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Physics 2102 Gabriela González Physics 2102 Magnetic fields produced by currents ? The Biot-Savart Law Jean-Baptiste Biot (1774-1862) dL Felix Savart (1791-1841) i • Quantitative rule for r computing the magnetic field from any electric current 0 idL r • Choose a differential element dB of wire of length dL and 3 4 r carrying a current i • The field dB from this element 0 =4x10-7 T.m/A at a point located by the vector (permeability constant) r is given by the Biot-Savart 1 dq r Law: Compare with: dE 4 0 r 3 Biot-Savart Law • An infinitely long straight wire carries a current i. • Determine the magnetic field generated at a point located at a perpendicular distance R from the wire. • Choose an element ds as shown • Biot-Savart Law: dB ~ ds x r points INTO the page • Integrate over all such elements 0 ids r dB 4 r 3 0 ids (r sin ) dB 4 r3 0i ds (r sin ) B 4 r3 Field of a straight wire (continued) sin R / r r (s R ) 2 2 1/ 2 i Rds 0i ds (r sin ) 0 B 3 2 2 3/ 2 4 4 r s R 0i Rds 2 0 s 2 R 2 3 / 2 0i 0iR s 2 2 1 / 2 2R 2 R s R 2 0 A Practical Matter A power line carries a current of 500 A. What is the magnetic field in a house located 100m away from the power line? 0i B 2R (4x107 T .m / A)(500 A) 2 (100m) = 1 T!! Recall that the earth’s magnetic field is ~10-4T = 100 T Biot-Savart Law • A circular loop of wire of radius R carries a current i. • What is the magnetic field at the center of the loop? 0 i ds r dB 3 4 r 0 idsR 0 iRd dB 3 2 4 R 4 R 0 id 0i 2 0i B 4 R 4R 2R ds R i Direction of B?? Not another right hand rule?! Curl fingers around direction of CURRENT. Thumb points along FIELD! Into page in this case. Forces between wires Magnetic field due to wire 1 where the wire 2 is, L I1 I2 F 0 I1 B1 2 a Force on wire 2 due to this field, F21 L I 2 B1 a 0 LI1 I 2 2 a Ampere’s law: Remember Gauss’ law? Given an arbitrary closed surface, the electric flux through it is proportional to the charge enclosed by the surface. q q Flux=0! Surface q E dA 0 Gauss’ law for magnetism: simple but useless No isolated magnetic poles! The magnetic flux through any closed “Gaussian surface” will be ZERO. This is one of the four “Maxwell’s equations”. B d A 0 q E d A 0 Always! No isolated magnetic charges Ampere’s law: a useful version of Gauss’ law. B d s 0 I i4 loop The circulation of B (the integral of B scalar ds) along an imaginary closed loop is proportional to the net amount of current traversing the loop. i2 i3 i1 ds B d s 0 (i1 i2 i3 ) loop Thumb rule for sign; ignore i4 As was the case for Gauss’ law, if you have a lot of symmetry, knowing the circulation of B allows you to know B. Ampere’s Law: Example • Infinitely long straight wire with current i. • Symmetry: magnetic field consists of circular loops centered around wire. • So: choose a circular loop C -- B is tangential to the loop everywhere! • Angle between B and ds = 0. C (Go around loop in same direction as B field lines!) R B ds 0i C Bds B(2R) i 0 0i B 2R Ampere’s Law: Example i • Infinitely long cylindrical wire of finite radius R carries a total current i with uniform current density • Compute the magnetic field at a distance r from cylinder axis for: – r < a (inside the wire) – r > a (outside the wire) r Current into page, circular field lines R B d s i 0 C Ampere’s Law: Example 2 (cont) B d s i 0 C r Current into page, field tangent to the closed amperian loop B(2r ) 0ienclosed 0ienclosed B 2r 2 i r 2 2 ienclosed J (r ) 2 r i 2 R R 0ir For r < R B 2 2R Ampere’s Law: Example 2 (cont) B d s i 0 r Current into page, field tangent to the closed amperian loop C B(2r ) 0ienclosed 0ienclosed 0i B 2 r 2 r 0i B 2 r For r > R 0ir B 2R 2 For r < R B(r) r Solenoids B ds 0ienc B ds 0 B h 0 0 ienc iN h i ( N / L)h inh B ds 0ienc Bh 0inh B 0in Magnetic field of a magnetic dipole A circular loop or a coil currying electrical current is a magnetic dipole, with magnetic dipole moment of magnitude =NiA. Since the coil curries a current, it produces a magnetic field, that can be calculated using Biot-Savart’s law: 0 0 B( z ) 2 2 3/ 2 2 ( R z ) 2 z 3 All loops in the figure have radius r or 2r. Which of these arrangements produce the largest magnetic field at the point indicated?