The Ampere-Maxwell law is Ampere's circuital law with one extra term added by Maxwell: the displacement current. It reads B.dl = mu0 ic + mu0 epsilon0 (dPhiE/dt), meaning a magnetic field is produced not only by moving charges (conduction current ic) but also by a changing electric field (displacement current id). Memory hook: "Ampere-Maxwell = old law + changing E-field", so even where no charge flows, a growing electric field still makes a magnetic field.
In a charging capacitor, conduction current ic flows in the wires while a changing electric field E between the plates creates the displacement current id = epsilon0(dPhiE/dt). The Ampere-Maxwell law adds id so the same Amperian loop gives one consistent magnetic field, whether its surface cuts the wire (ic) or the gap (id), with id = ic.
Your doubts, answered
Why did Maxwell modify Ampere's original circuital law?
The original law B.dl = mu0 ic gave two different answers for the same loop around a charging capacitor. If you cap the loop with a flat surface cutting the wire, current ic passes through, so B is non-zero. But if you cap it with a bulging surface that passes between the capacitor plates, no charge flows there, so it predicts B = 0. Both surfaces share the same loop, so B must be the same. Maxwell fixed this contradiction by adding the displacement current term, which is non-zero between the plates because the electric field there is changing.
What exactly does the extra term mu0 epsilon0 (dPhiE/dt) mean?
PhiE is the electric flux through the surface bounded by the loop. As a capacitor charges, the electric field E between its plates grows, so the flux PhiE increases with time. The rate dPhiE/dt multiplied by epsilon0 gives the displacement current id = epsilon0 (dPhiE/dt). This term acts as a real source of magnetic field, exactly like conduction current, even though nothing physically flows across the gap.
Does the changing electric field really make a magnetic field, or is it just a math trick?
It is real, not just algebra. NCERT states the magnetic field between the capacitor plates (point M) can be measured and equals the field just outside (point P). So a time-varying electric field genuinely produces a magnetic field. This is the key idea that lets electric and magnetic fields keep generating each other and travel as an electromagnetic wave through empty space.
When is the displacement current term zero?
It is zero whenever the electric flux does not change with time. In a steady DC current through a normal wire, E is constant, so dPhiE/dt = 0 and only conduction current matters. The displacement term only turns on when E (and hence PhiE) is changing, such as during charging of a capacitor or in an AC circuit.
Are ic and id both present in the same place?
Sometimes. In the ideal capacitor gap there is only id (no charge flows) and ic = 0; in the connecting wire there is only ic and id = 0. But in a real medium that is neither a perfect conductor nor a perfect insulator, both currents can exist together. What matters for Ampere-Maxwell is the total current ic + id passing through the surface.
⚠️ The NEET trap ✗ Thinking the Ampere-Maxwell law only adds a magnetic field outside the capacitor, and that inside the gap B = 0 because no charge flows there. ✓ Inside the gap the changing electric field creates a displacement current, so B is non-zero there too. In fact B grows with radius inside (B proportional to r) and falls outside (B proportional to 1/r), so it is maximum right at the plate edge, non-zero both inside and outside. 🧠 No charge flowing does NOT mean no magnetic field. A changing E-field is a current in disguise.
Real NEET questions
NEET 2025
A parallel plate capacitor made of circular plates is being charged such that the surface charge density on its plates is increasing at a constant rate with time. The magnetic field arising due to the displacement current is
A · Non-zero everywhere with maximum at the imaginary cylindrical surface connecting the peripheries of the plates ✓
B · Zero between the plates and non-zero outside
C · Zero at all places
D · Constant between the plates and zero outside the plates
Solution: The charging plates create a changing electric field between them, giving a displacement current id = epsilon0 (dPhiE/dt). By the Ampere-Maxwell law this id is a source of magnetic field. Applying B.dl = mu0 id to a circular loop of radius r about the axis: Inside the plates (r less than R) only the fraction of flux inside the loop counts, so B increases linearly, B proportional to r. Outside the plates (r greater than R) all the displacement current is enclosed, so B proportional to 1/r. Therefore B rises from the centre to a maximum at the edge r = R (the cylindrical surface joining the plate rims) and is non-zero both inside and outside. Correct option: A.
NEET 2019
A parallel plate capacitor of capacitance 20 microfarad is being charged by a voltage source whose potential is changing at the rate of 3 V/s. The conduction current through the connecting wires and the displacement current through the plates of the capacitor would be, respectively,
A · Zero, 60 microampere
B · 60 microampere, 60 microampere ✓
C · 60 microampere, zero
D · Zero, zero
Solution: Conduction current in the wire: ic = C (dV/dt) = 20e-6 x 3 = 60e-6 A = 60 microampere. By the Ampere-Maxwell continuity of current, the displacement current in the gap must equal the conduction current in the wire so the circuit is unbroken: id = ic = 60 microampere. So both are 60 microampere. Correct option: B.
Solved Electromagnetic Waves NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It is B.dl = mu0 ic + mu0 epsilon0 (dPhiE/dt): a magnetic field is produced by conduction current plus the displacement current from a changing electric field.
What is the only difference from the original Ampere's law?
The original law had just mu0 ic. Maxwell added the second term mu0 epsilon0 (dPhiE/dt) so the law also works for a charging capacitor where no charge crosses the gap.
Is displacement current a real flow of charge?
No charge actually crosses the gap. It is the effect of a changing electric field, but it produces a magnetic field just like real current and keeps the total current continuous.
Why is this law important for electromagnetic waves?
It shows a changing electric field creates a magnetic field. Combined with Faraday's law (a changing magnetic field creates an electric field), the two fields keep generating each other and propagate as an EM wave.
Is the Ampere-Maxwell law one of Maxwell's four equations?
Yes. It is the fourth of Maxwell's equations and the only one Maxwell corrected by adding the displacement current term.
