1. Magnetic field B (Tesla - College of Engineering ?· 1. Magnetic field B (Tesla) 2. Force acting…

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  • 1

  • 2

    1. Magnetic field B (Tesla)

    2. Force acting on a moving charge in B field: Fq

    3. Cyclotron motion

  • 3

    Historical milestones 5-th century B.C. Greeks knew about rocks in Magnesia

    (Western Turkey) that attract each other. They named rocks magnetite, chemical formula

    ~110 A.D. Chinese use compass to navigate Late 12-th century. Europeans invent loadstone=leading

    stone ~1600 William Gilbert. The Earth is a giant magnet ~1800 Hans Christian Oerstead. Compass is sensitive to

    electric current >1800 Ampere, Faraday, Maxwell. Universal theory of

    magnetism

    43OFe

  • 4

    Magnetic field

    We recognize the fact that the magnetic field is established around us by placing a compass needle (test magnet) that would point to magnetic SOUTH

    Postulate: Needles North pole shows the direction of magnetic field (towards South Pole)

    Compass cannot say anything about the magnetic field magnitude

  • 5

    Bar magnet (Permanent magnet)

    North pole is marked by RED color South pole is marked by BLUE color

    Magnetic field flux lines can be visualized with iron filings acting as little magnets Magnetic field flux lines are continuous loops, they emerge from N pole, arrive to S pole and continue inside the magnet Magnetic flux lines do not touch and cross each other One cannot get a magnetic monopole say S Pole; N and S poles always coexist Magnetic flux lines show the direction of magnetic field at every spot Magnetic flux lines act as rubber bends try to squeeze themselves

  • 6

    Earth magnetic field Permanent magnet

    North Magnetic Pole is located on Canadian territory

    North Magnetic Pole is off the geographic Pole; the separation is ~17 deg

    North Magnetic Pole should be actually called South Magnetic Pole, for doing physics

  • 7

    North (South) magnetic pole

    This figure shows the path of the North Magnetic Pole since its discovery in 1831 to the last observed position in 2001. During the last century the Pole has moved a remarkable 1100 km. What is more, since about 1970 the NMP has accelerated and is now moving at more than 40 km per year. If the NMP maintains its present speed and direction it will reach Siberia in about 50 years. Such an extrapolation is, however, tenuous. It is quite possible that the Pole will veer from its present course, and it is also possible that the pole will slow down sometime in the next half century.

  • 8

    Properties

    Like poles repel each other

    Unlike poles attract

    each other

  • 9

    Why North magnetic Pole should be South magnetic Pole?

    Because unlike poles attract each other, the needles North end points towards the South Pole in the Earth magnetic field, but we call this South Pole the North Pole, nevertheless

  • 10

    Force on a moving charge in B field

    sin||];[ qVBFBVqF qq ==

    (Vector product)

    This equation can be used as a definition of B field

  • 11

    Direction of the vector product

    Use ALWAYS right hand (right hand rule)

    3 2[ ]= 1V V V

  • 12

    Direction of the vector product

    Use ALWAYS right hand (right hand rule)

    3 2[ ]= 1V V V

  • 13

    Force on a moving charge in B field

    sin||];[ qVBFBVqF qq ==

    F=0 F=qVB F=qVBsin()

    Examples:

  • 14

    Important points regarding magnetic force Fq

    1) POLARITY of magnetic force depends on charge; the force acts in opposite directions for electrons

    (-charge) and protons (+ charge) 2) Magnetic force is zero for - non-moving charge - for motion along the B field direction 3) Notation for directions: away, towards you, down to the page out of page B B

  • 15

    Nikola Tesla Inventor (1856 1943)

    Units of B field

    Unit for B field magnitude is Tesla (T), after Nikola Tesla, Croatian-born US engineer

    B=F/ (q v sin ) [1 T ] =N/C/(m/s) or since 1 C/s= 1Amper [1 T ]= N/(Am) Non-SI unit for B is Gauss (G): I G=10-4 T

  • 16

    Work done by B field Magnetic field does

    not do work over a particle and does not change its kinetic energy as the force is always perpendicular to the velocity (scalar product of force and displacement vectors or F and velocity is zero)

    /

    | | | | cos( / 2) 0

    Work second F VFor mag field forceV B

    =

    V

    B

    [ ]F q V B=

    Is q positive here?

