}{dt} \big( m \gamma v \big) = F. Here mm is its rest mass, γ=(1−∣v∣2/c2)−1/2\gamma = (1 - \abs{v}^2 / c^2)^{-1/2} and cc is the speed of light.

Remarks

In matter the equations are often written with the fields DD and HH and with constitutive relations that describe the medium.

Example: a charge qq at rest at the origin, with no other sources, gives ρ=qδ0\rho = q \delta_0, J=0J = 0, the Coulomb field E(x)=qx/(4πε0∣x∣3)E(x) = q x / (4 \pi \varepsilon_0 \abs{x}^3) and B=0B = 0. This pair solves Maxwell's equations in the sense of distributions.

Non-example: Newtonian gravity is not part of the theory; it has no field like BB and no finite speed of propagation.

The coupled system of fields and point charges has a known difficulty. The field of a point charge is infinite at the charge, so the force of a charge on itself is not defined without further assumptions.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptElectric charge, concept, no evidence levelElectric chargeconceptElectric field, concept, no evidence levelElectric fieldconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptMagnetic field, concept, no evidence levelMagnetic fieldconceptMaxwell's equations, concept, no evidence levelMaxwell's equationsconceptLorentz force, statement, no evidence levelLorentz forcestatementVector-valued function, concept, no evidence levelVector-valued functionconceptVector field, concept, no evidence levelVector fieldconceptClassical electromagnetism, definition, no evidence levelClassical electromagn…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cross product concept
  2. Electric charge concept
  3. Electric field concept
  4. Euclidean space object
  5. Function concept
  6. Magnetic field concept
  7. Maxwell's equations concept
  8. Lorentz force statement
  9. Vector-valued function concept
  10. Vector field concept
  11. Classical electromagnetism definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Classical_electromagnetism
About this page
Tag42NMMARPPG
URIm2://42NMMARPPG
Kinddefinition
WritingW0 stub
MSC78A25, 35Q61
DefinesClassical electromagnetism
UsesMaxwell's equations, Lorentz force, Electric field, Magnetic field, Electric charge, Vector field
Changed3 November 2026

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