Displacement Current — Maxwell's Master Fix — Notes

Cue Column	Notes Column
The problem	In a circuit charging a capacitor: conduction current I_c flows in the wire but no charge crosses the gap between plates. Ampere's law (∮B·dl = μ_{0}I_c) gives different results depending on which surface you choose — a fatal inconsistency.
Maxwell's fix	Introduce displacement current: $I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \quad [A]$ This "current" arises from the changing electric field in the gap.
Modified law	$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0(I_c + I_d) = \mu_0\left(I_c + \varepsilon_0 \frac{d\Phi_E}{dt}\right)$
Is it real?	NO — displacement current is NOT actual charge movement. It is the ε_{0}(dΦ_E/dt) term. Yet it produces a magnetic field exactly as a real current would.
Continuity	In any circuit loop, I_c (wire) = I_d (gap) at every instant. Current continuity is preserved through the entire closed circuit.
Key check	For capacitor: Φ_E = EA → dΦ_E/dt = A·(dE/dt). So I_d = ε_{0}·A·(dE/dt).

Summary Row: Displacement current I_d = ε_{0}(dΦ_E/dt) is a fictitious current due to a changing E field. It equals the conduction current, ensuring Ampere-Maxwell law is consistent and current is continuous.