1. Davide Tommasini TitleEvent/Date Introduction to MAGNETS II Davide Tommasini JUAS February 22 nd 2016

2. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Fundamentals 1 : Maxwell

3. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 CURL / ROTOR The speed of water S is rotational around an axis determinined by a driving force F, its amplitude depends on the distance from the axis and on the driving force. Put in mathematics we get: Remark: a whirlpool is turbulent, the analogy is for didactics purposes only The curl of a vector field is the closed circulation field through an infinitesimal area

4. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 DIVERGENCE The Divergence of a vector is the amount of flux of vector entering or leaving a point. This is the 1st Maxwell equation, corresponding to the Gauss Law.

5. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Basic Principles 1820 Hans Christian Ørsted An electrical current produces a circular magnetic field around the wire This discovery pushed scientists to understand the mathematics behind this evidence

6. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Generating a magnetic field strength Maxwell Ampère 1826 André-Marie Ampère 1861 James Clerk Maxwell

7. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Let’s use this formula !

8. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Let’s use this formula ! cont r We consider the boundary along the circumference at radius «r». Keeping the same radius, due to symmetry, H remains constant.

9. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 The Magnetic Field Induction The «magnetic field induction» is created by the «magnetic field strength» In certain materials (ferromagnetic) we just need a small strength to produce a large induction, in most materials we need a large strength to produce a large induction. We define the following constitutive equation:

10. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 The Fantastic Four Gauss law for electricity Gauss law for magnetism Ampère law with correction Faraday-Lenz law of induction

11. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Continuity conditions n The flux which enters shall be equal to the flux which exits The integral of the field strength

12. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Fundamentals 2 : Field Harmonics

13. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Decomposition of magnetic field  r In cartesian coordinates In polar coordinates In case of a combination of uniform vertical k and uniform horizontal field h we have: The coefficients caracterizing the vertical field (producing horizontal beam deflection) are called «normal», the ones caracterizing the horizontal field are called «skew».

14. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 To go ahead we need …The Potential With then: In air, in a volume with no currents:

15. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Why we need the (vector) potential For a further insight I recommend checking the Feynman Lectures on Physics, now «free to read online» at http://www.feynmanlectures.caltech.edu r r0r0  in polar coordinates becomes: The solution of this equation is: … and the field components are :

16. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Field Harmonics «normal» dipole component

17. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Warning on the Sextupole Component When you reconstruct the field amplitude as a function of the radius you obtain: However, if you express the magnetic field by a polynomial expansion:

18. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Relative Field Harmonics Solution :

19. Davide Tommasini TitleEvent/Date D.TommasiniIntroduction to Magnets JUAS February 22 nd, 2016 Scaling of Relative Field Harmonics Let’s consider the dependency vs radius of a given field harmonic amplitude, does not matter normal or skew The field harmonic relative to a «reference order m» scales as: Solution :

20. Davide Tommasini TitleEvent/Date Thanks