Magnetic tension force

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The magnetic tension force is a tension that exists along magnetic field lines. It is due to the existence of magnetic pressure and the linear nature of the magnetic field. The magnetic field has an energy density; this takes the form of a pressure transverse to the field, because compressing field lines close to one another is equivalent to strengthening the magnetic field. However, shortening a field line reduces the total volume of space occupied by strong magnetic field, without changing the field strength in the rest of the field. Interplay between magnetic tension and magnetic pressure is responsible for the shape of magnetic field lines in the absence of constraining forces.

Magnetic tension is particularly important in plasma physics and magnetohydrodynamics, where it controls dynamics of some systems and the shape of magnetized structures. In magnetohydrodynamics, the magnetic tension force can be derived from the momentum equation

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

using the relation $\mu_0\mathbf{J}=\nabla\times\mathbf{B}$ . The first term on the right hand side of the above equation represents electromagnetic forces and the second term represents pressure gradient forces. From this we see that

$\mathbf{J}\times\mathbf{B}= \frac{1}{\mu_0} \left(\nabla\times\mathbf{B}\right)\times\mathbf{B} = \frac{\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}}{\mu_0} -\nabla\left(\frac{B^2}{2\mu_0}\right)$

The first term on the right hand side represents magnetic tension forces, and the second term on the right hand side represents magnetic pressure forces.