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Chapter 26: Magnetism Permanent magnets For magnets, like poles repel and opposite poles attract. no monopoles! A permanent magnet will attract a metal like iron with either the north or south pole. Q Magnetic force • The magnetic field, designated , exerts a force on moving electric charges. • The magnetic force may be written in terms of the vector cross product: Clicker question • A proton (charge +e) sits at rest in a uniform magnetic field, B, that is directed to the right. What is the direction of the magnetic force on the charge? 1. to the right 2. up 3. into the screen 4. zero 5. none of the above CT 32.3b A neg. and a pos. particle move with certain velocities in a constant, uniform magnetic field (which points to the right.) The (+) particle moves directly left; the (–) particle moves directly up. The force on the (-) particle due to the Bfield is… A: into page C: 0 B + - B: out of page D: right E: left CT 32.5 Here is an event display from a high energy experiment. (You are seeing millions of tracks of charged particles leaving the central region) There is a 1 Tesla uniform magnetic field coming out of the page. What sign is the electric charge for the highlighted (yellow) charged track? A: + B: - positron Invisible Highenergy photon B More energetic e + e- pair electron Scattered atomic electron CT 32.24 Charges (+q) flows right through a narrow wire, which sits in a uniform B field pointing INTO the page. V between top and bottom of the tube is A: + (top is higher) B: - (top lower) C: 0 Field lines are not lines of force The lines tracing the magnetic field crossed through the velocity vector of a moving charge will give the direction of force by the RHR. Motion of charge in uniform magnetic field • Since the magnetic force is always at right angles to a charged particle’s velocity… • A particle moving in a plane perpendicular to the field undergoes uniform circular motion. • Centripetal force: • The cyclotron frequency: v qB f= = 2πr 2πm • independent of the particle’s speed • useful for particle accelerator • If velocity has a component along B field: spiral motion Mass spectrometer velocity selector: � � � + �v × B � =0 F� = q E E v= B The magnetic force on a current-currying wire • An electric current consists of moving charges, so a current-carrying conductor experiences a magnetic force. � F� = (nAL)(q�v × B) � F� = (qn�v AL) × B � � F� = (JAL) ×B � ×B � F� = I L Magnetic Fields Exert Torques on Current Loops τ = IBA sin(θ) = µB sin(θ) The Electric Motor Slide 24-43 disk drive motor toyota hybrid motor Clicker question • In what direction is the net magnetic force on the current loop (current is counter-clockwise as seen from above)? S N far side 1. up 2. down 3. zero 4. none of the above I near side Loudspeaker engineering • To create music, we need longitudinal pulses in the air. The speaker cone is a very clever combination of induced and permanent magnetism arranged to move the cone to create compressions in the air. Sources of Magnetic Fields Electric Currents Create Magnetic Fields A long, straight wire A current loop A solenoid Slide 24-14 • Current produces a magnetic field • Current produced by moving charges • Moving charges produce magnetic field! • The Biot-Savart law gives the magnetic field arising from an infinitesimal current element: • The field of a finite current follows by integrating: Magnetic field of a straight current carrying conductor: • all components of Idl produce field in same direction (-z axis) sin(φ) = sin(π − φ) = � µ0 I Bz = − 4π µ0 I Bz = − 4π � � a a a a x x2 + y 2 sin(φ[y]) dy 2 r x dy 2 2 3/2 (x + y ) µ0 I 2a √ Bz = − 4π x x2 + a2 • as a goes to infinity: µ0 I Bz = − 2πx CT 32.14b What is the direction of the Force acting on I I the Red Wire? A) Up B) Right C) Left D) Into the Page E) Out of the Page ©University of Colorado, Boulder (2008) Clicker question • A flexible wire is wound into a flat spiral as shown in the figure. If a current flows in the direction shown, will the coil tighten or become looser? A. The coil will tighten. B. The coil will become looser. Using the Biot-Savart law: a current loop • Only component of dB in x direction doesn’t cancel out µ0 I dL dBx = cos θ 2 2 4π x + a • For large distances (x >> a), this reduces to CT 32.16 What is the direction of the B- Field at point P? A: B: C: 0 D: down E: Other ©University of Colorado, Boulder (2008) Magnetic dipoles • The 1/x3 dependence of the current-loop’s magnetic field is the same as the inverse-cube dependence of the electric field of an electric dipole. • In fact, a current loop constitutes a magnetic dipole. • Its dipole moment is µ = IA, with A the loop area. • For an N-turn loop, µ = NIA. • The direction of the dipole moment vector is perpendicular to the loop area. • The fields of electric and magnetic dipoles are similar far from their sources, but differ close to the sources. Dipoles and monopoles: Gauss’s law for magnetism • There do not appear to be any magnetic analogs of electric charge. • Such magnetic monopoles, if they existed, would be the source of radial magnetic field lines beginning on the monopoles, just as electric field lines begin on point charges. • Instead, the dipole is the simplest magnetic configuration. • The absence of magnetic monopoles is expressed in Gauss’s law for magnetism: O • Gauss’s law for magnetism is one of the four fundamental laws of electromagnetism. • Gauss’s law ensures that magnetic field lines have no beginnings or endings, but generally form closed loops. • If monopoles are ever discovered, the right-hand side of Gauss’s law for magnetism would be nonzero. CT 32.14c Two loops of wire have current going around in the same direction. The forces between the loops is: A: Attractive B: Repulsive C: Net force is zero. i2 i1 ©University of Colorado, Boulder (2008) Ampère’s law • Analogous to Gauss’s law, • For steady currents, Ampère’s law is O where the integral is taken around any closed loop, and Iencircled is the current encircled by that loop. • Useful only for current distributions with symmetry • Just as general as BiotSavart law Field inside a long cylindrical conductor µ0 I outside conductor: Iencl=I 2πBr = µ0 I =⇒ B = 2π r 2 r inside conductor: 2πBr = µ Jπr 2 = µ I 0 0 2 R µ0 Ir =⇒ B = 2π R2 Solenoids • A solenoid is a long, tightly wound coil of wire. • When a solenoid’s length is much greater than its diameter, the magnetic field inside is nearly uniform except near the ends, and the field outside is very small. • In the ideal limit of an infinitely long solenoid, the field inside the solenoid is uniform everywhere, and the field outside is zero. • Application of Ampère’s law shows that the field of an infinite solenoid is B = µ0nI, where n is the number of turns per unit length. The Magnetic Field of a Solenoid A short solenoid A long solenoid Slide 24-22 Magnetism in matter • Magnetism in matter arises from atomic current loops associated with orbiting and spinning electrons. Classical picture of • In ferromagnetic materials magnetic dipole moment like iron, strong interactions arising from orbiting electron among individual magnetic dipoles result in large-scale magnetic properties, including strong attraction to magnets. • Paramagnetic materials exhibit much weaker magnetism. • Diamagnetic materials respond oppositely, and are repelled by magnets. Origin of a permanent magnet Electron Magnetic Moments: Ferromagnetism A nonmagnetic solid (copper) A ferromagnetic solid (iron) Inducing a Magnetic Moment in a Piece of Iron Electromagnet • Use current to induce magnetic moment in iron