Difficulty: Intermediate | Prerequisites: Gauss's Law, Faraday's Law, Ampere's Law (PHY 212 Midterms 1–3)
This topic closes a gap in classical electromagnetism. You already know that changing magnetic fields produce electric fields (Faraday's Law). The missing piece is that changing electric fields also produce magnetic fields, and the quantity that describes this is the displacement current. Once Maxwell added the displacement current term to Ampere's Law, all four of Maxwell's Equations were complete, and they predicted something extraordinary: electromagnetic waves that travel at the speed of light. This is the bridge between electrostatics/magnetostatics and the physics of light, radio, and radiation.
Maxwell fixed Ampere's Law by adding a "displacement current" term that accounts for changing electric flux. The corrected set of four equations (Maxwell's Equations) predicts self-sustaining electromagnetic waves that propagate at c = 3 × 10⁸ m/s, derived purely from the constants ε₀ and μ₀.
Real current (conduction current)
The flow of actual charge Q through an area per unit time: I = dQ/dt. This is ordinary current in a wire.
Displacement current (I_D)
A term with the units of current but involving no moving charge. It equals ε₀ times the rate of change of electric flux: I_D = ε₀ dΦ_E/dt. Think of it as the "current" that a changing electric field mimics, keeping Ampere's Law consistent even in gaps like the space between capacitor plates.
Electric flux (Φ_E)
The product of the electric field and the area it passes through: Φ_E = EA (for uniform fields perpendicular to the area). In simple terms, it measures how much electric field "threads" through a given surface.
Maxwell's Equations
The four fundamental equations governing all classical electromagnetic phenomena: Gauss's Law for E, Gauss's Law for B, Faraday's Law, and the modified Ampere's Law. Together they unify electricity, magnetism, and optics.
Wave equation (Maxwell's wave equation)
The differential equation ∂²E_x/∂z² = μ₀ε₀ ∂²E_x/∂t² that emerges from Maxwell's Equations. It shows that electric (and magnetic) fields propagate as waves whose speed is fixed by μ₀ and ε₀.
(1) Gauss's Law for electric fields: ∮ E · dA = Q_enclosed / ε₀
Electric flux through a closed surface is proportional to the enclosed charge.
(2) Gauss's Law for magnetic fields: ∮ B · dA = 0
No magnetic monopoles exist; magnetic field lines always form closed loops.
(3) Faraday's Law: ∮ E · dl = −dΦ_B/dt
A changing magnetic flux induces an electric field (and hence an EMF).
(4) Ampere's Law (original): ∮ B · dl = μ₀ I_enclosed
Current produces a circulating magnetic field.
Consider a charging capacitor. Draw an Amperian loop between the plates.
No conduction current passes through that surface, so I_enclosed = 0.
The original Ampere's Law would give B = 0 between the plates, which contradicts experiment.
The electric field between the plates is changing as charge builds up, so something is missing.
Inside the capacitor: E = σ/ε₀ = Q/(ε₀A)
Electric flux: Φ_E = EA = Q/ε₀
Therefore Q = ε₀Φ_E
Differentiating: dQ/dt = ε₀ dΦ_E/dt = I_D
The displacement current I_D has the same value as the conduction current feeding the capacitor.
Ampere-Maxwell Law: ∮ B · dl = μ₀(I + I_D) = μ₀I + μ₀ε₀ dΦ_E/dt
This restores consistency: the magnetic field is produced by real current and by changing electric flux.
After the correction, the four equations stand as:
Gauss (E): ∮ E · dA = Q_enclosed / ε₀
Gauss (B): ∮ B · dA = 0
Faraday: ∮ E · dl = −d/dt ∫ B · dA
Ampere-Maxwell: ∮ B · dl = μ₀ε₀ d/dt ∫ E · dA
Note the beautiful symmetry: Faraday's Law and the Ampere-Maxwell Law are near-mirror images, with a changing B producing E and a changing E producing B.
From the complete equations, one can derive: ∂²E_x/∂z² = μ₀ε₀ ∂²E_x/∂t²
This is a standard wave equation of the form ∂²f/∂z² = (1/v²) ∂²f/∂t²
Comparing terms: v² = 1/(μ₀ε₀)
Therefore the wave speed is: v = 1/√(μ₀ε₀) = c = 3 × 10⁸ m/s
The speed of light falls out of two lab-measurable constants. This was Maxwell's great triumph.
