Magnetic flux from vector potential

Magnetic flux density follows from the vector potential as B = ∇×A, and the flux through a surface is Ψ = ∫ B·dS.

Step 1: find B. A = A_z a_z with A_z = −ρ²/4, a function of ρ only. In cylindrical coordinates:

B = ∇×A = (1/ρ ∂A_z/∂φ) a_ρ − (∂A_z/∂ρ) a_φ
        = −∂/∂ρ (−ρ²/4) a_φ
        = (ρ/2) a_φ   Wb/m²

Step 2: surface. The surface φ = π/2 is a plane (the yz-plane, y > 0) with 1 ≤ ρ ≤ 2 and 0 ≤ z ≤ 5. Its normal is a_φ and dS = dρ dz a_φ.

Step 3: integrate.

Ψ = ∫₀⁵ ∫₁² (ρ/2) a_φ · (dρ dz a_φ)
  = ∫₀⁵ dz × ∫₁² (ρ/2) dρ
  = 5 × [ρ²/4]₁²
  = 5 × (4 − 1)/4
  = 5 × 0.75
  = 3.75 Wb

Answer: Ψ = 3.75 Wb (in the +a_φ direction).