Electromagnetic studio: pump controls

Maxwell unifies local field relations

Electric charge sources electric divergence, magnetic fields have no observed isolated sources, changing magnetic fields circulate electric fields, and currents plus changing electric fields circulate magnetic fields. The equations are local bookkeeping for sources and curls.

Read the structure, not the list

In differential form, ∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −∂B/∂t, and ∇×B = μ₀J + μ₀ε₀∂E/∂t. The displacement-current term is required for charge conservation: taking divergence gives 0 = μ₀∇·J + μ₀ε₀∂(∇·E)/∂t, which becomes ∇·J + ∂ρ/∂t = 0.

In a charging capacitor, conduction current exists in the wires but not across the dielectric gap; the changing electric field supplies equal magnetic-circulation effect, preserving the loop description.

The macroscopic form hides material response

In matter, polarization and magnetization alter the effective source terms. Boundary conditions and constitutive relations are needed before Maxwell’s equations predict a device.

Work it: Write the charge-continuity equation from Maxwell’s equations. Explain in one paragraph why a capacitor would violate Ampère’s law if displacement current were omitted.