# Magnets Held Up by Length Contraction?

The effects of magnetism follow from the fact that moving charges see current-carrying wires as charged, due to relativistic length contraction which upsets the wire's apparent charge balance. Hence from special relativity and the fact that opposite charges attract, one can show that like currents in neutral wires attract. This is illustrated with a couple of x-ct plots below. The effect has practical applications because the electrostatic force is noticeable with only very slight (in this case apparent) charge imbalances. To see if relativity is at work, one thus might have to look no further than your refrigerator door!

Dan Shroeder's talk on this subject and our derivation.
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For source, cite URL at http://newton.umsl.edu/~run/magnets.html
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If you prefer algorithms over pictures, the Biot-Savart relation (for magnetic fields from current elements) AND the Lorentz force law (for the effect of magnetic fields on a charge) follow simply from only Coulomb's law and the equation for relativistic length contraction. Such a derivation, which somehow reminds me of ``The Cleveland Wrecking Yard'' section of Richard Brautigan's Trout Fishing in America (Houghton Mifflin, Boston, 1967) about foot-long segments of trout stream for sale, and waterfalls stacked against walls in the used-plumbing-department, is posted here.

By comparison, the x-ct plots here conjure up visions of gender-imbalance between lines in a New-England style contra-dance. In the first diagram below, elapsed-time progresses in the vertical direction for a neutral wire, with blue (e.g. negative) charges flowing to the right and with an equal number of red (e.g. positive) charges not moving at all. Observers in the frame of the wire and the stationary positive charges (e.g. some fixed external test charge) will see no charge imbalance in the wire and thus no electric field as a result.

Fig. 1: xct-plot from the viewpoint of the neutral wire.

Note in this plot that snapshots of the motion from the viewpoint of the moving negative charges correspond to tilted lines running upward to the right. The balance of red and blue dots along these tilted lines (like the tilted blue line through the origin) has shifted due to contraction of the now moving red charges, so that from the vantage point of observers moving along with the blue dots, the wire appears to have a net red (e.g. positive) charge. For a blue electron moving along with the other blue dots, an electrostatic attraction will pull it toward the wire even though a blue electron not moving with respect to the wire would feel no attraction at all. Thus like currents attract.

Putting this into dance terms, imagine a gender-balanced line of men and women, with the men only moving to the left. This analysis suggests that a stationary observer would see a gender-balanced line, while an observer moving along with the men would see a line with women slightly more abundant. If opposites attract (as in the case of electric charge), then a gender-based analog to magnetism might be expected as well, with "like currents" attracting, and "opposing currents" pushing lines apart! Don't expect this effect to be noticeable, however, unless the band is playing reallllllly fast!

This second plot contains exactly the same information as the first, except that it has been drawn with perpendicular x-ct axes for the blue (e.g. negative) charges.

Fig. 2: xct-plot from the viewpoint of the moving charges.

Note: This page contains original material, so if you choose to echo in your work, in print, or on the web, a citation would be cool. `(Thanks. /philf :)`