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Concept# Magnetic pressure

Summary

In physics, magnetic pressure is an energy density associated with a magnetic field. In SI units, the energy density of a magnetic field with strength can be expressed as
where is the vacuum permeability.
Any magnetic field has an associated magnetic pressure contained by the boundary conditions on the field. It is identical to any other physical pressure except that it is carried by the magnetic field rather than (in the case of a gas) by the kinetic energy of gas molecules. A gradient in field strength causes a force due to the magnetic pressure gradient called the magnetic pressure force.
In SI units, the magnetic pressure in a magnetic field of strength is
where is the vacuum permeability and has units of energy density.
In ideal magnetohydrodynamics (MHD) the magnetic pressure force in an electrically conducting fluid with a bulk plasma velocity field , current density , mass density , magnetic field , and plasma pressure can be derived from the Cauchy momentum equation:
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, , and the vector identity
to give
where the first term on the right hand side is the magnetic tension and the second term is the magnetic pressure force.
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.
The magnetic pressure force is readily observed in an unsupported loop of wire. If an electric current passes through the loop, the wire serves as an electromagnet, such that the magnetic field strength inside the loop is much greater than the field strength just outside the loop. This gradient in field strength gives rise to a magnetic pressure force that tends to stretch the wire uniformly outward. If enough current travels through the wire, the loop of wire will form a circle.

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