Mstat2 - Physics 54 Sources of the Magnetic Field Caution Cape does not enable user to y Label on Batman Costume The Biot-Savart law Soon after

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Physics 54 Sources of the Magnetic Field Caution: Cape does not enable user to fy. — Label on Batman Costume The Biot-Savart law Soon after they learned of Oersted's discovery that a current produces magnetic effects, Biot and Savart undertook careful experiments to determine the details. In modern notation and terminology, their Fnding concerns the B-Feld set up by inFnitesimal segment of a current-carrying wire. In the situation shown, we are interested in the B-Feld at Feld point P due to an inFnitesimal bit of wire. The current is I , in the direction speciFed by the vector d l . The displacement of P relative to the bit of wire is r , which makes angle θ with the direction of d l . The experimental answer found by Biot and Savart is a fundamental law: Biot-Savart law d B = μ 0 4 π I d l × r r 3 In this formula we have introduced another universal constant, 0 = 4 × 10 7 in SI units (exactly, by deFnition). The choice of this constant makes the ampere the deFning quantity for SI electromagnetic units. The direction of the B-Feld comes from the vector product. In the case shown it is out of the page. If the Feld point were at angle below the horizontal, the direction would be into the page. One can see that the Feld lines of this B-Feld form circles of radius r sin about a point on the horizontal line. This is an example of a general property: Lines of the B-±eld always form closed curves. The magnitude, in the case shown, is dB = 0 4 I sin r 2 . We see that the Feld strength falls off like 1 / r 2 , as in the case of the E-Feld of a point charge. d l r r sin P PHY 54 1 Magnetostatics 2
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Of course there are never isolated inFnitesimal bits of wire; one must integrate over all the current segments in the system in order to Fnd the total B-Feld at P . In general such an integration is complicated; we will do it only in a few very simple cases. Nevertheless, the Biot-Savart law gives us the magnetic equivalent of the E-Feld of a point charge. In principle, the B-Feld of any set of currents could be calculated using it. Two examples The simplest case geometrically is that of a straight wire carrying current I . This is also unphysical by itself, because steady currents cannot start and stop at the ends of a Fnite piece of wire. But circuits of rectangular shape are made of straight pieces, and the B-
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This note was uploaded on 10/19/2011 for the course PHYSICS 54L taught by Professor Thomas during the Summer '09 term at Duke.

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Mstat2 - Physics 54 Sources of the Magnetic Field Caution Cape does not enable user to y Label on Batman Costume The Biot-Savart law Soon after

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