Scalar potential, Laplace, magnetic flux

Scalar magnetic potential

In a region where there is no current (J = 0), Ampere's law gives ∇×H = 0. Since the curl of the gradient of any scalar is zero, H can be written as the negative gradient of a scalar function V_m:

H = −∇V_m

V_m is the scalar magnetic potential. It is the work done per unit pole strength in moving a magnetic pole between two points: V_m(ab) = −∫ H·dl. Its unit is the ampere (A). It is valid only in current-free regions, and is multi-valued if the path encircles a current.

Proof that ∇²V_m = 0

From the magnetic Gauss law, ∇·B = 0. In a linear homogeneous medium B = μH:

∇·B = μ ∇·H = 0  ⇒  ∇·H = 0

Substitute H = −∇V_m:

∇·(−∇V_m) = 0
⇒ ∇²V_m = 0

So the scalar magnetic potential satisfies Laplace's equation, exactly like the electrostatic potential in a charge-free region. The magnetic flux through a surface can then be found from B = −μ∇V_m, using the boundary conditions to solve for V_m.