Lecture 9 Magnetic Fields due to Currents Ch. 2006. 7. 17.¢  Torques on current loops Electric motors

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Text of Lecture 9 Magnetic Fields due to Currents Ch. 2006. 7. 17.¢  Torques on current loops...

  • Lecture 9 Magnetic Fields due to Currents Ch. 30 • Cartoon - Shows magnetic field around a long current carrying wire and a loop of wire

    • Opening Demo - Iron filings showing B fields around wires with currents

    • Warm-up problem

    • Topics

    – Magnetic field produced by a moving charge

    – Magnetic fields produced by currents. Big Bite as an example.

    – Using Biot-Savart Law to calculate magnetic fields produced by currents.

    – Examples: Field at center of loop of wire, at center of circular arc of wire, at center of segment of wire.

    – Amperes’ Law : Analogous to Gauss’ L:aw in electrostatics, Useful in symmetric cases.

    – Infinitely long straight wire of radius a. Find B outside and inside wire.

    – Solenoid and Toroid Find B field.

    – Forces between current carrying wires or parallel moving charges

    • Demos

    – Torque on a current loop(galvanometer)

    – Iron filings showing B fields around wires with currents.

    – Compass needle near current carrying wire

    – Big Bite as an example of using a magnet as a research tool.

    – Force between parallel wires carrying identical currents.

    Reminders

    Ungraded problem solutions are on website?

    UVa email activated

    Quiz tomorrow

    Class in afternoon 1:00 PM

    Grades are on WebAssign

  • Magnetic Fields due to Currents

    Torque on a coil in a magnetic field demo– left over from last time

    • So far we have used permanent magnets as our source of magnetic

    field. Historically this is how it started.

    • In early decades of the last century, it was learned that moving charges

    and electric currents produced magnetic fields.

    • How do you find the Magnetic field due to a moving point charge?

    • How do you find the Magnetic field due to a current?

    – Biot-Savart Law – direct integration

    – Ampere’s Law – uses symmetry

    • Examples and Demos

  • Torques on current loops

    Electric motors operate by connecting a coil in a magnetic field to a current supply, which produces a torque on the coil causing it to rotate.

    P

    F

    i a

    b F

    B

    B

    Above is a rectangular loop of wire of sides a and b carrying current i.

    B is in the plane of the loop and ̂ to a.

    Equal and opposite forces are exerted on the sides a. No forces exerted on b since

    Since net force is zero, we can evaluate t (torque) about any point. Evaluate it about P (cm).

    iaBF =

    Bi

    ! = NiabBsin" = NiABsin"

    i

    .

    A=ab=area of loop

    ! = Fbsin"

    F = iaB

    ! = iabBsin"

    For N loops we mult by N

  • Torque on a current loop

    ! = NiABsin"

    µ = NiA = magnetic dipole moment

    ! = µBsin"

    ! = µ # B

    !

    B

    !n̂

    Must reverse current using

    commutator to keep loop

    turning.

    Torque !!tends to rotate loop until plane is " to B

    ( parallel to B).n̂

  • http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/motdc.html#c4

    Split ring

    commutator +

    -

  • Electron moving with speed v in a crossed electric

    and magnetic field in a cathode ray tube.

    ! F = qE + q

    ! v x ! B

    y

  • Discovery of the electron by J.J. Thompson in 1897

    2

    2

    mv2

    qEL y =

    1. E=0, B=0 Observe spot on screen

    2. Set E to some value and measure y the deflection

    3. Now turn on B until spot returns to the original position

    B

    E v

    qvBqE

    =

    =

    4 Solve for yE2

    LB

    q

    m 22 =

    This ratio was first measured by Thompson

    to be lighter than hydrogen by 1000

    Show demo of CRT

  • Topic: Moving charges produce magnetic fields

    q

    r

    " r̂

    2

    0 ˆ

    4 r

    rv qB

    ! =

    ! !

    "

    µ

    2

    7 0 104

    A

    N! "= #µ

    Determined fom

    Experiment

    ! v

    1. Magnitude of B is proportional to q, , and 1/r2.

    2. B is zero along the line of motion and proportional to sin at other

    points.

    3. The direction is given by the RHR rotating into

    4. Magnetic permeability

    ! v

    ! v

  • Example:

    A point charge q = 1 mC (1x10-3C) moves in the x direction with

    v = 108 m/s. It misses a mosquito by 1 mm. What is the B field

    experienced by the mosquito?

    108 m/s 90o

    r̂ 2

    0

    4 r

    v qB

    !

