Magnetic pressure

Magnetic pressure is an energy density associated with a magnetic field. 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.

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 cgs) F exerted on a coil by its own current is

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

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.

If any currents present are parallel to a magnetic field, the field lines follow shapes in which the magnetic pressure gradient is balanced by the magnetic tension force. Such a field configuration is called force-free because there is no Lorentz force ($j\times B=0$). The familiar potential magnetic field is a special case of a force-free field: potential field configurations occupy space that contains no electric current at all.

The magnetic pressure $P_{B}$ is given in SI units (P in Pa, B in T, μ₀ in H/m) by

$P_{B}={\frac {B^{2}}{2\mu _{0}}}$

and in cgs units (P in dyn/cm², B in G) by

$P_{B}={\frac {B^{2}}{8\pi }}$.

In practical units,

$P_{B}{\text{ [bar]}}\approx \left({\frac {B{\text{ [T]}}}{0.501}}\right)^{2}.$

See also

• Magnetic tension force
• Maxwell stress tensor
• Electromagnetically induced acoustic noise and vibration

References

• Garren & Chen (1994). "Lorentz Self Forces on Curved Current Loops". Physics of Plasmas. 1 (10): 3425–3436. Bibcode:1994PhPl....1.3425G. doi:10.1063/1.870491.