Magnetostatics & Magnet Design 1 1 Magnetostatics & Magnet Design Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU 1 1 1 1 1 1

Vacuum Magnetostatics 2 2 2 2 2 2 2 2 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 2 2 2 2 2 2

Magnetostatics Assume all charges and currents are steady - There are no time-varying E & B fields. In a vacuum, the equations for B then become: This is an incompressible field, like idealized currents of water. 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 3 3 3 3

Calculating the Magnetic Field Using the vector potential A, we can make the calculation easier: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 4 4 4 4

Stokes’ Theorem & Ampere’s Law 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 5 5 5 5

Current Carrying Wire 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 6 6 6 6

Displaced Wire 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 7 7 7 7

Cosθ Superconducting Dipole LHC Dipole: Currents distributed along a circle, by cosθ: Superconducting magnets are current-dominated magnets. 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 8 8 8 8

Cos2θ Superconducting Quadrupole Currents distributed along a circle, by cos2θ: 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 9 9 9 9

NIST Neutron Storage Ring 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 10 10 10 10

Magnetic Fields in Media & 11 11 Magnetic Fields in Media & Magnetization 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 11 11 11 11 11 11

Applied Field H and Magnetic Materials H is the applied magnetic field, B is the magnetic response And Maxwell’s equations are updated to include H, These equations describe the continuity equations at the boundary: 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 12 12 12 12

Applied Field H and Magnetic Materials For μ  ∞, Magnetic fields normal to pole-tip surface: The poles act as sources of magnetic field. 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 13 13 13 13

Iron-Dominated Magnets In iron-dominated magnets the field profile is primarily determined by the shape of the pole-tips. Magnet pole-tip shapes are a trade-off between manufacturing cost, field-uniformity, and field-strength. Quad with low fringe field: High-gradient Quad: TRIUMF ESRF 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 14 14 14 14

Jackson 5.10: Magnetized Sphere 1. Decompose Φm in appropriate coordinate basis. 3.38, 3.70 2. Apply boundary conditions for magnet surface currents. 5.104 3. Use Φm to solve for Hmag due to magnetization. 5.105 4. Verify Hmag is proportional to the applied field Hext. 5.106 5. Find Btot and Htot. 5.112 6. Relate Btot and Htot with material permeability μ to solve for magnitude of magnetization in terms of applied field. 5.113-5.115 7. Find the Btot in terms of permeability μ. 5.112, 5.115 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 15 15 15 15

Hysteresis Loop The slope of the lines is connected to the aspect-ratio of the magnetized object: - disk-like is gradual, - needle-like is steeper. Area of the figure is connected to the saturation at peak field. - proportional to heating losses per magnet cycle. 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 16 16 16 16