Scalar potential for magnetic field

[Mr. Rho]

I have that ∇²∅ = 0 everywhere. ∅ is a scalar potential and must be finite everywhere.
Why is it that ∅ must be a constant?

I'm trying to understand magnetic field B in terms of the Debye potentials: B = Lψ+Lχ+∇∅. I get this from C.G.Gray, Am. J. Phys. 46 (1978) page 169. Here they found that Lχ=0 and ∇∅=0 and therefore gives no contribution to B.

any help?

 

[vanhees71]

First of all the Debye decomposition of an arbitrary vector field is given as
$$\vec{V}=\vec{L} \psi + \vec{\nabla} \times (\vec{L} \chi)+\vec{\nabla} \phi.$$
By definition
$$\vec{L}=\vec{x} \times \vec{\nabla}.$$
First of all you have
$$\vec{\nabla} \cdot \vec{V}=0,$$
because
$$\vec{\nabla} \cdot (\vec{L} \psi)=\partial_j (\epsilon_{jkl} r_k \partial_l \psi)=\epsilon_{jkl} (\delta_{jk} \partial_l \psi +r_k \partial_j \partial_l \psi)=0$$
and
$$\vec{\nabla} \cdot (\vec{\nabla} \times \vec{L} \chi)=0.$$
For the magnetic field you additionally have the absence of magnetic charges,
$$\vec{\nabla} \cdot \vec{B}=0.$$
This implies
$$\Delta \phi=0$$
everywhere. With the appropriate boundary conditions this implies that $\phi=0$ everywhere.