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mstat2 - Physics 54 Sources of the Magnetic Field Caution...

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Physics 54 Sources of the Magnetic Field Caution: Cape does not enable user to fly. — 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 finding concerns the B-field set up by infinitesimal segment of a current-carrying wire. In the situation shown, we are interested in the B-field at field point P due to an infinitesimal bit of wire. The current is I , in the direction specified 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, μ 0 = 4 π × 10 7 in SI units (exactly, by definition). The choice of this constant makes the ampere the defining quantity for SI electromagnetic units. The direction of the B-field comes from the vector product. In the case shown it is out of the page. If the field point were at angle θ below the horizontal, the direction would be into the page. One can see that the field lines of this B-field form circles of radius r sin θ about a point on the horizontal line. This is an example of a general property: Lines of the B-field always form closed curves. The magnitude, in the case shown, is dB = μ 0 4 π I sin θ r 2 . We see that the field strength falls off like 1/ r 2 , as in the case of the E-field of a point charge. d l r r sin θ θ P PHY 54 1 Magnetostatics 2
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Of course there are never isolated infinitesimal bits of wire; one must integrate over all the current segments in the system in order to find the total B-field 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-field of a point charge. In principle, the B-field 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 finite piece of wire. But circuits of rectangular shape are made of straight pieces, and the B- field of such a circuit can be obtained by adding their separate contributions.
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