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.
Gamma rays are the high-energy EM radiation released in nuclear disintegrations, and displacement current arises wherever the electric field (not charge flow) changes, e.g. between capacitor plates. …
γ-rays (from nuclei) are used in radiotherapy; a charging capacitor's gap carries displacement current only; a steady dc wire carries conduction current only.
(a) Nuclear disintegrations emit gamma rays, the most energetic EM radiation. One use: radiotherapy for treating cancer (also sterilisation of surgical instruments).
(b) Displacement vs conduction current. Maxwell defined displacement current Id=ε0dtdΦE, present wherever the electric flux changes.
(i) Displacement current but no conduction current: the vacuum/dielectric region between the plates of a charging capacitor — no charge crosses the gap, but the changing E gives Id. …
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
JKBOSE Class 12 Annual Regular Examination 2024Set SZ2 marks
Q.What is Maxwell's displacement current ? Is displacement current a source of magnetic field ?
›Reveal solutionSolution
Maxwell introduced displacement current to fix a flaw in Ampere's law and to keep it consistent everywhere, including places (like between capacitor plates) where no actual charge flows but the electric field is changing — and this displacement current does produce a magnetic field.
Maxwell's displacement current: Ampere's circuital law (∮B·dl = μ0 Ic) works fine for a simple current-carrying wire, but fails for a circuit containing a capacitor: no charge actually crosses the gap between the plates, yet a magnetic field is still observed around the connecting wires while the capacitor charges/discharges. Maxwell resolved this by proposing an additional current term, called the displacement current, associated with a changing electric flux ΦE through the surface:
Id = ε0 (dΦE/dt)
Ampere's law was then generalised (the Ampere–Maxwell law) to:
∮B·dl = μ0 (Ic + Id)
Between the capacitor plates, Ic = 0 but Id ≠ 0 (because the electric field, and hence the flux, is changing as the capacitor charges), so the law now gives a consistent, correct magnetic field on both sides of the gap.
JKBOSE Class 12 Annual Regular Examination 2021Set SZ2 marks
Q.What is meant by displacement current?
›Reveal solutionSolution
Displacement current is Maxwell's correction to Ampere's law — a changing electric field acts as a source of magnetic field, exactly as a conduction current does.
Consider a capacitor being charged. Conduction current flows in the connecting wires, but no charge actually crosses the gap between the plates, so no conduction current flows there — yet Ampere's circuital law, applied to a surface bulging into the gap, seemed to demand one, creating an inconsistency.
Maxwell resolved this by proposing that a changing electric field between the plates gives rise to an equivalent current, called the displacement current:
Id=ε0dtdΦE
where ΦE is the electric flux through the surface. As the capacitor charges, the electric field (and hence ΦE) between the plates increases, so Id is non-zero and exactly equals the conduction current in the wires, keeping current continuous around the circuit.
Maxwell generalised Ampere's law to include this term: …