# Electromagnetic fields and Green’s functions in elliptical vacuum chambers

## Abstract

In this paper, we discuss the electromagnetic interaction between a point charge travelling inside a waveguide of elliptical cross section, and the waveguide itself. By using a convenient expansion of the Mathieu functions, useful in particular for treating a variety of problems in applied mathematics and physics with elliptic geometry, we first obtain the longitudinal electromagnetic field of a point charge (Green's function) in free space in terms of elliptical coordinates. This expression allows, then, to calculate the scattered field due to the boundary conditions in our geometry. By summing the contribution of the direct or primary field and the indirect field scattered by the boundary, after a careful choice of some expansion expressions, we derive a novel formula of the longitudinal electric field, in any transverse position of the elliptical cross section, generated by the charge moving along the longitudinal axis of the waveguide. The obtained expression is represented in a closed form, it can be differentiated and integrated, it can be used to fully describe the radiation process of a particle beam travelling inside a waveguide of elliptical cross section, and it is valid for any elliptic geometry. The equations are used to evaluate the coupling impedance duemore »

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- European Organization for Nuclear Research (CERN), Geneva (Switzerland)
- European Organization for Nuclear Research (CERN), Geneva (Switzerland); Univ. of Rome La Sapienza, and INFN, Rome (Italy)
- Univ. of Rome La Sapienza, and INFN, Rome (Italy)
- Univ. of Napoli Federico II, and INFN, Napoli (Italy)

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1402111

- Alternate Identifier(s):
- OSTI ID: 1421835

- Grant/Contract Number:
- AC02-05CH11231

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- Physical Review Accelerators and Beams

- Additional Journal Information:
- Journal Volume: 20; Journal Issue: 10; Journal ID: ISSN 2469-9888

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Persichelli, S., Biancacci, N., Migliorati, M., Palumbo, L., and Vaccaro, V. G.
```*Electromagnetic fields and Green’s functions in elliptical vacuum chambers*. United States: N. p., 2017.
Web. doi:10.1103/PhysRevAccelBeams.20.101004.

```
Persichelli, S., Biancacci, N., Migliorati, M., Palumbo, L., & Vaccaro, V. G.
```*Electromagnetic fields and Green’s functions in elliptical vacuum chambers*. United States. doi:10.1103/PhysRevAccelBeams.20.101004.

```
Persichelli, S., Biancacci, N., Migliorati, M., Palumbo, L., and Vaccaro, V. G. Mon .
"Electromagnetic fields and Green’s functions in elliptical vacuum chambers". United States.
doi:10.1103/PhysRevAccelBeams.20.101004.
```

```
@article{osti_1402111,
```

title = {Electromagnetic fields and Green’s functions in elliptical vacuum chambers},

author = {Persichelli, S. and Biancacci, N. and Migliorati, M. and Palumbo, L. and Vaccaro, V. G.},

abstractNote = {In this paper, we discuss the electromagnetic interaction between a point charge travelling inside a waveguide of elliptical cross section, and the waveguide itself. By using a convenient expansion of the Mathieu functions, useful in particular for treating a variety of problems in applied mathematics and physics with elliptic geometry, we first obtain the longitudinal electromagnetic field of a point charge (Green's function) in free space in terms of elliptical coordinates. This expression allows, then, to calculate the scattered field due to the boundary conditions in our geometry. By summing the contribution of the direct or primary field and the indirect field scattered by the boundary, after a careful choice of some expansion expressions, we derive a novel formula of the longitudinal electric field, in any transverse position of the elliptical cross section, generated by the charge moving along the longitudinal axis of the waveguide. The obtained expression is represented in a closed form, it can be differentiated and integrated, it can be used to fully describe the radiation process of a particle beam travelling inside a waveguide of elliptical cross section, and it is valid for any elliptic geometry. The equations are used to evaluate the coupling impedance due to indirect space charge in case of elliptical geometry. In addition, they are useful as preliminary studies for the determination of the coupling impedance in different cases involving elliptic vacuum chambers, as, for example, the effect of the finite conductivity of the beam pipe wall or the geometrical variation of the vacuum chamber due to elliptic step transitions existing in some accelerators.},

doi = {10.1103/PhysRevAccelBeams.20.101004},

journal = {Physical Review Accelerators and Beams},

number = 10,

volume = 20,

place = {United States},

year = {Mon Oct 23 00:00:00 EDT 2017},

month = {Mon Oct 23 00:00:00 EDT 2017}

}