Our starting points along with the material relation are
∇⋅B∇×HB=0=J=μH
Under the assumption of no current flow (J=0) Gauss’s law for magnetism and Ampere’s law state
∇⋅B∇×H=0=0
The curl of H is zero, hence H is a conservative field.
Meaning there exists a scalar function φ such that
H=−∇φ
Here, φ is called the magnetic scalar potential.
Substituting (5) and the constitutive relation B=μH in Gauss’s law for magnetism we get
∇⋅μH∇⋅μ∇φ=0=0
Resulting in Laplace’s equation.
The partial differential equation is multiplied by the test function φ′ and integrated over the entire domain Ω.
∫Ω∇⋅(μ∇φ)φ′dΩ=0
Using the Leibniz rule for nabla operator on the divergence term we get
∫Ω∇⋅(φ′μ∇φ)dΩ+∫Ω−(∇φ′⋅μ∇φ)dΩ=0
Applying the divergence theorem on the divergence term we get
∫Γμ∇φ⋅n φ′dΓ+∫Ω−(μ∇φ⋅∇φ′)dΩ=0
Rearranging the terms and using relation B=μH=−μ∇φ on the Neumann term, we obtain
∫Ω(−μ∇φ⋅∇φ′)dΩ+∫Γ(−B⋅n)φ′dΓ=0