What magnetic pressure does a 10 T field produce?
In vacuum, a 10 T field has a magnetic energy density of 39.8 MJ/m³. The same quantity sets a 39.8 MPa magnetic-pressure scale—but it is not automatically the mechanical stress in a coil.
The distinction matters when translating a field requirement into a support structure. A pressure-like energy density is one input to that calculation, not a substitute for geometry, current distribution and boundary conditions.
39.8 MPa at 10 T.
μ0 ≈ 4π × 10−7 H/m
At B = 10 T:
pB ≈ 3.9789 × 107 Pa = 39.8 MPa
Calculated. The permeability approximation is sufficient for the displayed precision. A pascal is a joule per cubic metre, so the numerical magnetic energy density is 39.8 MJ/m³. The calculation uses a vacuum or approximately nonmagnetic region, not an arbitrary nonlinear magnetic material.
| FIELD | PRESSURE SCALE | ENERGY DENSITY |
|---|---|---|
| 1 T | 0.398 MPa | 0.398 MJ/m³ |
| 10 T | 39.8 MPa | 39.8 MJ/m³ |
| 20 T | 159 MPa | 159 MJ/m³ |
| 45 T | 806 MPa | 806 MJ/m³ |
| 100 T | 3.98 GPa | 3.98 GJ/m³ |
Doubling field multiplies this scale by four. It does not, by itself, specify a component's hoop stress, yield margin or fatigue life.
Energy density is local; stored energy is an integral.
Feynman, Leighton and Sands, Volume II, Chapter 27 gives the electromagnetic field-energy density. For the magnetic term in vacuum, integrate over the actual field distribution:
Uniform-region approximation: UB ≈ uB V
A uniform 10 T field occupying an assumed 1 litre (0.001 m³) contains approximately 39.8 kJ in that region. Specifying 10 T without a volume cannot establish a gigajoule energy inventory. A real magnet also stores energy outside the selected bore region.
The magnetic part of the Maxwell stress tensor is directional: Tij = (BiBj − ½δijB²)/μ0. Normal traction at a boundary depends on which field components are normal and tangential, and on the field on both sides. The familiar pressure magnitude B²/(2μ0) applies to the corresponding field-excluding boundary idealization. A uniform field does not exert an identical outward mechanical pressure on every possible surface.
The support still needs a structural calculation.
Coil loads follow the current distribution through J × B. A thin cylindrical pressure-vessel analogy gives a hoop-stress scale σθ ≈ p r/t, where r is radius and t is wall thickness, only for the assumptions of that analogy. Balancing forces on a thin half-cylinder of length L gives 2σθtL = 2prL, hence σθ = pr/t. An actual winding is an anisotropic conductor–insulation–reinforcement assembly with end forces, prestress and contact loads.
Three missing inputs change the answer.
Geometry: bore, length, winding dimensions and end restraint. Material state: temperature, processing, prestress and defects. Waveform: peak current, rise time, dwell, reversal and repetition. Without them, no yield threshold, service lifetime or safe operating point follows from the pressure table.
A superconducting conductor also has critical-current limits depending on temperature, field magnitude, field orientation and strain. Low DC winding resistance does not remove losses in joints, current leads, changing fields or refrigeration. Resistive and hybrid magnets remain valid alternatives: the MagLab 45 T DC hybrid combines resistive and superconducting coils.
Use the pressure scale to reject incomplete specifications.
The God Magnet repeated-pulse study specifies a 100 T target. Its 3.98 GPa field-pressure scale makes reinforcement and cyclic strain first-order requirements. That number does not establish a 3.98 GPa allowable stress, nor prove that the proposed coil survives one pulse.
The same study's energy calculation states the assumed active volume and recovery fraction separately. This is the appropriate progression: field → load and energy scales → actual geometry → material and circuit models → instrumented test.
Adjacent: stored energy
A narrow bore and a large bore can have the same peak field and radically different energy inventories.
Read the bore-energy estimateAdjacent: protection
Quench behavior depends on conductor and winding architecture; a low-resistance state is not a complete protection design.
Read the 45.5 T test and quench evidenceSources used in this note.
- Feynman, Leighton & Sands — Field Energy and Field Momentum. Vacuum field-energy basis; not a coil-design calculation.
- Richard Fitzpatrick — Momentum conservation, equation 1074. Maxwell stress and the field-boundary distinction.
- National MagLab — 45 Tesla, 32 mm Bore Hybrid Magnet. Documented external apparatus; not evidence of Highfield hardware availability.
- Hahn et al. — 45.5-tesla direct-current magnetic field generated with a high-temperature superconducting magnet (2019). Measured high-field test; the article also presents quench and post-mortem evidence.
See the reference index for source identity, regime, relevance and limitations. This note presents analytical calculations, not simulated or measured Highfield test results.