Maxwell’s equations in integral and differential form

Maxwell’s equations can be written in both integral and differential forms. Here, we will provide the equations in both forms:

  1. Gauss’s Law for Electricity:

Integral form:

∮ E • dA = (1/ε₀) ∫ ρ dV

Differential form:

∇ • E = ρ/ε₀

  1. Gauss’s Law for Magnetism:

Integral form:

∮ B • dA = 0

Differential form:

∇ • B = 0

  1. Faraday’s Law of Electromagnetic Induction:

Integral form:

∮ E • dl = -d(∫ B • dA)/dt

Differential form:

∇ × E = -∂B/∂t

  1. Ampere’s Law with Maxwell’s Addition (Ampere-Maxwell Law):

Integral form:

∮ B • dl = μ₀ ( ∫ J • dA + ε₀ * d(∫ E • dA)/dt )

Differential form:

∇ × B = μ₀(J + ε₀ ∂E/∂t)

In these equations:

  • E represents the electric field.
  • B represents the magnetic field.
  • ρ represents the electric charge density.
  • J represents the electric current density.
  • ε₀ is the vacuum permittivity.
  • μ₀ is the vacuum permeability.
  • ∇ (nabla) is a vector differential operator, used to calculate divergence (∇ •) and curl (∇ ×).
  • ∂/∂t represents the partial derivative with respect to time.

The integral form of Maxwell’s equations deals with closed surfaces (flux integrals) and closed loops (path integrals), whereas the differential form relates the local properties of electric and magnetic fields to the charge and current distributions at a point in space. Both forms of the equations are widely used in the analysis of electromagnetic phenomena.

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