How it works
Maxwell's four equations in differential form: (1) ∇·D = ρ_v (Gauss's law for electric fields, charge is source of D), (2) ∇·B = 0 (no magnetic monopoles, B field lines are closed loops), (3) ∇×E = −∂B/∂t (Faraday's law, changing B induces E), (4) ∇×H = J + ∂D/∂t (Ampere-Maxwell law, current and changing E produce H). The displacement current density J_d = ∂D/∂t in a capacitor dielectric has the same effect as conduction current J = σE for the magnetic field surrounding it.
Key points to remember
In free space (J = 0, ρ_v = 0), the equations simplify and combining curl equations gives the wave equation ∇²E = µ₀ε₀·∂²E/∂t², confirming EM wave propagation at c. In phasor form, ∂/∂t replaces with jω: ∇×E = −jωB and ∇×H = J + jωD. Boundary conditions derived from Maxwell's equations: E_t and H_t continuity from the curl equations; D_n and B_n continuity from the divergence equations. The Poynting vector S = E×H (W/m²) gives instantaneous power flow density — average power density is S_avg = ½Re(E×H*) in phasor notation.
Exam tip
Every Anna University EM theory paper asks you to state all four Maxwell's equations in both integral and differential form and explain the physical significance of each — memorise the word descriptions (Gauss, no-monopole, Faraday, Ampere-Maxwell) alongside the mathematics.