Frequency where conduction equals displacement

Displacement current (2 marks)

Displacement current is the current that exists in a dielectric or free space due to a time-varying electric flux density. Its density is J_d = ∂D/∂t = ε ∂E/∂t (A/m²). It was introduced by Maxwell into Ampere's law, ∇×H = J + ∂D/∂t, to make it consistent with the continuity of charge. It produces a magnetic field like a conduction current.

Numerical (5 marks)

Take E = E_m cos ωt (any time-harmonic field). Then:

J_c = σ E             → magnitude σ E_m
J_d = ε ∂E/∂t         → magnitude ω ε E_m

Ratio of magnitudes (this is also the loss tangent):

|J_c| / |J_d| = σ / (ω ε)

For the ratio to be unity: σ = ω ε = 2π f ε, so

f = σ / (2π ε)

Given: σ = 10⁻⁴ S/m, ε = 3ε₀ = 3 × 8.854×10⁻¹² = 2.656×10⁻¹¹ F/m.

f = 10⁻⁴ / (2π × 2.656×10⁻¹¹)
  = 10⁻⁴ / (1.669×10⁻¹⁰)
  = 5.99×10⁵ Hz

Answer: f ≈ 599 kHz (about 0.6 MHz). Below this frequency, conduction current dominates (soil behaves like a conductor); above it, displacement current dominates (soil behaves like a dielectric).