# Scalar potential for magnetic field

## Main Question or Discussion Point

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?

Related Classical Physics News on Phys.org
vanhees71
Gold Member
2019 Award
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.