Both wire-wound solenoids and cylindrical magnets can be approximated as ideal azimuthally symmetric solenoids. We present an exact solution for the magnetic field of an ideal solenoid in an easy to use form. The field is expressed in terms of a single function that can be rapidly computed by means of a compact efficient algorithm, which can be coded as an add-in function to a spreadsheet, making field calculations accessible to introductory students. These expressions are not only accurate but are also as fast as most approximate expressions. We demonstrate their utility by simulating the dropping of a cylindrical magnet through a nonmagnetic conducting tube and comparing the calculation with data obtained from experiments suitable for an undergraduate laboratory.
REFERENCES
1.
John T.
Conway
, “Exact solutions for the magnetic fields of axisymmetric solenoids and current distributions
,” IEEE Trans. Magn.
37
(4
), 2977
–2988
(2001
).2.
Velimir
Labinac
, Natasa
Erceg
, and Dubravka
Kotnik-Karuza
, “Magnetic field of a cylindrical coil
,” Am. J. Phys.
74
(7
), 621
–627
(2006
).3.
Milan Wayne
Garrett
, “Axially symmetric systems for generating and measuring magnetic fields. Part I
,” J. Appl. Phys.
22
(9
), 1091
–1107
(1951
).4.
Edmund E.
Callaghan
and Stephen H.
Maslen
, “The magnetic field of a finite solenoid
,” NASA TN D-465 (1960
), <ntrs.nasa.gov>.5.
Andreas
Sundquist
, “Dynamic line integral convolution for visualizing streamline evolution
,” IEEE Trans. Vis. Comput. Graph.
9
(3
), 273
–282
(2003
); available at <web.mit.edu/viz/soft/>.6.
For source code see <web.mit.edu/viz/soft/>.
7.
E. B.
Rosa
and F. W.
Grover
, “Formulas and tables for the calculation of mutual and self induction
,” Bur. Stand. (U.S.), Bull.
8
(1
), 53
–77
(1911
), <www.g3ynh.info/zdocs/refs/Rosa-Grover1911/Contents.html>.8.
G. R.
Olshausen
, “Absolute formulae for the mutual inductance of coaxial solenoids
,” Phys. Rev. (Series I)
35
(2
), 148
–152
(1912
).9.
Milan Wayne
Garrett
, “Calculations of fields, forces and mutual inductances of current systems by elliptic integrals
,” J. Appl. Phys.
34
, 2567
–2573
(1963
).10.
K. D.
Hahn
, E. M.
Johnson
, A.
Brokken
, and S.
Baldwin
, “Eddy current damping of a magnet moving through a pipe
,” Am. J. Phys.
66
(12
), 1066
–1076
(1998
).11.
D.
Amrani
and P.
Paradis
, “Faraday’s law of induction gets free-falling magnet treatment
,” Phys. Educ.
40
(4
), 313
–314
(2005
).12.
Yan
Levin
, Fernando L.
da Silveira
, and Felipe B.
Rizzato
, “Electromagnetic braking: A simple quantitative model
,” Am. J. Phys.
74
(9
), 815
–817
(2006
).13.
Jhules A. M.
Clack
and Terrence P.
Toepker
, “Magnetic induction experiment
,” Phys. Teach.
28
(4
), 236
–238
(1990
).14.
Reed College, Physics 100, <academic.reed.edu/physics/courses/phys100/Lab Manuals/Electromagnetic Induction/faraday.pdf>.
15.
Video lesson, falling magnets, <www.wikidebrouillard.org/index.php/Aimant_qui_tombe_au_ralenti>.
16.
J. A.
Pelesko
, M.
Cesky
, and S.
Huertas
, “Lenz’s law and dimensional analysis
,” Am. J. Phys.
73
(1
), 37
–39
(2005
).17.
M. K.
Roy
, Manoj K.
Harbola
, and H. C.
Verma
, “Demonstration of Lenz’s law: Analysis of a magnet falling through a conducting tube
,” Am. J. Phys.
75
(8
), 728
–730
(2007
).18.
B. A.
Knyazev
, I. A.
Kotel’nikov
, A. A.
Tyutin
, and V. S.
Cherkasskii
, “Braking of a magnetic dipole moving with an arbitrary velocity through a conducting pipe
,” Phys. Usp.
49
, 937
–946
(2006
).19.
J.
Iñiguez
, V.
Raposo
, A.
Hernández-López
, A. G.
Flores
, and M.
Zazo
, “Study of the conductivity of a metallic tube by analysing the damped fall of a magnet
,” Eur. J. Phys.
25
, 593
–604
(2004
).20.
C. S.
MacLatchy
, P.
Backman
, and L.
Bogan
, “A quantitative magnetic braking experiment
,” Am. J. Phys.
61
(12
), 1096
–1101
(1993
).21.
M.
Hossein Partovi
and Eliza J.
Morris
, “Eddy current damping of a magnet moving through a pipe
,” Can. J. Phys.
84
, 253
–274
(2006
).22.
Roland
Bulirsch
, “Numerical calculation of elliptic integrals and elliptic functions
,” Numer. Math.
7
, 78
–90
(1965
).23.
W.
Bartky
, “Numerical calculation of a generalized elliptic integral
,” Rev. Mod. Phys.
10
, 264
–269
(1938
).24.
The open source OpenOffice software suite is available at <www.openoffice.org>.
25.
FORTRAN code for evaluating the function in Eq. (1) is available at <docdb.fnal.gov/ILC/sid/stanitz/ilcsoft-latest/cernlib-2006/src/mathlib/gen/c/religc64.F>.
26.
William H.
Press
, Brian
Flannery
, Saul A.
Teukolsky
, and William T.
Vetterling
, Numerical Recipes in C
(Cambridge U. P.
, New York
, 1988
).27.
William H.
Press
, Brian
Flannery
, Saul A.
Teukolsky
, and William T.
Vetterling
, Numerical Recipes: The Art of Scientific Computing
, 3rd ed. (Cambridge U. P.
, New York
, 2007
).28.
K. -D.
Reinsch
and W.
Raab
, “Elliptic integrals of the first and second kind: Comparison of Bulirsch’s and Carlson’s algorithms for numerical calculation
,” in Special Functions, Proceedings of the International Workshop
, edited by Charles
Dunkl
, Mourad
Ismail
, and Roderick
Wong
(World Scientific
, Singapore
, 1999
), pp. 293
–308
, <www-m2.ma.tum.de/homepages/kladire/publications/bucavgl/bucavgl.html>.© 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.
