# Forces and Stored Energy in Thin Cosine (n0) Accelerator Magnets

## Abstract

We wish to compute Lorentz forces, equilibrium stress and stored energy in thin multipole magnets (Fig.1), that are proportional to cos(n{theta}) and whose strength varies purely as a Fourier sinusoidal series of the longitudinal coordinate z (say proportional to cos (2m-1){pi}z/L where L denotes the half-period and m = 1,2,3...). We shall demonstrate that in cases where the current is situated on such a surface of discontinuity at r = R (i.e. J = f({theta},z)), by computing the Lorentz force and solving the state of equilibrium on that surface, a closed form solution can be obtained for single function magnets as well as for any combination of interacting nested multi function magnets. The results that have been obtained, indicate that the total axial force on the end of a single multipole magnet n is independent (orthogonal) to any other multipole magnet i as long as n {ne} i. The same is true for the stored energy, the total energy of a nested set of multipole magnets is equal to the some of the energy of the individual magnets (of the same period length 2L). Finally we demonstrate our results on a nested set of magnets a dipole (n = 1)more »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- Accelerator& Fusion Research Division

- OSTI Identifier:
- 1004211

- Report Number(s):
- LBL-38500

TRN: US201104%%914

- DOE Contract Number:
- DE-AC02-05CH11231

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99; ACCELERATORS; DIPOLES; LORENTZ FORCE; MAGNETS; MULTIPOLES; PERIODICITY; STORED ENERGY

### Citation Formats

```
Caspi, S.
```*Forces and Stored Energy in Thin Cosine (n0) Accelerator Magnets*. United States: N. p., 1996.
Web. doi:10.2172/1004211.

```
Caspi, S.
```*Forces and Stored Energy in Thin Cosine (n0) Accelerator Magnets*. United States. https://doi.org/10.2172/1004211

```
Caspi, S. 1996.
"Forces and Stored Energy in Thin Cosine (n0) Accelerator Magnets". United States. https://doi.org/10.2172/1004211. https://www.osti.gov/servlets/purl/1004211.
```

```
@article{osti_1004211,
```

title = {Forces and Stored Energy in Thin Cosine (n0) Accelerator Magnets},

author = {Caspi, S},

abstractNote = {We wish to compute Lorentz forces, equilibrium stress and stored energy in thin multipole magnets (Fig.1), that are proportional to cos(n{theta}) and whose strength varies purely as a Fourier sinusoidal series of the longitudinal coordinate z (say proportional to cos (2m-1){pi}z/L where L denotes the half-period and m = 1,2,3...). We shall demonstrate that in cases where the current is situated on such a surface of discontinuity at r = R (i.e. J = f({theta},z)), by computing the Lorentz force and solving the state of equilibrium on that surface, a closed form solution can be obtained for single function magnets as well as for any combination of interacting nested multi function magnets. The results that have been obtained, indicate that the total axial force on the end of a single multipole magnet n is independent (orthogonal) to any other multipole magnet i as long as n {ne} i. The same is true for the stored energy, the total energy of a nested set of multipole magnets is equal to the some of the energy of the individual magnets (of the same period length 2L). Finally we demonstrate our results on a nested set of magnets a dipole (n = 1) and a quadmpole (n=2) that have an identical single periodicity {omega}{sub 1}. We show that in the limiting 2D case (period 2L tends to infinity), the force reduces to the commonly known 2D case.},

doi = {10.2172/1004211},

url = {https://www.osti.gov/biblio/1004211},
journal = {},

number = ,

volume = ,

place = {United States},

year = {1996},

month = {3}

}