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Chapter 21 Magnetic Force Clicker What is the potential difference across the clams when I take bulb out of circuit? A) 1.7 V B) 3.4 V C) 5.1 V D) 6.8 V E) 12 V Magnetic Field of a Moving Charge The Biot-Savart law for a moving charge 0 qv rˆ B 4 r 2 The Biot-Savart law for a short piece of wire: 0 Il rˆ B 4 r 2 How magnetic field affects other charges? Magnetic Force on a Moving Charge Fmagnetic qv B q – charge of the particle v – speed of the particle B – magnetic field N N T C m/s A m Direction of the magnetic force depends on: the direction of B the direction of v of the moving charge the sign of the moving charge A negative charge is placed at rest in a magnetic field as shown below. What is the direction of the magnetic force on the charge? B A. B. C. D. E. Up Down Into the page Out of the page No force at all. A negatively charged particle is moving horizontally to the right in a uniform magnetic field that is pointing in the same direction as the velocity. What is the direction of the magnetic force on the charge? V A. B. C. D. E. Up Down Into the page Out of the page No force at all. B Now, another negatively charged particle is moving upward and to the right in a uniform magnetic field that points in the horizontal direction. What is the direction of the magnetic force on the charge? V A. B. C. D. E. Left Up Down Into the page Out of the page B Magnetic Force on a Current-carrying Wire Fm qv B Current: many charges are moving Superposition: add up forces on individual charges Number of moving charges in short wire: nAl Total force: I Force of a short wire: Fm Il B In metals: charges q are negative. Will this equation still work? Forces Between Parallel Wires Definition of 1 Ampere: Ampere is defined as a current at which two very long parallel wires 1 m apart create a force on each other of 2.10-7 N per meter length. From this also follows that 0/(4) = 10-7 T.m/A Forces Between Parallel Wires For long wire: m0 2 I1 B1 = 4p d Magnetic force on lower wire: Fm = IDl ´ B F21 = I 2 LB1 sin 90 o m0 2I1 F21 = I 2 L 4p d Magnetic force on upper wire: m0 2 I 2 B2 = 4p d F12 = I1LB2 sin 90 o m0 2I 2 F12 = I1 L 4p d What if current runs in opposite directions? Electric forces: “likes repel, unlikes attract” Magnetic forces: “likes attract, unlikes repel” Effect of B on the Speed of the Charge Fmagnetic qv B What is the effect on the magnitude of speed? F dl 0 Kinetic energy does not change Magnetic field cannot change a particle’s energy! Magnetic field cannot change a particle’s speed! Magnetic force can only change the direction of velocity but not its magnitude Magnitude of the Magnetic Force Fmagnetic qv B Single electron in television tube: dp qvB sin dt dv q vB sin dt m (v<<c) e/me = 1.78.1011 C/kg Circular Motion at any Speed Fmagnetic qv B Any rotating vector: dX X …angular speed dt dp d p p qv B q vB sin 90o dt dt mv 1 v / c 2 2 q vB qB 1 v2 / c2 m Cyclotron Frequency Circular Motion at Low Speed qB 1 v2 / c2 m qB if v<<c: m independent of v! Alternative derivation: F ma 2 v q vBsin 90o m R q B m Period T: 2 T Circular motion: v2 v a R R qB m m T 2 qB Non-Relativistic A Cyclotron qB v<<c : m V V0 cos t Vequivalent 2 NV0 Vequivalent ~ 108 V v~c : need to adjust B or Exercise v|| What if v is not perpendicular to B? Fmagnetic qv B Direction? v Magnitude? Fm qv B Trajectory: helix Determining e/m of an Electron p q Br q v m Br mv2 q V 2 2 q v2 2 2 Br m q v 2V m 2 2 q 2 V q 2 2 m m B r e 2V 2 2 m Br Joseph John Thomson (1856-1940) 1897: m/e >1000 times smaller than H atom Fmagnetic qv B Fe qE F qE qv B Magnetic force can only change the direction of velocity but not its magnitude 2 v F ma a 2 R v o q vBsin 90 m R (high q – small R, large m large R) Applications: find p, e/m, separate particles (including 238U and 235U (natural abundance 0.7%) isotopes Proton energy Velocity, m/sec R, m 1MeV 1.3*107 0.15 m 7000 GeV 0.9999995c 4.2 km e = 1.6*10-19 C mproton = 1.7*10-27 kg Giga 109 What is the radius of circle of proton accelerated over 7000GeV in B =5.5 T? Proton speed up to 0.9999995c mv 1 R B q 1 v 2 / c2 Particle Accelerator Fermilab: Tevatron CERN: Large Hadron Collider Applications: study nucleus – accelerate particles (protons) and shoot at nucleus to observe nuclear reactions create new particles by colliding particles Synchrotron – source of synchrotron radiation To keep R constant B is increased up to 3 T (superconducting coils) Argonne National Laboratory