Local Form of Maxwell's Equations

From integrals to local equations

Applying the Green-Ostrogradsky (divergence) theorem and Stokes' (curl) theorem to the four integral equations yields a local form, valid at every point in space:

Name	Local form
Maxwell-Gauss	div(E) = rho / epsilon0
Maxwell-Thomson (flux)	div(B) = 0
Maxwell-Faraday	curl(E) = - dB/dt
Maxwell-Ampère	curl(B) = mu0J + mu0epsilon0*dE/dt

Where rho is the volumetric charge density, J the current density, and the time derivatives are partial derivatives.

Example

If at a point div(E) = 5,000 (in V/m per meter) is measured, with epsilon0 = 8.85e-12 F/m, then the local charge density is rho = epsilon0 * div(E) = 8.85e-12 * 5000 ~ 4.4e-8 C/m^3. The local form thus allows the sources to be traced at a single point, without knowing the field elsewhere.

Common pitfall

Do not confuse div (a scalar, a measure of a vector field's source/sink) with curl (a vector, a measure of local circulation). Another common mistake is forgetting that dE/dt and dB/dt denote partial derivatives with respect to time at a fixed position, not total derivatives along a trajectory.