Magnetic Forces and Fields (Chapters 29-30) freamamv/Secondary/PHYS155/L05.pdf · PDF...

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Transcript of Magnetic Forces and Fields (Chapters 29-30) freamamv/Secondary/PHYS155/L05.pdf · PDF...

  • Magnetic Forces and Fields

    (Chapters 29-30)

    • Magnetism – Magnetic Materials and Sources

    • Magnetic Field, B

    • Magnetic Force

    • Force on Moving Electric Charges – Lorentz Force

    • Force on Current Carrying Wires

    • Applications

    • Electromagnetic motors

    • Torques on magnetic dipole moments μ

    • Sources of Magnetic Field

    • Magnetic field of a moving charge

    • Current Carrying Wires – Biot-Savart law

    • Loops, Coils and Solenoids – Ampère’s Law

    • Microscopic Nature of Magnetism

  • How can one magnetize objects?

    • Magnetism can be induced: either by stroking an unmagnetized piece of

    magnetizable material with a magnet, or by placing it near a strong permanent magnet

    • Soft magnetic materials, such as iron, are easily magnetized

    - They also tend to lose their magnetism easily

    • Hard magnetic materials, such as cobalt and nickel, are difficult to magnetize

    - They tend to retain their magnetism

    • Magnets are objects exhibiting magnetic behavior or magnetism

    • A magnet exhibits the strongest magnetism at extremities called

    magnetic poles: any magnet has two poles, conventionally

    dubbed north and south

    • Like poles repel each other and unlike poles attract each other

    • Unlike charges, magnetic poles cannot be isolated into

    monopoles: if a permanent magnetic is cut in half repeatedly, the

    parts will still have a north and a south pole

    • The region of space surrounding a moving charge includes a magnetic field as well

    as by an electric field: so, magnetism and electricity cannot be separated

    • They are interrelated into the integrated field of electromagnetism: the first

    breakthrough in the great effort of developing unified theories about fundamental

    interactions (Maxwell, beginning of the XIX-th century).

    Magnetism – Magnets and relation to electricity

  • Magnetic Field – Operational definition and field lines

    • Like the sources of electric field, any magnetic material produces a magnetic field

    that surrounds it and extends to infinity.

    • The symbol used to represent this vector is

    • Let’s first describe this vector using an operational definition:

    B

    Def: The magnetic field in each location in the surroundings of a magnetic source

    is a vector with the direction given by the direction of the north pole of a compass

    needle placed in the respective location

    • Similar to the electric field, a magnetic field can be patterned using field lines: the

    vector B in a point is tangent to the line passing through that point, and the density of

    lines represents the strength of the field

    • However, while electric field lines start and end on

    electric charges (electric monopoles), the magnetic

    field lines form closed loops (since there are no

    magnetic monopoles)

    • Thus, the magnetic field lines should be seen as

    closing the loops through the body of the magnet:

    that is, the magnetic field inside magnets is not zero

    Ex: A compass can be used to

    trace the magnetic field lines

  • • A compass can be used to probe the magnetic field lines produced by various

    source, and they will always form closed loops

    • Later we’ll look at these sources more systematically:

    Magnetic Field – Magnetic field lines for various magnetic sources

    Notice the similar

    pattern

  • Magnetic Field – Example: Earth’s Magnetic Field

    • The Earth’s geographic north pole is closed to

    a (slowly migrating) magnetic south pole

    • The Earth’s magnetic field resembles that of a

    huge bar magnet deep in the Earth’s interior

    slightly tilted with respect to the axis of

    rotation of the planet

    • The mechanism of Earth’s magnetism is not

    very well understood

    • There cannot be large masses of permanently

    magnetized materials since the high

    temperatures of the core prevent materials from

    retaining permanent magnetization

    • The most likely source is believed to be electric currents in the liquid part of the

    planetary core

    • The direction of the Earth’s magnetic field reverses every few million years

    • The origin of the reversals is not well understood in detail, albeit there are models

    describing how it may happen

  • Magnetic Force – On a moving charge

    • Magnetic fields act on moving charges with magnetic forces. We’ll study this effect

    in two (related) cases:

