Questions: Displacement Current and Maxwell's Equations

Question 1

Maxwell added the displacement current term ε₀∂E/∂t to Ampère's law primarily because:

• A. Experiments had directly measured this new form of current flowing through capacitor gaps
• B. Applying Ampère's original law to a charging capacitor produced a mathematical inconsistency — different surfaces bounded by the same loop gave different results
• C. Faraday's law required a symmetric partner to conserve electromagnetic energy
• D. He needed an additional term to correctly calculate the force on a moving charge

Answer: (B)

The problem was mathematical consistency, not experimental discovery. Applying Ampère's law ∮B·dl = μ₀I_enc to a loop around a wire feeding a capacitor: if you choose a surface cutting the wire, I_enc is nonzero; if you choose a surface through the gap between plates, no charge crosses it, giving I_enc = 0. The same integral cannot equal two different things. Maxwell's displacement current resolves this: ε₀∂E/∂t in the gap acts as an effective current density, making the result consistent for any surface choice. The experimental confirmation came later through electromagnetic wave predictions.

Question 2

The displacement current in a charging capacitor gap is best described as:

• A. A real flow of electrons tunneling through the dielectric between the plates
• B. An effective current arising from the changing electric field, with no actual charge motion through the gap
• C. A magnetic field that mimics the effect of conventional current in Ampère's law
• D. A polarization current in the dielectric material that stores energy

Answer: (B)

No charge crosses the gap between capacitor plates — the displacement current is not a flow of electrons. It is the term ε₀∂E/∂t, which has the units of current density and produces a magnetic field exactly as if a real current were present. Maxwell postulated it to maintain mathematical consistency in Ampère's law. This is a conceptual distinction with deep implications: the 'current' is the rate of change of electric flux, not charge transport.

Question 3

Maxwell's addition of the displacement current created a symmetric relationship: just as Faraday showed that a changing B field produces a circulating E field, the displacement current shows that a changing E field produces a circulating B field.

• T. True
• F. False

Answer: True

This symmetry is the deep insight. Faraday's law: ∇ × E⃗ = −∂B⃗/∂t. Modified Ampère's law: ∇ × B⃗ = μ₀J⃗ + μ₀ε₀∂E⃗/∂t. In free space (J⃗ = 0), the two laws are mirrors: changing B drives circulating E, and changing E drives circulating B. This mutual induction between the two fields is what produces self-sustaining electromagnetic waves.

Question 4

The displacement current was confirmed by direct experimental measurement before Maxwell included it in his equations.

• T. True
• F. False

Answer: False

Maxwell added the displacement current term purely for reasons of mathematical consistency — it was a theoretical postulate, not an experimental discovery. The experimental confirmation came indirectly: the modified equations predicted electromagnetic waves traveling at speed 1/√(μ₀ε₀) ≈ 3×10⁸ m/s, matching the known speed of light. Hertz's experiments (1887) directly confirmed electromagnetic waves. The displacement current was justified after the fact by the enormous predictive success of the complete Maxwell equations.

Question 5

Why did Maxwell's addition of the displacement current term lead to the prediction that light is an electromagnetic wave?

Model answer: With both Faraday's law (∇ × E⃗ = −∂B⃗/∂t) and the modified Ampère's law (∇ × B⃗ = μ₀ε₀∂E⃗/∂t in free space), you can take the curl of one equation and substitute the other. This produces the wave equation ∇²E⃗ = μ₀ε₀ ∂²E⃗/∂t², whose solutions are waves traveling at speed v = 1/√(μ₀ε₀). Plugging in the known electromagnetic constants gives v ≈ 3×10⁸ m/s — exactly the measured speed of light, revealing that light is an electromagnetic wave.

Without the displacement current term, no such wave equation emerges — the system is inconsistent and the fields cannot propagate in vacuum. The displacement current is what 'closes the loop': E changing drives B, B changing drives E, and the result is a self-reinforcing wave that can travel through empty space at the speed of light.