Ampere's circuital law, in its original form, links the magnetic field around a closed loop to the conduction current (moving charges) threading that loop:
∮B⋅dl=μ0Ic
Maxwell realised this law is incomplete. The classic illustration is a charging capacitor. Consider an Amperian loop encircling the wire that feeds one plate.
If you cap that loop with a flat surface cut by the wire, a real conduction current Ic passes through it.
If you instead cap the SAME loop with a bulging surface that passes between the two capacitor plates, no charge crosses the gap — the space between the plates is an insulator. So Ic=0 through this surface.
Ampere's law now gives two different answers for ∮B⋅dl for the same loop, depending on which surface you choose. That is a contradiction — the law cannot be right as it stands.
Maxwell's Fix: A Current Made of Changing Field
Between the plates there is no moving charge, but there is a growing electric field, because charge is piling up on the plates. Maxwell proposed that a changing electric flux acts like a current for the purpose of producing a magnetic field. He called it the displacement current, Id.
Id=ε0dtdΦE
where ΦE=∫E⋅dA is the electric flux through the surface, and ε0=8.85×10−12C2N−1m−2 is the permittivity of free space.
Check with the capacitor. For a parallel-plate capacitor of area A and plate charge q, the field between the plates is E=ε0Aq, so the flux is ΦE=EA=ε0q. Then
Id=ε0dtdΦE=ε0⋅ε01dtdq=dtdq=Ic
So the displacement current in the gap is exactly equal to the conduction current in the wire. The two surfaces now give the same answer — the contradiction is gone.
The Complete (Ampere–Maxwell) Law
Maxwell rewrote Ampere's law so that the total current is conduction plus displacement current:
∮B⋅dl=μ0(Ic+Id)=μ0Ic+μ0ε0dtdΦE
Important
The deep meaning: a changing electric field produces a magnetic field, just as (by Faraday's law) a changing magnetic field produces an electric field. This symmetry is what makes self-sustaining electromagnetic waves possible — the changing E-field of the wave generates the B-field and vice versa.
Key Points to Remember …
Why this formula?
Displacement Current: Why the Formula Holds
The displacement current is one of the most elegant corrections in physics — it fixed a logical flaw in Maxwell's equations and predicted electromagnetic waves. Let's understand why its formula emerges.
1. The Problem That Demanded a Fix
Consider a capacitor being charged in a circuit. Ampère's law (in its original form) states:
∮B⋅dl=μ0Ienc
where Ienc is the current passing through any surface bounded by the loop.
Now take two different surfaces bounded by the same loop:
Surface S₁: Cuts the wire — current I passes through.
Surface S₂: Passes between the capacitor plates — no current passes through.
Surface
Current through it
S₁ (cuts wire)
I
S₂ (between plates)
0
This is a contradiction: the same loop gives two different values for ∮B⋅dl. Ampère's law is inconsistent for time-varying fields.
2. The Insight: Changing Electric Field
Between the capacitor plates, there is no conduction current, but there is a changing electric field as charge builds up.
The electric field between plates: E=ε0σ=ε0AQ
As Q changes, E changes: dtdE=ε0A1dtdQ
Maxwell realized: a changing electric field should produce a magnetic field, just like a current does.
3. Deriving the Displacement Current Formula
Step 1: Relate charge to electric flux
The electric flux through the capacitor plates is:
ΦE=∫E⋅dA=E⋅A=ε0Q
Step 2: Differentiate with respect to time
dtdΦE=ε01dtdQ=ε0I
Step 3: Define displacement current
Maxwell defined the displacement currentId as:
Id=ε0dtdΦE
From Step 2, this equals I — the same conduction current in the wire. The displacement current "bridges" the gap.
4. The Corrected Ampère-Maxwell Law
The full law becomes:
∮B⋅dl=μ0(Ienc+Id)
Or equivalently:
∮B⋅dl=μ0Ienc+μ0ε0dtdΦE
Why this works:
For surface S₁: Ienc=I, dtdΦE=0 → result = μ0I
For surface S₂: Ienc=0, dtdΦE=ε0I → result = μ0ε0⋅ε0I=μ0I
Both surfaces give the same answer. The contradiction is resolved.
Displacement current is the current-like term Id=ε0dtdΦE that exists wherever the electric flux is changing with time (e.g. in the gap between charging capacitor plates), even though no actual charge crosses that region; it was introduced by Maxwell to keep Ampere's law consistent, and it produces a magnetic field exactly like a real conduction current. …
Displacement current is Maxwell's term ε0dΦE/dt for a changing electric flux, which acts exactly like a real current in producing a magnetic field and keeping current continuous through regions (like a capacitor gap) with no actual charge flow.
While a capacitor is being charged (or discharged), conduction current flows in the connecting wires, but no charge physically crosses the gap between the plates — yet a magnetic field is still observed around the wires and even in the gap, consistent with a current flowing there.
Maxwell resolved this by proposing that a time-varying electric flux ΦE through a surface constitutes an additional 'current', called the displacement current:
Id=ε0dtdΦE
This displacement current is not due to any actual motion of charge; it arises purely from the changing electric field, but it produces a magnetic field exactly as a conduction current would, as required by the (Maxwell-corrected) Ampere's circuital law:
∮B⋅dl=μ0(Ic+ε0dtdΦE) …