Motivated by an observation of a popular children’s toy, I investigate the behavior of a bar magnet with a steel ball bearing held on one of its poles as it approaches another bar magnet. Mapping the problem onto electrostatics, I give an explanation of my observations based on the behavior of point charges near an isolated, uncharged sphere. This magnetic analog offers a simple demonstration of the familiar “method of images” in electrostatics.

Topics
Electrostatics, Electrodynamics, Educational aids
1.
The toy is manufactured by PlastWood, and comes in kits with various number of pieces.
2.
A. E. Kip, Fundamentals of Electricity and Magnetism (McGraw–Hill, New York, 1969). Kip treats “fictitious magnetization charges” in bar magnets.
3.
D. R. Frankl, Electromagnetic Theory (Prentice–Hall, Englewood Cliffs, NJ, 1986).
4.
For a complete discussion of B, H, and M in bar magnets, see A. Sommerfeld, Electrodynamics, English translation by E. G. Ramberg (Academic, New York, 1952), Sec. 12.
5.
This argument applies only if the high permeability material remains linear, isotropic, and homogeneous under all conditions. The linearity requirement is certainly violated by a material like steel at high enough fields, where saturation and hysteresis occur (see, e.g., Ref. 2, Chapter 10). The analogy between high-μ materials in magnetostatics and conductors in electrostatics is therefore only approximate. The degree of agreement between experiments and our calculations using the electrostatic analogy suggests that this approximation is a reasonable one.
6.
An extensive discussion of point-charge-and-sphere image problems is given by J. D. Jackson, Electrodynamics (Wiley, New York, 1998), 3rd ed., Secs. 2.2–2.7. As a reminder, consider a single charge Q at distance r from a sphere (radius R) centered at the origin O. As far as the field outside the sphere and the charge distribution on the sphere itself is concerned, the sphere can be replaced by an image charge of size Q′=−(R/r)Q placed at distance R2/r from O along the line OQ and a second image charge of size −Q′ at O.
7.
With p̂=q̂=1 and r̂p=1.1, we find that the total charge on the “patch” facing q is ∼−0.3 at the point when the force on q changes sign.
8.
The two problems treated in this appendix are worked examples in Frankl (Ref. 3); I follow his treatment closely.
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© 2003 American Association of Physics Teachers.
2003
American Association of Physics Teachers
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