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{{Short description|Energy density associated with a magnetic field}}
[[File:Magnetic pressure force diagram.svg|thumb|Gradients in magnetic field strength result in a magnetic pressure force perpendicular to the magnetic field in the direction of decreasing magnetic field strength.]]
In [[physics]], '''magnetic pressure''' is an [[energy density]] associated with a [[magnetic field]]. In [[SI]] units, the energy density <math>P_B</math> of a magnetic field with strength <math>B</math> can be expressed as
:<math>P_B = \frac{B^2}{2\mu_0}</math>
where <math>\mu_0</math> 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'''.

==Mathematical statement==
In [[SI]] units, the magnetic pressure <math>P_B</math> in a magnetic field of strength <math>B</math> is
:<math>P_B = \frac{B^2}{2\mu_0}</math>
where <math>\mu_0</math> is the [[vacuum permeability]] and <math>P_B</math> has units of [[energy density]].

===Magnetic pressure force===
In ideal [[magnetohydrodynamics]] (MHD) the magnetic pressure force in an electrically conducting fluid with a bulk plasma [[velocity field]] <math>\mathbf{v}</math>, [[current density]] <math>\mathbf{J}</math>, [[mass density]] <math>\rho</math>, magnetic field <math>\mathbf{B}</math>, and plasma [[pressure]] <math>p</math> can be derived from the [[Cauchy momentum equation]]:
:<math> \rho\left(\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla\right)\mathbf{v} = \mathbf{J}\times\mathbf{B} - \nabla p,</math>
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]], <math>\mu_0\mathbf{J} = \nabla \times \mathbf{B}</math>, and the vector identity
:<math> \tfrac12\nabla(\mathbf{B}\cdot \mathbf{B})=(\mathbf{B}\cdot\nabla)\mathbf{B}+\mathbf{B}\times(\nabla\times \mathbf{B})</math>
to give
:<math> \mathbf{J} \times \mathbf{B} = {(\mathbf{B} \cdot \nabla)\mathbf{B} \over \mu_0} - \nabla\left(\frac{B^2}{2\mu_0}\right), </math>
where the first term on the right hand side is the [[magnetic tension]] and the second term is the magnetic pressure force.<ref>{{cite book |last1=Bellan |first1=Paul Murray |title=Fundamentals of Plasma Physics |date=2006 |publisher=Cambridge University Press |location=Cambridge |isbn=9780511807183 |pages=268–272}}</ref><ref name=hood>{{cite web |last1=Hood |first1=Alan |title=The Lorentz Force - Magnetic Pressure and Tension |url=http://www-solar.mcs.st-andrews.ac.uk/~alan/sun_course/Chapter2/node15.html |website=www-solar.mcs.st-andrews.ac.uk |access-date=14 May 2022}}</ref>

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.

==Wire loops==
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]]. At even higher currents, the magnetic pressure can create [[tensile stress]] that exceeds the [[tensile strength]] of the wire, causing it to fracture, or even explosively fragment. Thus, management of magnetic pressure is a significant challenge in the design of ultrastrong electromagnets.

The force (in [[Centimeter gram second system of units|cgs]]) {{math|'''F'''}} exerted on a coil by its own current is<ref>{{cite journal
| author = Garren & Chen
| title = Lorentz Self Forces on Curved Current Loops
| year = 1994
| journal = [[Physics of Plasmas]]
| volume = 1
| issue = 10
| pages = 3425–3436
| doi = 10.1063/1.870491|bibcode = 1994PhPl....1.3425G }}</ref>{{rp|3425}}

:<math>
 \mathbf{F} =\dfrac{I^2}{c^2R} \left[\ln\left(\dfrac{8R}{a}\right) - 1 + Y\right]
</math>

where ''Y'' is the internal inductance of the coil, defined by the distribution of current. ''Y'' is 0 for high frequency currents carried mostly by the outer surface of the conductor, and 0.25 for DC currents distributed evenly throughout the conductor. See [[inductance]] for more information.

Interplay between magnetic pressure and ordinary gas pressure is important to [[magnetohydrodynamics]] and [[plasma physics]]. Magnetic pressure can also be used to propel [[projectiles]]; this is the operating principle of a [[railgun]].

==Force-free fields==
{{Main|Force-free magnetic field}}
When all [[electric current]]s present in a conducting fluid are parallel to the magnetic field, the magnetic pressure gradient and magnetic tension force are balanced, and the Lorentz force vanishes. If non-magnetic forces are also neglected, the field configuration is referred to as ''[[Force-free magnetic field|force-free]]''. Furthermore, if the current density is zero, the magnetic field is the gradient of a [[magnetic scalar potential]], and the field is subsequently referred to as ''potential''.{{Citation needed|date=August 2022}}

==See also==
* [[Magnetic tension force]]
* [[Maxwell stress tensor]]
* [[Electromagnetically induced acoustic noise and vibration]]
* [[Alfvén wave#Relativistic case|Alfvén wave]]

== References ==
{{Reflist}}

[[Category:Plasma parameters]]
[[Category:Electromagnetism]]

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