  • 17

    Comparing effects of E field and B field

    Positive charge is shifted along E field towards negatively charged plate. The trajectory is a parabola

    Positive charge is

    shifted upwards. The trajectory is a circle

  • 18

    Velocity separator and making a straight trajectory for a charged particle

    E/BVBVqEq ==

    Particle detector/ collector

  • 19

    Circular motion in B field

    1) Centripetal acceleration, centripetal force 2) Radius of the orbit 3) Radial frequency

  • 20

    Note on centripetal (center seeking) acceleration

    Even though (circular motion with constant speed), the velocity changes its direction so that

    0| ( ) | | ( ) |V t V t=

    0( ) ( ) 0V t V t V =

  • 21

    Centripetal acceleration

    carV

    tV

    rtV

    VV

    ==

    =

    2

    For circular motion, the resultant force should always be directed toward the center

    Direction of ac: Toward the center 0( ) ( )V t V t V =

  • 22

    Derivation of expressions for the radius of gyration and angular (cyclotron) frequency

    2

    ; ;c qV mVma F m qVB rr qB

    = = =

    mqB

    =sec

    r VSo much of a distancecovered per

    =

    ( / )V V qBr mV qB m

    = = =

    22 ( ) [ ]( ) sec

    . . sec( 2 s in )

    radianf angular frequencyT period

    ang freq so many radians perthere are radian one circle

    = = =

    =

    1 1( ) [ ]( )sec

    f frequencyT period s

    so many turns per

    =

  • 23

    Cyclotron frequency of gyration

    mVrqB

    =

    Radius of the orbit

    mqB

    =

  • 24

    Aurora Borealis

  • 25

    Aurora borealis is a global phenomenon

  • 26

    Plasma (electrons & protons) is ejected from the Sun toward the Earth

  • 27

    Particles from the Sun are diverted by the Earth magnetic field, but some of them precipitate into the upper atmosphere, causing aurora borealis

  • 28

    Example of a problem: aurora Aurora borealis occurs because energetic

    electrons and protons bombard the atmospheric gases at the height of ~ 100-150 km. Assume that the precipitating protons and electrons have energy of 5 eV and move almost perpendicular the Earths magnetic field (0.5 G). (a) Find the radius of proton (electron) gyration and the radial frequency of their motion. (b) If you are looking along the Earths magnetic field, what is the direction of the proton (electron) gyration, clockwise or counterclockwise?

  • 29

    Solution: Electron motion

  • 30

    Solution: Proton motion

  • 31 Of two particles accelerated to the same velocity, m & m1, which one is lighter?

    Applications: The mass spectrometer

    Ions are accelerated in electric field

    Ions travel in B field along trajectories of different radius, depending on mass

    Detector selects particles of proper mass, certainly you have to calibrate the instrument

  • 32

    Example of a problem: A proton is released from rest

    at point A. The proton is then accelerated toward the negative plate of a capacitor. It leaves the capacitor at B through a small hole in the plate. The electric potential of the positive plate with respect to the negative plate is 2.1 kV. Outside the capacitor, there is constant magnetic field of 0.1 T, directed perpendicular to the plates. Find: (a) the speed of the proton at point B and (b) the largest separation of the proton from B.

    Point B

    A

  • 33

    Solution: Mass spectrometer

  • 34

    Comments of particle motion in electric field

    chargeForce Electric

    q= =

    electrFE

    1) Electric field is directed from + charge towards charge:

    2) Positive charge (free of other forces) moves in the direction of E field

    3) Negative charge (free of other forces) moves against the direction of E field

    + -

    E

    E E

    - +

    Positive q Negative q

  • 35

    Conceptual analogy between E & B fields

    Electric field Magnetic field

    In both cases, the field characterizes force acting on a particle

    (positive)

    (shown is force on a negative particle)

  • 36

    Summary of equations

    sin||];[ qVBFBVqF qq ==

    rVac

    2

    =

    qBmVr =

    mqB

    =

    ( & )EVel Crossed E BB

    =

    Slide Number 1Magnetic field B (Tesla)Historical milestonesMagnetic fieldBar magnet (Permanent magnet)Earth magnetic field Permanent magnetNorth (South) magnetic polePropertiesWhy North magnetic Pole should be South magnetic Pole?Force on a moving charge in B field Direction of the vector product Direction of the vector product Force on a moving charge in B field Important points regarding magnetic force FqUnits of B field Work done by B fieldComparing effects of E field and B fieldVelocity separator and making a straight trajectory for a charged particleCircular motion in B field Note on centripetal (center seeking) accelerationCentripetal accelerationDerivation of expressions for the radius of gyration and angular (cyclotron) frequencyCyclotron frequency of gyrationSlide Number 24Aurora borealis is a global phenomenonPlasma (electrons & protons) is ejected from the Sun toward the EarthParticles from the Sun are diverted by the Earth magnetic field, but some of them precipitate into the upper atmosphere, causing aurora borealisExample of a problem: aurora Solution: Electron motion Solution: Proton motion Applications:The mass spectrometerExample of a problem:Solution: Mass spectrometerComments of particle motion in electric fieldConceptual analogy between E & B fieldsSummary of equations

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