PHY132_L15 - Classical Physics II Classical Physics II...

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lassical Physics II Classical Physics II PHY132 Lecture 15 Ampère’s Law Lecture 15 1 03/05/2010
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Magnetic Field from a Wire Current We now return to the GENERATION of the magnetic field B by a current. .. he implest case is the generation of a ield by a The simplest case is the generation of a B field by a infinitely long straight wire carrying a constant current I Biot-Savart Law : d l 0 2 4 I r d d l B r I d B d B’ right-hand rule! • gives the contribution d B to the field due to the current I in a (infinitesimally) short piece of the wire d l curr nt n a ( nf n t s ma y) short p c of th w r in point r away from d l . Lecture 15 2 03/05/2010
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Magnetic Field from a Long Wire Calculate the Magnetic Field at point P , at a distance x from a long wire carrying a current I : – Consider a segment dy of the wire, at position y above the origin… – Using the right-hand-rule, all contributions to the B -field are in the same z direction! = = B 0 2 4 z Idy d r yr B a 0 2 sin ' () 4 Idy r k 0 2 cos 4 a z a I dy Bx r  B ( x ) = B ( x )( k ) = dB z ( x ) Choose α as the integration variable; then: 2 tan cos x yx d y  dy r P α B : right-handed circles around I I 2 1c o s x Thus: O x y α d B z 22 cos r rx 0 a I d    0 sin aax I d  a 4 a x  0 2 4 I a x x ax Bk 4 x  0 2 1 I x k Lecture 15 4 limit for a>>x ! 03/05/2010 lim ( ) 00 II x x 
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Symmetry! The symmetry of the long straight wire problem, allows the use of a powerful tool, SYMMETRY, to constrain the possible shapes of the magnetic field: 1. the wire has a rotational symmetry: rotating the wire around itself doesn’t change the current (direction and magnitude), and thus the B -field must be equal to itself as well under such rotations… he wire has anslational ymmetry: oving he (infinitely long 2. The wire has translational symmetry: moving the (infinitely long
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This note was uploaded on 04/14/2010 for the course PHY 132 taught by Professor Rijssenbeek during the Spring '04 term at SUNY Stony Brook.

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PHY132_L15 - Classical Physics II Classical Physics II...

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