Simple lessons about static magnetic fields are often taught with the model of an “infinite” solenoid, outside of which the fields vanish. Just outside a very long but finite solenoid of length *L*, the field must be a decreasing function of *L*. We show that this external field is approximately uniform and decreases as $L\u22122.$ Furthermore, we show that the study of this external field provides interesting and surprisingly simple illustrations of techniques for analyzing magnetic fields.

Topics

Electromagnetic coils
## REFERENCES

1.

See, for example, D. C. Giancoli,

*Physics*, 3rd ed. (Prentice–Hall, Englewood Cliffs, NJ, 1991), Sec. 20-12.2.

See, for example, L. S. Lerner,

*Physics for Scientists and Engineers*(Jones and Bartlett, Boston, 1996), Sec. 29.4. Here it is argued that the external field must be weak since it fills an infinte space.3.

See, for example, Fig. 30.14 of R. Serway,

*Physics for Scientists and Engineers*(Saunders, Philadelphia, 1990), or Fig. 30-28 in R. Wolfson and J. M. Pasachoff,*Physics*, 3rd ed. (Addison–Wesley, Reading, MA, 1999).4.

In the special case of a circular cross section solenoid, the symmetry allows an analytic treatment of the fields just outside the solenoid. See J. D. Jackson,

*Classical Electrodynamics*, 3rd ed. (Wiley, New York, 1998), Prob. 5.5.5.

This symmetry of mutual inductance is usually stated in introductory texts, but no proof is given. See for example, R. Serway,

*Physics for Scientists and Engineers*(Saunders, Philadelphia, 1990), Sec. 32.4;R. Wolfson and J. M. Pasachoff,

*Physics*, 3rd ed. (Addison–Wesley, Reading, MA, 1999), Sec. 32.1;D. C. Giancoli,

*Physics*, 3rd ed. (Prentice–Hall, Englewood Cliffs, NJ, 1991), Sec. 21-9;L. S. Lerner,

*Physics for Scientists and Engineers*(Jones and Bartlett, Boston, 1996), Sec. 31.2. The symmetry is usually proven in junior-level texts by the presentation of the Neumann formula. See, for example, D. J. Griffiths,*Introduction to Electrodynamics*, 3rd ed. (Prentice–Hall, Englewood Cliffs, NJ, 1999), Sec. 7.2.3J. R. Reitz and F. J. Milford,*Foundations of Electromagnetic Theory*(Addison–Wesley, Reading, MA, 1960), Sec. 9.4;W. K. H. Panofsky and M. Phillips,

*Classical Electricity and Magnetism*(Addison-Wesley, Reading, MA, 1955), Sec. 10-3.6.

Since the inner boundary of the region of integration will not be at constant

*r*this form of the integral is not really justified, but the use to which it will be put in the subsequent steps involves only the outer boundary.
This content is only available via PDF.

© 2001 American Association of Physics Teachers.

2001

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.