Total current through surface, H field

The current through a surface equals the flux of the current density J = ∇×H, or equivalently the circulation of H around the boundary of the surface (Stokes' theorem). The surface is part of the cylinder ρ = 2, so we need the a_ρ component of J = ∇×H.

H = (2ρ + z) a_ρ + (2/z) a_z, i.e. H_ρ = 2ρ + z, H_φ = 0, H_z = 2/z.

Step 1: a_ρ component of the curl.

(∇×H)_ρ = (1/ρ) ∂H_z/∂φ − ∂H_φ/∂z
        = (1/ρ)(0) − 0
        = 0

(H_z = 2/z does not depend on φ, and H_φ = 0.)

Step 2: current.

I = ∬ J_ρ dS = ∫₃⁵ ∫_{π/4}^{π/2} 0 · ρ dφ dz = 0

Step 3: check with the line integral ∮ H·dl around the boundary of the patch (corners at φ = π/4, π/2 and z = 3, 5, all at ρ = 2):

Along the φ-edges (z = 3 and z = 5): H_φ = 0 ⇒ contribution 0.
Along the z-edges (φ = π/4 and φ = π/2): H_z = 2/z, same function on both
edges but traversed in opposite directions ⇒ ∫₃⁵ (2/z) dz − ∫₃⁵ (2/z) dz = 0.

Both methods agree.

Answer: I = 0 A