    µ =

    26

    83

    2

    7

    10

    1 101010

    m CB

    s

    m

    A

    N

    !

    !! """=

    TB 4

    10=

  • Recall the E field of a charge distribution

    “Coulombs Law”

    r̂ r

    kdq Ed

    2 =

    !

    To find the field of a current distribution use:B !

    2

    0

    r

    r̂sid

    4 Bd

    !

    "

    µ =

    ! !

    Biot-Savart Law

    Topic: A current produces a magnetic field

    This Law is found from experiment

  • Find B field at center of loop of wire lying in a plane with radius R

    and total current i flowing in it.

    B = µ0

    4!" i d ! l # r̂

    r 2

    R

    i

    r̂ld ! !

    is a vector coming out of the paper The angle

    between dl and r is constant and equal to 90

    degrees.

    ! B = d

    ! B! =

    µ0

    4" k̂ idl

    R 2! =

    µ0

    4" k̂ i

    R 2 dl! =

    µ0

    4" k̂ i

    R 2 2"R

    k̂ R2

    i B

    0µ = !

    Magnitude of B field at center of

    loop. Direction is out of paper.

    R k̂ i

    d ! l ! r̂ = dl sin" k̂

    sin" = sin90 = 1

    d ! l ! r̂ = dl k̂

    ld !

    P

    "

    Magnitude is µ0i

    2R

    Direction is k̂

  • ! B = d

    ! B! =

    µ0

    4" k̂ idl

    R 2!

    = µ0

    4"

    i

    R 2 k̂ dl li

    l f

    ! = µ0

    4"

    i

    R 2 k̂ l

    li

    l f

    = µ0

    4"

    i

    R 2 k̂(l f # li )

    = µ0

    4"

    i

    R 2 k̂(2"R # 0)

    = µ 0 i

    2R k̂

    ld !

    P

    "

    Integration Details

  • Example

    Loop of wire of radius R = 5 cm and current i = 10 A. What is B at the

    center? Magnitude and direction

    R

    i B

    2

    0µ =

    )05(.2

    10 104

    2

    7

    m

    A B

    A

    N! "= #

    TB 26

    10102.1 !"= #

    GaussTB 2.1102.1 4

    =!= " Direction is out of

    the page.

    i

  • What is the B field at the center of a segment or circular

    arc of wire?

    "0 R

    i ld !

    Total length of arc is S.

    Why is the contribution to the B field at P equal to zero from

    the straight section of wire?

    P

    !0 = 100 deg=100 deg " 6.28 rad

    360 deg = 100 " 0.0174 = 1.74 rad

    ! B = d

    ! B! =

    µ0

    4"

    i

    R 2 k̂ dl!

    = µ0

    4"

    i

    R 2

    k̂ S

    ! B =

    µ0

    4!

    i

    R "

    0 k̂

    where S is the arc length S = R"0

    "0 is in radians (not degrees)

    B = µ0

    4!" i d ! l # r̂

    r 2

    d ! l ! ! r = rdl sin" k̂

    sin" = sin0 = 0

    d ! l " r̂

    d ! l " r̂

  • Find magnetic field at center of arc length

    0

    0

    4 !

    "

    µ

    R

    i B =

    Suppose you had the following loop.

    What is the magnitude and direction of B at the origin?

    S R

    i B

    2

    0

    4!

    µ =

    0 0

    00 0

    2 )

    2/ (

    4 !

    "

    µ !!

    "

    µ

    R

    i

    R

    i

    R

    i B #=#=

  • Next topic: Ampere’s Law

    Allows us to solve certain highly symmetric current problems for the

    magnetic field as Gauss’ Law did in electrostatics.

    Ampere’s Law is cIldB 0µ=!" !! Current enclosed by the path

    ! E"! "d ! A =

    qenc

    # 0

    Gauss’s Law Charge enclosed by surface

  • Example: Use Ampere’s Law to find B near a very long, straight

    wire. B is independent of position along the wire and only depends

    on the distance from the wire (symmetry).

    rld !

    i i

    By symmetry ldB !!

    irBdlBBdlldB 02 µ! ====" # ## !!

    r

    i B

    !

    µ

    2

    0

    = Suppose i = 10 A

    R = 10 cm T 102

    10

    10102

    5

    1

    7

    !

    !

    !

    "=

    "" =

    B

    B

    Show Fe fillings around a straight wire with current,

    current loop, and solenoid.

    2

    7 0 104

    A

    N! "= #µ

  • Rules for finding direction of B field from a current

    flowing in a wire

    r2