Displacement Current
I_d = ε₀·dΦ_E/dt. Maxwell's missing piece for the Ampère-Maxwell law.
Key Notes
Maxwell noticed Ampère's law (∮B·dl = μ₀I_c) breaks down when applied to a charging capacitor — no real current crosses the gap, yet B clearly exists around it.
He postulated a NEW current, the displacement current I_d = ε₀(dΦ_E/dt), produced by a changing electric flux — not by moving charges.
The complete Ampère–Maxwell law: ∮B·dl = μ₀(I_c + I_d). With this, the magnetic field is continuous everywhere — inside the wire AND inside the capacitor gap.
I_d has the SAME units (ampere) and produces the SAME magnetic field as a real conduction current of equal magnitude.
Inside a charging parallel-plate capacitor, dE/dt is uniform between the plates, so I_d = ε₀·A·(dE/dt) = C·(dV/dt) = I_c. The two currents are exactly equal.
The conception of displacement current closed Maxwell's equations and predicted that ANY changing E creates B and vice-versa — the seed of electromagnetic waves.
In a conductor, I_c dominates; in vacuum or a perfect dielectric, only I_d exists.
Formulas
Displacement current
Where Φ_E = ∫E·dA is the electric flux through the chosen surface.
Ampère–Maxwell law
Generalised Ampère's law; valid in all situations including time-varying fields.
Inside capacitor (uniform E)
For parallel plates of area A, capacitance C, voltage V across them.
B between plates (radius r ≤ R)
Magnetic field inside the gap, at distance r from the central axis (R = plate radius).
B outside the plates (r ≥ R)
Matches the field of an equivalent conduction current — same dependence on r.
Important Points
The name 'displacement current' is historical — there is NO physical displacement of charge. It is a flux-rate term, nothing more.
Without I_d, applying Ampère's law to two different surfaces bounded by the same loop (one through the wire, one through the gap) gives two different answers — a logical contradiction Maxwell resolved.
I_d ≠ 0 only when E changes with time. In steady DC, dE/dt = 0 ⇒ I_d = 0; in DC capacitor circuits at steady state, no B exists in the gap.
For sinusoidal AC, I_d (rms) inside a capacitor equals I_c (rms) in the wires — Kirchhoff's current law is preserved.
Displacement current is the mechanism that makes EM waves self-sustaining: changing E creates B, changing B creates E, and the wave propagates in vacuum.
Common pitfall: thinking I_d requires a medium. It works in vacuum — ε₀ × (dE/dt) needs no charges.
Displacement Current notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.