In physics, magnetic tension is a restoring force with units of force density that acts to straighten bent magnetic field lines. In SI units, the force density f T {\displaystyle \mathbf {f} _{T}} exerted perpendicular to a magnetic field B {\displaystyle \mathbf {B} } can be expressed as
f T = ( B ⋅ ∇ ) B μ 0 {\displaystyle \mathbf {f} _{T}={\frac {\left(\mathbf {B} \cdot \nabla \right)\mathbf {B} }{\mu _{0}}}}
where μ 0 {\displaystyle \mu _{0}} is the vacuum permeability. Magnetic tension forces also rely on vector current densities and their interaction with the magnetic field. Plotting magnetic tension along adjacent field lines can give a picture as to their divergence and convergence with respect to each other as well as current densities. Magnetic tension is analogous to the restoring force of rubber bands.
Mathematical statement In ideal magnetohydrodynamics (MHD) the magnetic tension force in an electrically conducting fluid with a bulk plasma velocity field v {\displaystyle \mathbf {v} } , current density J {\displaystyle \mathbf {J} } , mass density ρ {\displaystyle \rho } , magnetic field B {\displaystyle \mathbf {B} } , and plasma pressure p {\displaystyle p} can be derived from the Cauchy momentum equation:
ρ ( ∂ ∂ t + v ⋅ ∇ ) v = J × B − ∇ p , {\displaystyle \rho \left({\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla \right)\mathbf {v} =\mathbf {J} \times \mathbf {B} -\nabla p,}
where the first term on the right hand side represents the Lorentz force and the second term represents pressure gradient forces. The Lorentz force can be expanded using Ampère's law, μ 0 J = ∇ × B {\displaystyle \mu _{0}\mathbf {J} =\nabla \times \mathbf {B} } , and the vector identity
1 2 ∇ ( B ⋅ B ) = ( B ⋅ ∇ ) B + B × ( ∇ × B ) {\displaystyle {\tfrac {1}{2}}\nabla (\mathbf {B} \cdot \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {B} +\mathbf {B} \times (\nabla \times \mathbf {B} )}
to give
J × B = ( B ⋅ ∇ ) B μ 0 − ∇ ( B 2 2 μ 0 ) , {\displaystyle \mathbf {J} \times \mathbf {B} ={(\mathbf {B} \cdot \nabla )\mathbf {B} \over \mu _{0}}-\nabla \left({\frac {B^{2}}{2\mu _{0}}}\right),}
where the first term on the right hand side is the magnetic tension and the second term is the magnetic pressure force. The force due to changes in the magnitude of B {\displaystyle \mathbf {B} } and its direction can be separated by writing B = B b {\displaystyle \mathbf {B} =B\mathbf {b} } with B = | B | {\displaystyle B=|\mathbf {B} |} and b {\displaystyle \mathbf {b} } a unit vector:
… excerpt ends here. Continue reading the full article.
