On frozen-in and resistive force-free magnetic fields

Abstract

By taking the variation of the magnetic energy of a given system $\text{1}\text{8π}\text{∫}{}_{\mathrm{\upsilon }}\text{(∇ ×}\text{A}\text{)}{}^2\text{dτ}$ with the constraint that $\text{∫}{}_{\mathrm{\upsilon }}\text{A}\text{· ∇ ×}\text{A}\text{dτ =}\text{const.}$ and under the condition that the potential part ∇φ of E is defined as $\text{∇φ =}\text{c}\text{4πσ}\text{∇ ∫ α}\text{B}\text{.d1}$ which ensures that $\text{∂}\text{A}\text{∂t}\text{.}\text{B}\text{= 0}$, it will be shown that the force-free factor α is a constant for a stable magnetic field.