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Scalar vs vector magnetic potential

Short answer

Scalar magnetic potential V_m satisfies H = −∇V_m and applies only in current-free regions. Vector magnetic potential A satisfies B = ∇×A everywhere, governed by ∇²A = −μJ, and has units of Wb/m.

Differentiate between scalar and vector magnetic potential.

Both potentials are defined to find H or B more easily. The scalar magnetic potential V_m is defined only where J = 0, with H = −∇V_m. The vector magnetic potential A is defined everywhere, with B = ∇×A.

PointScalar magnetic potential V_mVector magnetic potential A
DefinitionH = −∇V_mB = ∇×A
NatureScalar (amp)Vector (Wb/m)
Region of validityOnly where J = 0 (∇×H = 0)Everywhere, with or without current
Governing equation∇²V_m = 0 (since ∇·B = 0)∇²A = −μJ
SourceCurrent-free regionCurrent density J
Single-valued?Not necessarily (multi-valued around a current)Yes
ExpressionV_m = −∫H·dlA = (μ/4π)∫ J dv / R
UnitA (ampere)Wb/m

Note that B = ∇×A is valid because ∇·B = 0, and ∇·(∇×A) = 0 always.

✓ Answer checked by Claude Opus 5.5 (high effort)

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