Quantity | Formula |
|---|---|
Real current | I = dQ/dt |
Displacement current | I_D = ε₀ dΦ_E/dt |
Electric flux (uniform field) | Φ_E = EA |
Ampere-Maxwell Law | ∮ B · dl = μ₀(I + I_D) |
Speed of light from constants | c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s |
Maxwell's wave equation | ∂²E_x/∂z² = μ₀ε₀ ∂²E_x/∂t² |
The displacement current concept is why radio transmitters work. An oscillating voltage across an antenna creates a rapidly changing electric field, which generates a changing magnetic field, which regenerates the electric field, and so on, launching an electromagnetic wave into space. Without the displacement current term, Maxwell's Equations would not predict radiation, and we would have no theoretical basis for wireless communication, radar, or microwave ovens.
"Displacement current is a flow of charge." It is not. No charges move through the vacuum between capacitor plates. The name is historical; it behaves like a current in Ampere's Law but involves only a changing electric field.
"The electric field between capacitor plates is static while charging." The field is growing as charge accumulates. That time-varying field is exactly what produces the displacement current.
"Maxwell's wave equation was an assumption." It was not assumed. It is derived directly from the four equations. The speed c = 1/√(μ₀ε₀) is a prediction, not an input.
"Faraday's Law and Ampere's Law are unrelated." After Maxwell's correction they are symmetric partners: each says that a changing field of one type produces the other type.
⚠️ You must be able to derive I_D = ε₀ dΦ_E/dt starting from E = Q/(ε₀A) for a parallel-plate capacitor.
⚠️ Know all four Maxwell's Equations in integral form and be able to identify each by name.
⚠️ Be prepared to show how c = 1/√(μ₀ε₀) follows from the wave equation, and to compute c from given values of μ₀ and ε₀.
⚠️ Expect a question where you must explain why original Ampere's Law fails for a capacitor and how the displacement current fixes it.
True or False: Displacement current involves the physical movement of charge across a gap. (False)
Fill in the blank: The displacement current is defined as I_D = ____. (ε₀ dΦ_E/dt)
True or False: Gauss's Law for magnetic fields states that the net magnetic flux through a closed surface is always zero. (True)
Fill in the blank: The speed of electromagnetic waves equals 1/√(____). (μ₀ε₀)
True or False: Maxwell's wave equation was an experimental observation, not a theoretical derivation. (False)
Q: A parallel-plate capacitor is being charged. Explain why the original form of Ampere's Law gives an inconsistent result, and how the displacement current resolves this.
A: If you draw an Amperian loop in the gap between the plates, no conduction current passes through the enclosed surface, so the original law predicts B = 0. However, a magnetic field does exist there. The resolution is that the electric field between the plates is changing with time, producing a displacement current I_D = ε₀ dΦ_E/dt that enters the modified Ampere's Law and restores the correct B field.
Q: Derive the speed of electromagnetic waves from Maxwell's wave equation.
A: The wave equation is ∂²E_x/∂z² = μ₀ε₀ ∂²E_x/∂t². Comparing with the standard form ∂²f/∂z² = (1/v²) ∂²f/∂t², we identify 1/v² = μ₀ε₀, giving v = 1/√(μ₀ε₀). Substituting known values yields v ≈ 3 × 10⁸ m/s = c.
Q: Write down all four of Maxwell's Equations in integral form and name each one.
A: (1) Gauss's Law for E: ∮ E · dA = Q_enc/ε₀. (2) Gauss's Law for B: ∮ B · dA = 0. (3) Faraday's Law: ∮ E · dl = −dΦ_B/dt. (4) Ampere-Maxwell Law: ∮ B · dl = μ₀(I + ε₀ dΦ_E/dt).
Q: What is the physical significance of the symmetry between Faraday's Law and the Ampere-Maxwell Law?
A: Faraday's Law says a changing magnetic field produces an electric field; the Ampere-Maxwell Law says a changing electric field produces a magnetic field. Together, these allow self-sustaining oscillations: each field regenerates the other, producing a wave that propagates through free space.
This material connects directly to the electromagnetic spectrum and the properties of EM waves (covered in the companion notes). The displacement current concept also links back to capacitor physics from earlier in the course; the same Q = CV and I = dQ/dt relationships reappear here in a new context. In more advanced courses, Maxwell's Equations in differential form become the starting point for special relativity and quantum electrodynamics.
displacement current, Maxwell's Equations, Ampere-Maxwell Law, modified Ampere's Law, electric flux, magnetic flux, Gauss's Law, Faraday's Law, electromagnetic wave derivation, wave equation, speed of light, ε₀, μ₀, permittivity, permeability, capacitor gap, PHY 212, UIUC physics, midterm 4 review