Difficulty: Intermediate | Prerequisites: Introductory electrostatics, Gauss's law, Faraday's law, basic calculus (derivatives and integrals).
This topic sits at the boundary between classical electrostatics/magnetostatics and the full dynamic picture of electromagnetism. Maxwell noticed that Ampere's law was incomplete for time-varying fields and added the displacement current term, which completed the set of equations that predict electromagnetic waves. If you are comfortable with electric flux, magnetic fields from currents, and Faraday's law of induction, you have the pieces you need. The payoff is large: this is the theoretical foundation for light, radio, and every other electromagnetic wave.
Maxwell's addition of the displacement current to Ampere's law completed the theoretical picture and showed that changing electric and magnetic fields sustain each other as waves travelling at the speed of light. Those electromagnetic waves carry energy (described by the Poynting vector and intensity) and exhibit a relativistic Doppler shift when source and observer move relative to each other.
Displacement current (I_D)
The term ε₀ dΦ_E/dt that Maxwell added to Ampere's law. It is not a flow of charge; it is the effect of a time-changing electric flux behaving, for the purposes of generating a magnetic field, as though a current were present.
In simple terms: when the electric field between capacitor plates changes over time, it produces a magnetic field just as a real current would.
Ampere-Maxwell law
The generalised form of Ampere's law: ∮ B · dl = μ₀(I + I_D). It accounts for both conduction current and displacement current as sources of magnetic fields.
Think of it as: the full version of Ampere's law that works even when the electric field is changing.
Harmonic plane wave
An electromagnetic wave described by h(x, t) = A cos(kx − ωt), where k = 2π/λ is the wave number and ω = 2π/T is the angular frequency. The electric and magnetic fields oscillate sinusoidally, in phase, perpendicular to each other and to the direction of propagation.
Poynting vector (S)
Defined as S = (E × B) / μ₀ = cε₀E². It gives the direction and magnitude of electromagnetic energy flow per unit area per unit time.
In simple terms: it tells you which way the wave's energy is travelling and how much energy passes through a given area each second.
Intensity (I)
The time-averaged power per unit area delivered by an electromagnetic wave: I = ⟨Power⟩ / area = ½ cε₀E₀². Measured in watts per square metre.
Energy density (u)
The energy stored per unit volume in the electromagnetic field. The electric part is u_E = ½ ε₀E², the magnetic part is u_B = ½ B²/μ₀, and the total instantaneous density is u = ε₀E². The time-averaged total is ⟨u⟩ = ½ ε₀E₀².
Doppler shift (electromagnetic)
The change in observed frequency of an EM wave when source and observer move relative to each other. For decreasing separation: f' = f √((1+β)/(1−β)). For increasing separation: f' = f √((1−β)/(1+β)). Here β = v/c.
In simple terms: approaching objects see higher frequencies (blueshift), receding objects see lower frequencies (redshift), analogous to the sound Doppler effect but with a relativistic formula.
Ampere's original law (∮ B · dl = μ₀I) fails for time-varying electric fields, e.g. the gap between capacitor plates during charging.
Maxwell introduced the displacement current I_D = ε₀ dΦ_E/dt to fix this.
The corrected law becomes ∮ B · dl = μ₀(I + I_D), meaning a changing electric flux generates a magnetic field even where no charges flow.
This symmetry (changing E makes B, changing B makes E via Faraday's law) is what allows self-sustaining electromagnetic waves.
General form: h(x, t) = A cos(kx − ωt)
k = 2π/λ (wave number)
ω = 2π/T (angular frequency)
For an EM wave propagating in the z-direction:
E_x = E₀ cos(kz − ωt)
B_y = (k/ω) E₀ cos(kz − ωt)
The ratio gives B₀ = E₀/c
E and B are perpendicular to each other and to the direction of travel.
The wave speed in vacuum is c = ω/k = 1/√(μ₀ε₀).
Unlike the acoustic Doppler effect, the EM version is relativistic and depends only on relative velocity, not on which party is moving.
Source and observer approaching (decreasing separation): f' = f √((1+β)/(1−β))
Source and observer receding (increasing separation): f' = f √((1−β)/(1+β))
β = v/c, where v is the relative speed.
Electric energy density: u_E = ½ ε₀E²
Magnetic energy density: u_B = ½ B²/μ₀
In an EM wave, these two are equal at every instant.
Total instantaneous energy density: u = ε₀E²
Time-averaged energy density: ⟨u⟩ = ½ ε₀E₀²
The Poynting vector S = (E × B)/μ₀ = cε₀E² points in the direction of wave propagation and has units of W/m².
Intensity is the time-averaged magnitude of S: I = ⟨Power⟩/area = ½ cε₀E₀².
To connect intensity to energy density: I = ⟨u⟩ · c (the energy density times the speed at which it sweeps past).
