1. Magnetostatics & Magnet Design1 1
Magnetostatics & Magnet Design
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. Vacuum Magnetostatics 2 2 2 2 2 2 2 2 2
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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3. 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
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4. 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
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5. Stokes’ Theorem & Ampere’s Law5
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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6. Current Carrying Wire 6
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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7. Displaced Wire 7
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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8. 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
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9. Cos2θ Superconducting Quadrupole
Currents distributed along a circle, by cos2θ: 9
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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10. NIST Neutron Storage Ring10
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11. Magnetic Fields in Media &11 11
Magnetic Fields in Media &
Magnetization 11
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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12. 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

13. 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
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14. 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
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15. 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.
7. Find the Btot in terms of permeability μ.
5.112, 5.115 15
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16. 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
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