- #1

ergospherical

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- Homework Statement
- Show that a force-free, axisymmetric field has ##B_{\phi} = f(\psi)/R## in cylindrical polars, where ##f## is an arbitrary function and ##\psi(R,z) = rA_{\phi}(R,z)## is the poloidal flux function, and find the equation satisfied by ##\psi##.

- Relevant Equations
- ##-\nabla^2 \mathbf{B} = \lambda^2 \mathbf{B}##

Force free: ##\mathbf{J} \times \mathbf{B} \sim (\nabla \times \mathbf{B}) \times \mathbf{B} = 0##

(N.B. MHD applies so ##\epsilon_0|\partial \mathbf{E}/\partial t|/|\mathbf{J}| \ll 1##).

Axisymmetric: can write ##\mathbf{B} = \nabla \psi \times \nabla \phi + B_{\phi} \mathbf{e}_{\phi}##

(##\phi## is the azimuthal coordinate i.e. ##\nabla \phi = \mathbf{e}_{\phi}/r##)

Inserting into ##(\nabla \times \mathbf{B}) \times \mathbf{B} = 0## gives a bit of a mess. Is there an easier route?

(N.B. MHD applies so ##\epsilon_0|\partial \mathbf{E}/\partial t|/|\mathbf{J}| \ll 1##).

Axisymmetric: can write ##\mathbf{B} = \nabla \psi \times \nabla \phi + B_{\phi} \mathbf{e}_{\phi}##

(##\phi## is the azimuthal coordinate i.e. ##\nabla \phi = \mathbf{e}_{\phi}/r##)

Inserting into ##(\nabla \times \mathbf{B}) \times \mathbf{B} = 0## gives a bit of a mess. Is there an easier route?