It is true (but not obvious) which any vector field V(r) which satisfies ∇ · V(r) = 0 can be written uniquely in the form

where L = −ir × ∇ is the angular momentum operator and ψ(r) and γ (r) are scalar fields. T(r) = Lψ(r) is called a toroidal field and P(r) = ∇ × Lγ (r) is called a poloidal field. This decomposition is widely used in laboratory plasma physics.

(a) Confirm that ∇ · V(r) = 0.
(b) Show that a poloidal current density generates a toroidal magnetic field and vice versa.
(c) Show that B(r) is toroidal for a toroidal solenoid.
(d) Suppose there is no current in a finite volume V . Show that ∇²B(r) = 0 in V.
(e) Show that A(r) in the Coulomb gauge is purely toroidal in V when ψ(r) and γ (r) are chosen so that ∇²B(r) = 0 in V.

V(r) = T(r) + P(r) = L(r) + V x Ly(r),