Superconducting Solenoid: Field, Stress and Margin

What is modelled

A thick solenoid is a tube of current. Its field in the bore grows with the current density and with the thickness of the winding, and the formula of Fabry gives the exact centre field of a uniformly filled tube of finite length. The field presses the windings outwards with a magnetic pressure of B squared over twice mu zero, and that pressure appears as a hoop stress in the conductor. The superconductor itself can carry only so much current: the critical current density falls when the field or the temperature rise, so the coil quenches when the load line of the winding meets this critical surface. The model sums the field of many current loops with elliptic integrals to get the field on the winding, and the current is ramped from zero to the set value so that you can see where a coil that is set too high would quench.

Equations

• Centre field B0 = μ0 J a1 F(α, β), F = β ln[(α + √(α2 + β2)) / (1 + √(1 + β2))], with α = a2 / a1, β = L / (2 a1).
• Hoop stress σ ≈ J B̄ a, with the average field about B₀ / 2; thin shell σ = B2 a / (2 μ0 t).
• NbTi (illustrative fit): Bc2(T) = 14.5 (1 − (T / 9.2)1.7), Jc = K (1 − B / Bc2) / B with 3000 A/mm² at 5 T and 4.2 K; quench when J / λ ≥ Jc.

Checks run in the page

• The centre field from summed current loops against the Fabry formula.
• The long-solenoid limit against the Ampere law.
• The critical margin and the quench limit by a search.
• The hoop stress of a thin shell from the integrated Lorentz force against the magnetic-pressure formula.

Honest simplifications

A uniform current density, a simple fit of the critical surface of NbTi with no strain or copper stabiliser, no quench propagation or heating, a hoop stress estimate that ignores supporting structure and shear, and no screening currents. A real research magnet is designed with grading, reinforcement and a quench protection circuit, so these numbers show the trends and are not a design.

• Press Pushed past the limit and watch the ramp end in a quench before the full current.
• Press Cooled to superfluid helium and see how the lower temperature lets the same winding carry a higher current.
• Press Long thin solenoid and compare the field with the Ampere law estimate.