Magnetic tension

In ideal magnetohydrodynamics (MHD) the magnetic tension force in an electrically conducting fluid with a bulk plasma velocity field ${\displaystyle \mathbf {v} }$, current density ${\displaystyle \mathbf {J} }$, mass density ${\displaystyle \rho }$, magnetic field ${\displaystyle \mathbf {B} }$, and plasma pressure ${\displaystyle p}$ can be derived from the Cauchy momentum equation:

${\displaystyle \rho \left({\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla \right)\mathbf {v} =\mathbf {J} \times \mathbf {B} -\nabla p,}$

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, ${\displaystyle \mu _{0}\mathbf {J} =\nabla \times \mathbf {B} }$, and the vector identity

${\displaystyle {\tfrac {1}{2}}\nabla (\mathbf {B} \cdot \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {B} +\mathbf {B} \times (\nabla \times \mathbf {B} )}$

to give

${\displaystyle \mathbf {J} \times \mathbf {B} ={(\mathbf {B} \cdot \nabla )\mathbf {B} \over \mu _{0}}-\nabla \left({\frac {B^{2}}{2\mu _{0}}}\right),}$

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 ${\displaystyle \mathbf {B} }$ and its direction can be separated by writing ${\displaystyle \mathbf {B} =B\mathbf {b} }$ with ${\displaystyle B=|\mathbf {B} |}$ and ${\displaystyle \mathbf {b} }$ a unit vector:

${\displaystyle {(\mathbf {B} \cdot \nabla )\mathbf {B} \over \mu _{0}}={\frac {B^{2}}{\mu _{0}}}(\mathbf {b} \cdot \nabla )\mathbf {b} ={\frac {B^{2}}{\mu _{0}}}{\boldsymbol {\kappa }}}$

where the spatial constancy of the magnitude has been assumed ${\displaystyle \nabla B=0}$ and

${\displaystyle {\boldsymbol {\kappa }}=(\mathbf {b} \cdot \nabla )\mathbf {b} }$

has magnitude equal to the curvature, or the reciprocal of the radius of curvature, and is directed from a point on a magnetic field line to the center of curvature. Therefore, as the curvature of the magnetic field line increases, so too does the magnetic tension force resisting this curvature.[2][1]

Magnetic tension and pressure are both implicitly included in the Maxwell stress tensor. Terms representing these two forces are present along the main diagonal where they act on differential area elements normal to the corresponding axis.

Magnetic tension is particularly important in plasma physics and MHD, where it controls dynamics of some systems and the shape of magnetic structures. For example, in a homogeneous magnetic field and an absence of gravity, magnetic tension is the sole driver of linear Alfvén waves.[3]

• Magnetic pinch
• Magnetosonic wave
• List of plasma (physics) articles