3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A)
Related Concepts: 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation | 3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable) | 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect
3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A)
Overview
Ampere’s Circuital Law provides a rapid algebraic method to calculate magnetic fields for symmetric current distributions. For general non-symmetric fields, Vector Magnetic Potential A serves as the magnetic analogue to electric potential V.
1. Vector Magnetic Potential (A) Definition & Identity Proof [PYQ: 2016, 2018, 2019, 2021]
Since ∇ ⋅ B = 0, and the divergence of any vector curl is identically zero (∇ ⋅ (∇ × A) ≡ 0), we define A as:
B = ∇ × A [Wb/m² or Tesla]
2. Vector Poisson’s Equation (∇² A = -μ₀ J) [PYQ: 2019, 2024]
Substitute B = μ₀ H and B = ∇ × A into Ampere’s postulate ∇ × H = J:
∇ × (∇ × A) = μ₀ J
Using vector identity ∇ × (∇ × A) = ∇ (∇ ⋅ A) - ∇² A, and choosing the Coulomb Gauge (∇ ⋅ A = 0):
∇² A = -μ₀ J (Vector Poisson’s Equation)
2.1 Integral Expressions for A
- Volume Current (J): A = (μ₀ / 4π) ∫_V (J / R) dv’
- Line Current (I): A = (μ₀ I / 4π) ∫_L (dl’ / R)