    1. Moving Charged Particles

    2. Current Carrying Wires

    1. Magnetic Force on a Moving Point Charge

    • Consider a test charge q moving in a field B with velocity

    v making an angle θ with B: the particle will be acted by a

    magnetic force F (sometimes called Lorentz force) of

    Magnitude:

    Direction: given by a right hand rule (let’s call it #1):

    F

    v

    B

    θ q>0

    F q>0

    F q

  • Notation: Vectors perpendicular on page/board/slide:

    Outward Inward

    Exercises:

    1. Force direction: Find the direction of the force on an electron moving through the magnetic

    fields represented below.

    Problem:

    1. Charge moving in a magnetic field: What velocity would a proton need to circle Earth 800

    km above the magnetic equator, where Earth's magnetic field is directed horizontally north and

    has a magnitude of 4.0010-8 T?

    2. Field direction: Find the direction of the magnetic

    field acting on a proton moving as represented by the

    adjacent velocity and force vectors. (Assume that the

    velocity is perpendicular on the magnetic field.)

  • • Any moving charge not only that is acted by a magnetic field but it also produces a

    magnetic field that surrounds it and extends to infinity

    • A test charge q moving in an electric field E and a magnetic field B, with velocity

    making an angle θ with B will be acted by a net electromagnetic force (sometimes

    called Lorentz force):

    Magnetic Force – Charge in an electromagnetic field

     electric magneticF BF q vEF     parallel to the

    direction of E

    perpendicular on

    the direction of B

    Ex: One type of velocity selector

    • Consider an electric field perpendicular on a magnetic field

    • Then only the particles entering the fields with velocity perpendicular

    of both will be allowed to pass, which corresponds to the following

    condition that the particles are supposed to obey:

    +

    +

    0 E

    qE qvB v B

        

  • Magnetic Force – Trajectory of a point charge in a magnetic field

    • Let’s look at two particular trajectories that a charged

    particle may have in a magnetic field

    1. Consider a particle moving into an external magnetic field

    so that its velocity is perpendicular to the field

    • In this case, the particle will move in a circle, with the

    magnetic force always directed toward the center of the

    circular path

    • Equating the magnetic and centripetal forces, we can find the radius of the circle:

    2v F qvB m

    r   

    m r

    B

    v

    q 

    +

    +

    +

    : called cyclotron equation

    2. If the particle’s velocity is not perpendicular to

    the field, the path followed by the particle is a spiral

    called a helix

    • The helix spirals along the direction of the field

    with a velocity given by the component of the

    velocity parallel with B

    v

    v

    v

    v B

    F

    FF

    +

  • • A current is a collection of many drifting charged particles, such that a magnetic

    force is expected to act on a current-carrying wire placed in a magnetic field

    • This magnetic force is the resultant of the forces acted on the individual microscopic

    electric carriers, but it makes more sense to integrate its effects into a unique

    magnetic force acted on the macroscopic current

    I = 0  F = 0 I ↑  F ← I ↓  F→

    Magnetic Force – Currents in magnetic field

    Ex: Experimental observations:

    A current carrying vertical wire

    placed in a magnetic field pointing

    perpendicular into the slide, will be

    acted by a magnetic force

    perpendicular on the current and

    magnetic field: either to the left, or

    to the right, depending on the

    direction of the current

  • 2. Magnetic Force on Current Carrying Wire

    • Consider a straight current carrying wire of length ℓ

    immersed in field B, making an angle θ with B: the

    portion dℓ of wire will be acted by a magnetic force dF

    Magnitude:

    Direction: Given by right hand rule #1, but instead of

    aligning the fingers with the velocity, one aligns the

    fingers with the direction of the current

    • Since the cu