Exterior calculus is a powerful tool for finding solutions to the electromagnetic field equations. Its strength can be better appreciated when applied to nontrivial configurations. We show how to exploit this tool to obtain the TM and TE modes in hollow cylindrical waveguides. The use of exterior calculus and Lorentz boosts leads straightforwardly to the solutions and their respective power transmitted along the waveguide.
REFERENCES
1.
B. F.
Schutz
, Geometrical Methods of Mathematical Physics
(Cambridge U. P.
, Cambridge
, 1990
).2.
H.
Flanders
, Differential Forms with Applications to the Physical Sciences
(Dover
, New York
, 1989
).3.
R. W. R.
Darling
, Differential Forms and Connections
(Cambridge U. P.
, Cambridge
, 1994
).4.
C.
von Westenholz
, Differential Forms in Mathematical Physics
(North-Holland
, Amsterdam
, 1979
).5.
P.
Russer
, Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering
(Artech House
, Boston
, 2003
).6.
7.
P.
Bamberg
and S.
Sternberg
, A Course in Mathematics for Students of Physics
(Cambridge U. P.
, Cambridge
, 1991
), Vols. 1
and 2.8.
W. D.
Curtis
and F. R.
Miller
, Differential Manifolds and Theoretical Physics
(Academic
, New York
, 1986
).9.
V. I.
Arnol’d
, Mathematical Methods of Classical Mechanics
(Springer
, New York
, 1989
).10.
J.
Baez
and J. P.
Muniain
, Gauge Fields, Knots and Gravity
(World Scientific
, Singapore
, 2001
).11.
12.
T.
Frankel
, The Geometry of Physics: An Introduction
(Cambridge U. P.
, Cambridge
, 1997
).13.
C. W.
Misner
, K. S.
Thorne
, and J. A.
Wheeler
, Gravitation
(Freeman
, San Francisco
, 1973
).14.
V.
Guillemin
and S.
Sternberg
, Symplectic Techniques in Physics
(Cambridge U. P.
, Cambridge
, 1993
).15.
S.
Kobayashi
and K.
Nomizu
, Foundations of Differential Geometry
(Wiley-Interscience
, New York
, 1996
).16.
It is and in Gaussian units.
17.
R.
Ferraro
, Einstein’s Space-Time: An Introduction to Special and General Relativity
(Springer
, New York
, 2007
).18.
Green’s first identity can be stated as , where is the length element on the boundary and is the normal derivative. See the Appendix for a proof using exterior calculus.
19.
M.
Kaku
, Introduction to Superstrings and M-Theory
(Springer
, New York
, 1999
).© 2010 American Association of Physics Teachers.
2010
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