Displacement current: I_D = ε₀ dΦ_E/dt
Ampere-Maxwell law: ∮ B · dl = μ₀(I + I_D)
Plane wave: h(x,t) = A cos(kx − ωt), with k = 2π/λ, ω = 2π/T
EM wave fields: E_x = E₀ cos(kz − ωt), B_y = (E₀/c) cos(kz − ωt)
Amplitude relation: B₀ = E₀/c
Electric energy density: u_E = ½ ε₀E²
Magnetic energy density: u_B = ½ B²/μ₀
Total instantaneous energy density: u = ε₀E²
Time-averaged energy density: ⟨u⟩ = ½ ε₀E₀²
Poynting vector: S = (E × B)/μ₀ = cε₀E²
Intensity: I = ½ cε₀E₀²
Doppler (approaching): f' = f √((1+β)/(1−β)), β = v/c
Doppler (receding): f' = f √((1−β)/(1+β))
Displacement current is the reason capacitors work in AC circuits: even though no charge crosses the gap, the changing electric flux maintains a continuous magnetic field around the circuit. Every wireless technology, from radio to Wi-Fi to satellite communication, depends on the self-sustaining EM waves that Maxwell's equations predict. The Doppler shift of EM waves is the principle behind police radar guns and the astronomical measurement of stellar velocities (redshift/blueshift).
Students often think displacement current involves actual charge carriers moving through the dielectric. It does not; it is purely the effect of a changing electric flux.
A common error is applying the acoustic Doppler formula (with separate source/observer terms) to light. The EM Doppler effect is relativistic and depends only on relative velocity.
Students sometimes assume u_E and u_B are unequal in a wave. In a plane EM wave in vacuum, they are always equal.
Confusing intensity (time-averaged power per unit area) with instantaneous energy density. Intensity involves a time average and a factor of c; energy density does not.
⚠️ Be ready to compute the displacement current through a capacitor given a rate of change of electric flux.
⚠️ Know how to relate E₀ and B₀ via B₀ = E₀/c and use that relationship in energy/intensity calculations.
⚠️ The Doppler formulas look similar; know which square-root expression goes with approaching vs receding.
⚠️ Intensity problems often give you E₀ and ask for I, or vice versa. Drill the formula I = ½ cε₀E₀² until it is automatic.
⚠️ Expect at least one question connecting Poynting vector direction to the cross product E × B.
True or false: displacement current involves the physical movement of charges across a capacitor gap.
Fill in the blank: in an EM wave, B₀ = E₀ / ______.
True or false: the electric and magnetic energy densities in a plane EM wave are equal.
Fill in the blank: when source and observer approach each other, the observed frequency is ______ (higher/lower) than the emitted frequency.
True or false: the Poynting vector points in the direction of wave propagation.
Q: A parallel-plate capacitor has a uniform electric field that is increasing at a rate dE/dt = 2.0 × 10¹⁰ V/(m·s). The plate area is 0.04 m². What is the displacement current?
A: I_D = ε₀ × A × dE/dt = (8.85 × 10⁻¹²)(0.04)(2.0 × 10¹⁰) ≈ 7.1 × 10⁻³ A (about 7.1 mA).
Q: An EM wave has an electric field amplitude E₀ = 600 V/m. What is B₀?
A: B₀ = E₀/c = 600 / (3 × 10⁸) = 2.0 × 10⁻⁶ T.
Q: Calculate the intensity of an EM wave with E₀ = 600 V/m.
A: I = ½ cε₀E₀² = ½ (3 × 10⁸)(8.85 × 10⁻¹²)(600²) ≈ 478 W/m².
Q: A star is moving away from Earth at 0.1c. If it emits light at frequency f, what frequency does an Earth observer measure?
A: f' = f √((1 − 0.1)/(1 + 0.1)) = f √(0.9/1.1) ≈ 0.905 f. The observed frequency is about 9.5% lower (redshifted).
Q: In which direction does the Poynting vector point for a wave with E in the x-direction and B in the y-direction?
A: S = (E × B)/μ₀. x̂ × ŷ = ẑ, so the wave propagates in the +z direction.
Displacement current completes Maxwell's equations, which unify everything from Coulomb's law to Faraday's law into one framework. The energy carried by EM waves connects directly to the radiation pressure and momentum of photons covered in modern physics. The Doppler shift formulae here are a direct application of special relativity, bridging classical E&M into relativistic kinematics.
Displacement current, Maxwell's equations, Ampere-Maxwell law, electromagnetic waves, EM waves, harmonic plane wave, wave number, angular frequency, Poynting vector, energy flux, intensity of EM wave, energy density of electromagnetic field, Doppler effect for light, relativistic Doppler shift, blueshift, redshift, E₀ and B₀ relationship, time-averaged energy density, PHYS 212, University Physics Electricity and Magnetism