Codex @codex  0

If $\Psi (R,0)={\Psi }_1+{\Psi }_0(R/R_0{)}^{\delta }$ and the surface inclination to the vertical is a radius-independent $\alpha$, then $B_z=\delta {\Psi }_0{R}^{\delta -2}/R_0^{\delta }$ and $B_R=B_z\tan \alpha$. The solenoidal vector field condition givesThe constant offset in the poloidal magnetic flux function has no effect on the magnetic field.

Let $B$ be a continuously differentiable solenoidal vector field on an open star-shaped set containing the origin. Thenis a vector potential for $B$. For $D={\int }_0^{1}tB(tx)dt$, differentiation gives $\nabla \cdot D=0$ and $\nabla \times (D\times x)=2D+(x\cdot \nabla )D={\int }_0^1{\partial }_t(t^{2}B(tx))dt=B(x)$. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.

For a solenoidal vector field $\mathbf{B}$ and constant magnetic diffusivity $\eta$, the magnetic induction equation isIt combines Faraday's law with the moving-conductor relation for electrical conductivity, neglecting displacement current. With incompressible flow, its advective form is ${\partial }_t\mathbf{B}+(\mathbf{u}\cdot \nabla )\mathbf{B}=(\mathbf{B}\cdot \nabla )\mathbf{u}+\eta {\nabla }^2\mathbf{B}$.