Why Ampere's Circuital Law Fails for a Charging Capacitor
Physics · Electromagnetic Waves · NEET
Ampere's circuital law says the magnetic field around a wire depends on the current passing through any surface bounded by your loop. But for a charging capacitor, one surface (flat, cutting the wire) has current I, while a bag-shaped surface passing between the plates has zero current, so the same law gives two different answers. This contradiction shows the original law is incomplete. MEMORY HOOK: same loop, two surfaces, two answers means the law is broken.
The same Amperian loop bounds two different surfaces: a flat one cutting the wire (current I) and a bag-shaped one passing through the capacitor gap (current 0). Ampere's original law gives two different magnetic fields, proving it is incomplete.
Your doubts, answered
Why does Ampere's law give two different answers for the same loop?
Ampere's law is B.dl around a closed loop = mu0 times the current crossing ANY surface that has that loop as its edge. For a wire charging a capacitor, pick a flat surface cutting the wire: current = I, so the law predicts a magnetic field. Now pick a balloon-shaped surface with the same edge but bulging out to pass between the two plates: no charge crosses the gap, so current = 0, and the law predicts zero field. Same loop, two surfaces, two answers. A correct law must never depend on which surface you imagine, so this proves the original law is incomplete.
Is there really no current between the capacitor plates?
No conduction current (no moving charges) crosses the empty gap between the plates. Charge piles up on one plate and leaves the other, but nothing physically jumps across. That is exactly why the flat surface (through the wire) sees current I but the surface passing through the gap sees zero. The wire has conduction current; the gap does not.
What is actually changing inside the gap while the capacitor charges?
As charge builds up on the plates, the electric field E between the plates grows with time. So even though no charges cross the gap, there is a changing electric flux there. Maxwell realised this changing electric flux acts like a current (he called it displacement current) and fixed the inconsistency by adding it to Ampere's law. See what-is-displacement-current.
Does this mean Ampere's original law is wrong?
It is not wrong, it is incomplete. For steady currents with no changing electric fields (like a plain current-carrying wire) the original law works perfectly. It only fails when an electric field is changing with time, as in the capacitor gap. Maxwell's added term is zero in the steady case, so the corrected law reduces to the old one there.
Why can I choose any surface for the loop in the first place?
For a truly consistent law, the current through every surface sharing the same boundary loop must be identical. This works for steady currents because charge is conserved and current in equals current out. The capacitor breaks this because charge accumulates on the plates instead of flowing straight through, so the current is not the same for the wire-cutting surface and the gap-cutting surface.
⚠️ The NEET trap ✗ Thinking that because no charge crosses the gap, the magnetic field just outside the capacitor must be zero. ✓ The magnetic field outside a charging capacitor is the same as around the wire. The changing electric field in the gap (displacement current) produces the field, keeping the answer consistent whichever surface you choose. 🧠 No charge in the gap does NOT mean no magnetic field. The changing E-field takes over the job of the current.
Real NEET questions
2024
A parallel plate capacitor is charged by connecting it to a battery through a resistor. If I is the current in the circuit, then in the gap between the plates:
A · Displacement current of magnitude equal to I flows in the same direction as I ✓
B · Displacement current of magnitude equal to I flows in a direction opposite to that of I
C · Displacement current of magnitude greater than I flows but can be in any direction
D · There is no current
Solution: Step 1: In the connecting wire the conduction current is I (moving charges). Step 2: In the gap there are no moving charges, so conduction current there is zero. This is exactly the inconsistency Ampere's law faces. Step 3: Maxwell's fix says a displacement current Id = eps0 (dPhiE/dt) flows in the gap. Step 4: Since charge conservation demands the current be continuous, Id in the gap must exactly equal I in the wire, and in the same direction, so the magnetic field is consistent for any surface. Answer: A.
2019
A parallel plate capacitor of capacitance 20 microF 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 microA
B · 60 microA, 60 microA ✓
C · 60 microA, zero
D · Zero, zero
Solution: Step 1: Conduction current in the wire, I = dQ/dt. Since Q = C V, I = C (dV/dt). Step 2: I = 20e-6 F x 3 V/s = 60e-6 A = 60 microA. Step 3: The whole point of Maxwell's fix is that the displacement current in the gap equals the conduction current in the wire, so Id = 60 microA too. Step 4: Both are 60 microA (option C, which claims zero displacement current, is the trap based on the failed original law). Answer: B.
Solved Electromagnetic Waves NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
In one line, why does Ampere's circuital law fail for a charging capacitor?
Because a flat surface cutting the wire sees current I but a surface passing through the gap sees zero current, so the same loop gives two different magnetic fields, which a valid law can never do.
Who discovered this inconsistency and how was it fixed?
James Clerk Maxwell noticed it while applying Ampere's law near a capacitor. He fixed it by adding the displacement current term Id = eps0 (dPhiE/dt), giving the Ampere-Maxwell law.
Does current physically flow between the plates?
No conduction current (no charges) crosses the gap. Only the electric field changes there, and this changing electric flux behaves like a current, called displacement current.
When does the original Ampere's law work fine?
For steady currents with no time-changing electric field, such as a plain current-carrying wire. Maxwell's extra term is zero there, so the old law is exact.
Why is this important for NEET?
It is the reasoning that leads to displacement current and Maxwell's equations, a favourite one-mark conceptual question, and it directly sets up the idea of electromagnetic waves.
