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Chapter 21 Magnetic Force

Magnetic Field of a Moving Charge The Biot-Savart law for a moving charge The Biot-Savart law for a short piece of wire: How magnetic field affects other charges?

Magnetic Force on a Moving Charge q – charge of the particle v – speed of the particle B – magnetic field 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 Electrons are accelerated and deflected vertically by a magnetic field

A negative charge is placed at rest in a magnetic field as shown below 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 Up Down Into the page Out of the page No force at all. 73 of 140

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 B Up Down Into the page Out of the page No force at all. 72 of 140

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 B Left Up Down Into the page Out of the page E 70 of 140

Magnetic Force on a Current-carrying Wire Current: many charges are moving Superposition: add up forces on individual charges Number of moving charges in short wire: I Total force: Electrons drift in a direction OPPOSITE to the conventional current. The term qv has 2 minus which cancel Force of a short wire: 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 Andre Marie Ampere 1775-1836, never attended school (father educated him). Read Ancyclopedia through starting from A etc. First paper at age 13 – not accepted… . Maxwell, writing about this Memoir in 1879, says:- We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shown us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html

Forces Between Parallel Wires For long wire: Magnetic force on lower wire: Magnetic force on upper wire: Note that in this case, F12 and F21 are equal and opposite, obeying reciprocity. But this is not true of all velocity-dependent cases. 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 What is the effect on the magnitude of speed? 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 Single electron in television tube: (v<<c) e/me = 1.78.1011 C/kg Effect is not significant for charge tape but is significant for electrone.

Circular Motion at any Speed Any rotating vector: …angular speed Cyclotron Frequency

Circular Motion at Low Speed if v<<c: independent of v! Alternative derivation: Circular motion: Red oval – can determine particle momentum Period T: Non-Relativistic

A Cyclotron Vequivalent ~ 108 V  v~c : need to adjust B or  Used to study nucleus – accelerate particles (protons) and shoot at nucleus to observe nuclear reactions Also create new particles – collide particles Synchrotron – source of synchrotron radiation D-shaped boxes (Dees)

Exercise What if v is not perpendicular to B? Direction? Magnitude? Trajectory: helix

Determining e/m of an Electron

Joseph John Thomson (1856-1940) 1897: m/e >1000 times smaller than H atom

(high q – small R, large m large R) Magnetic force can only change the direction of velocity but not its magnitude Coulomb law and Biot-Savart law have coefficients 1/(40) and 0/(4) to make the field and force equations consistent with each other (high q – small R, large m large R) Applications: find p, e/m, separate particles (including 238U and 235U (natural abundance 0.7%) isotopes

Clicker: Radius of circle of 1MeV proton in B =1T magnetic field is Proton energy Velocity, m/sec R, m 1MeV Proton energy Velocity, m/sec R, m 1MeV 1.3*107 0.15 m e = 1.6*10-19 C mproton = 1.7*10-27 kg Clicker: Radius of circle of 1MeV proton in B =1T magnetic field is 0.01 m 0.15 m 1 m 1 km

e = 1.6*10-19 C mproton = 1.7*10-27 kg Giga 109 Proton energy Velocity, m/sec R, m 1MeV 1.3*107 0.15 m 7000 GeV Proton energy Velocity, m/sec R, m 1MeV 1.3*107 0.15 m 7000 GeV 0.9999995c 4.2 km What is the radius of circle of proton accelerated over 7000GeV in B =5.5 T? Proton speed up to 0.9999995c Large Hadron collider 27 km, 17 miles

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