Displacement current density in transformer

In air σ = 0, so there is no conduction current (J = 0). From Maxwell's equation ∇×H = J + ∂D/∂t, the displacement current density equals the curl of H:

J_d = ∂D/∂t = ∇×H

Given: H = H_y a_y with H_y = 10⁶ cos(377t + 1.2566×10⁻⁶ z) A/m. H depends only on z.

Step 1: curl of H.

∇×H = (∂H_z/∂y − ∂H_y/∂z) a_x + (∂H_x/∂z − ∂H_z/∂x) a_y + (∂H_y/∂x − ∂H_x/∂y) a_z

Only H_y(z) exists, so only the a_x component survives:

∇×H = −∂H_y/∂z a_x
    = −10⁶ × (−sin(377t + 1.2566×10⁻⁶ z)) × 1.2566×10⁻⁶ a_x
    = 1.2566 sin(377t + 1.2566×10⁻⁶ z) a_x   A/m²

Step 2: interpretation. In air the conduction part is zero, so the whole of ∇×H is displacement current. Since sin θ = cos(θ − 90°), J_d lags H by 90° in time.

Answer: J_d = 1.2566 sin(377t + 1.2566×10⁻⁶ z) a_x A/m², so the amplitude is 1.2566 A/m² (about 1.26 A/m²).