Without the displacement current term, what problem arises when applying Ampère's law to a circuit with a charging capacitor?
AThe magnetic field around the wire becomes infinite as the capacitor charges
BThe value of ∮ B · dl depends on which surface bounded by the Amperian loop you choose — a flat disk through the wire gives current I, but a balloon surface through the gap gives zero
CGauss's law for B gives a non-zero result inside the capacitor, implying magnetic monopoles
DThe electric field inside the capacitor cannot be calculated because the boundary conditions are incomplete
Ampère's original law, ∮ B · dl = μ₀I_enc, requires a surface bounded by the Amperian loop. For a loop around the wire feeding a capacitor, two valid surfaces exist: a flat disk that the wire pierces (current I passes through) and a balloon surface passing between the plates (no current passes through). These give different answers for the same path integral — a contradiction that signals the law is inconsistent for time-varying fields. Maxwell's displacement current term ε₀ dΦ_E/dt fixes this: the changing electric field between the plates contributes as if it were a current, giving the same answer for both surfaces.
Question 2 Multiple Choice
How does Maxwell's addition of the displacement current lead directly to the prediction of electromagnetic waves?
AThe displacement current provides a physical medium through which light can propagate, replacing the aether
BWith the displacement current, Faraday's law and Ampère-Maxwell form a coupled system: changing E creates B and changing B creates E, producing a wave equation in vacuum with speed 1/√(μ₀ε₀)
CThe displacement current increases the effective speed of electric currents in conductors, and light is simply very fast current propagation
DMaxwell's equations require accelerating charges to radiate, and this radiation was identified with light
The key is taking the curl of Faraday's law and substituting Ampère-Maxwell. Faraday: ∇ × E = −∂B/∂t. Ampère-Maxwell in free space: ∇ × B = μ₀ε₀ ∂E/∂t. Combining these gives ∇²E = μ₀ε₀ ∂²E/∂t² — a wave equation with speed v = 1/√(μ₀ε₀). Computing from known constants gives exactly the measured speed of light. Without the displacement current, Ampère's law has no ∂E/∂t term, the symmetry between E and B is broken, and no wave equation emerges. The displacement current is not optional — it is what makes electromagnetic waves possible.
Question 3 True / False
The displacement current ε₀ dΦ_E/dt is not an actual electric current — no charge moves when it is nonzero.
TTrue
FFalse
Answer: True
Despite the name 'current,' the displacement current involves no motion of electric charges. It represents a changing electric flux — the rate at which the electric field strength is changing through a surface. Maxwell called it a 'current' because it plays the same mathematical role in Ampère's law as real current does, and because it has units of amperes. But physically, it is a field quantity, not a flow of charge. This is why displacement current exists in a vacuum between capacitor plates where no charges are present.
Question 4 True / False
Maxwell discovered most four of the equations that bear his name.
TTrue
FFalse
Answer: False
This is a common misconception the topic explicitly addresses. Gauss's law for E was developed by Gauss; Gauss's law for B (no magnetic monopoles) was known before Maxwell; Faraday's law was Faraday's discovery; and the original Ampère's law was Ampère's. Maxwell's contribution was recognizing that Ampère's law was inconsistent for time-varying fields, adding the displacement current term to correct it, and synthesizing all four equations into a unified system — which revealed that light is an electromagnetic wave.
Question 5 Short Answer
Explain why the displacement current term was logically necessary — what would go wrong mathematically without it — and what physical insight it encodes.
Think about your answer, then reveal below.
Model answer: Without the displacement current, Ampère's law ∇ × B = μ₀J is only mathematically consistent for steady currents. The inconsistency appears by taking the divergence of both sides: ∇ · (∇ × B) = 0 always (divergence of curl is zero), but ∇ · (μ₀J) = −μ₀ ∂ρ/∂t ≠ 0 when charge is accumulating (as in a charging capacitor). Maxwell's term ε₀ ∂E/∂t has divergence ∂ρ/∂t (from Gauss's law), which exactly cancels the charge accumulation term and restores mathematical consistency. The physical insight is the symmetry: just as Faraday showed that a changing B creates E, Maxwell's correction shows that a changing E creates B. This symmetric coupling is what allows electromagnetic fields to sustain themselves through space — the mechanism for wave propagation.
The displacement current serves a double role: it is both a mathematical necessity (required by charge conservation) and a physical discovery (a new relationship between changing electric fields and magnetic fields). These two aspects are not coincidental — the mathematical inconsistency in Ampère's law was pointing toward a genuine physical truth that Maxwell uncovered.