B²/2μ₀ — the pressure exerted by a magnetic field on a conducting plasma, providing the confining force that keeps hundred-million-degree fuel away from material walls
A magnetic field in the presence of a conducting medium exerts a pressure given by pmag = B²/(2μ0), where B is the magnetic field strength and μ0 is the permeability of free space. In SI units, a 1 T field exerts roughly 0.4 MPa — about four atmospheres. At the 5.3 T on-axis field of ITER, the magnetic pressure reaches approximately 11 MPa, comparable to the pressure at the bottom of the Mariana Trench.[1]
This pressure acts perpendicular to the field lines and tends to push the field lines apart, analogous to the pressure inside an inflated balloon. In a plasma confinement device the magnetic pressure must exceed the kinetic (thermal) pressure of the plasma to maintain equilibrium and prevent the hot fuel from expanding into the vessel wall.
The ratio of plasma kinetic pressure to magnetic pressure is called beta (β): β = 2μ0pkin/B². Beta is the single most important dimensionless figure of merit for magnetic confinement efficiency. A higher beta means the device extracts more confinement from a given magnet investment, which translates directly into smaller, cheaper reactors. Conventional tokamaks typically operate at β ≈ 2–5%, while spherical tokamaks and field-reversed configurations can reach β values exceeding 20%.[2]
In a confined plasma the total pressure balance at every point must satisfy the MHD equilibrium condition: ∇p = J × B. Decomposing the magnetic force into its pressure-gradient and tension components shows that the magnetic pressure gradient pushes the plasma inward while the magnetic tension along curved field lines provides an additional restoring force. Both effects contribute to confinement, but it is the pressure term that dominates in regions of strong field gradient, such as the outboard midplane of a tokamak.[3]
The enormous magnetic pressures in fusion devices impose severe structural demands on the magnet support structures. ITER’s toroidal field coils must withstand cumulative electromagnetic forces exceeding 400 MN — the weight of an aircraft carrier — transmitted through massive intercoil structures and a gravity support system designed for the lifetime of the machine.[4]