Physics 112 Magnetism

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Physics 112Magnetism

• Walker, Chapter 22
• Spring 2004

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Magnetism

• Known since antiquity
• Pieces of Magnetite, also called lodestone
  (Fe3O4) known by Greeks to exert both forces of
  attraction and repulsion on each other
• Chinese invented compass for navigation
• The earth exerts a force on magnetite.

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Basic model of Magnetic Materials

• All magnetic materials have two poles
• Labeled North and South poles
• Just as in electrostatics,
• Like repels like and opposites attract.
• N repels N, S repels S, N attracts S

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Magnetic Monopoles

• Unlike electrostatics
• Magnetic monopoles have never been detected.
• But magnetic monopoles would resolve many puzzles
  in particle physics and cosmology

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Modern View of Magnetism(Oersted, Faraday,
Maxwell, 19th Centuryplus Quantum Mechanics
20th Century)

• Magnetism is associated with charges in motion
• These currents can be microscopic currents in the
  atoms of magnetic materials.
• These currents can be macroscopic currents in the
  windings of an electromagnet.

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Magnetic Fields

• The region around a moving charge is disturbed by
  the charges motion. We call this disturbance a
  magnetic field B.
• If an isolated charge is moving, the space
  contains both an electric field AND a magnetic
  field. If the charge is stationary, only an
  electric field is present.

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Magnetic Field Lines

• We can plot the magnetic field lines surrounding
  a magnetic object.
• Magnetic field lines (outside the object) always
  go from the N pole to the S pole

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The Earths Magnetic Field

• The spinning iron core of the earth produces a
  magnetic field.
• The magnetic north pole corresponds to the
  geographic south pole...
• Intense magnetic fields on the surface of the sun
  are associated with sun-spots.

earth
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Magnetic Fields
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Magnetic Forces1. Forces on moving charges2.
Forces on currents in wires or fluids

• A charged particle in a static (not changing with
  time) magnetic field will experience a magnetic
  force only if the particle is moving.
• If a charge q with velocity v moves in a magnetic
  field B and v makes an angle q w.r.t. B, then the
  magnitude of the force on the charge is
• F q v B sinq q v?B

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Magnetic Units

• We can determine the magnetic field by measuring
  the force on a moving charge
• The mks unit of magnetic field is the Tesla (T).
• Dimensional analysis 1 T 1 Ns / (Cm) 1 V
  s / m2
• Sometimes we use a unit called a Gauss (G)
• 1 T 104 G
• The earths magnetic field is about 0.5 G

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Magnetic Forces Lots of Vectors!

• Consider the magnetic force on a current I1
  resulting from its interaction with another
  current I2.
• How many vectors in the problem
• Directions of currents I1 and I2.
• Direction of shortest separation between I1 and
  I2.
• Direction of Magnetic Force
• How can two or more vectors combine to produce
  the resulting force vector?
• Vector addition cannot account for the behavior
  of magnetism.
• The geometry of two vectors defines a third
  vector
• Two vectors define a plane, and define the
  perpendicular to that plane

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Direction of Magnetic Forces

• The magnetic force is in a direction
  perpendicular to the plane formed by B and v.
• (Two vectors determine the a third vector via the
  cross-product
• To determine the direction, you must apply the
  right hand rule.

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Right Hand Rule

• Draw vectors v and B with their tails at the
  location of the charge q.
• Point fingers of right hand along velocity vector
  v.
• Curl fingers towards Magnetic field vector B.
• Thumb points in direction of magnetic force F on
  q, perpendicular to both v and B.

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Walker, Problem 22-8 p. 743

• An electron moving with speed 9.1105 m/s in x
  direction experiences zero magnetic force.
• When it moves in y direction it experiences a
  force 2.0 10-13 N in (-z) direction.
• What is the direction and magnitude of magnetic
  field?

Magnetic field B is along /- x direction (no
force when v and B parallel or antiparallel) Guess
that B along x direction. Then direction of
force on electron travelling in y direction is
in z direction (electron qlt0). B is in x
direction. F q v B B F/(qv) (2.0
10-13 N )/(1.6 10-19 C 9.1105 m/s )1.4 (N s
/ C m) 1.4 T
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Quiz 1
z
v

• A positive charge q is moving in the z direction
  with velocity v.
• The magnetic field B is in the y direction
• What is the direction of the force on q?
• y direction
• -y direction
• Force is zero
• x direction
• -x direction

B
y
x
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Force on a Current Carrying Wire

• Recall that a current in a wire is a collection
  of moving charges therefore, a current carrying
  wire in a magnetic field also experiences a
  force.
• If a wire of length L, carrying a current I,
  makes an angle q with a magnetic field B, then
  the total force on the wire is
• F I L B sin q

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Magnetic Force on a Current

• N charges q move with velocity v along a segment
  of wire of length L.
• Current in wire N q v / L charge flowing
  in/out of wire segment per unit time.
• The wire is in a region of space with magnetic
  field B.
• Force of each charge q v B sinq
• Force on wire segment N q v B sinq I L B
  sinq
• Result is independent of N, q, v
• depends only on I, L

B
q
I
q
F
?
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Right-Hand-Rule for Magnetic Force on a Wire

• Direction of magnetic force I L B sinq on wire
  is perpendicular to direction of I and to
  direction of B.
• Orientation of Force is determined by curling
  fingers from direction to I to direction of B.

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Motion of Charges in B Fields

• If a charged particle is moving in a direction
  perpendicular to a uniform magnetic field, then
  its trajectory will be a circle because the force
  FqvB is always perpendicular to the motion, and
  therefore centripetal.

Recall that so
From which we find the radius of the circular
trajectory is
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Walker, Problem 22-17, p. 744

• An electron accelerated from rest through a
  voltage of 310 V enters a region of constant
  magnetic field.
• The electron follows a circular path with a
  radius of 0.17 m. What is the magnitude of the
  magnetic field?

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Crossed E and B Fields
B
E
q

• Charge q travels at velocity v in perpendicular
  electric and magnetic fields.
• Electric force qE (up)
• Magnetic force qvB (down).

v
If vB E, then net force is 0. Velocity
selector, Charge moving with velocity v
E/B travels in straight line
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Electromagnetic Flowmeter
Moving ions in the blood are deflected by
magnetic force. Positive ions are deflected down,
negative ions are deflected up. This separation
of charge creates an electric field E pointing
up. There is therefore a potential difference V
Ed between the two electrodes. The velocity of
blood flow is measured by v E/B.
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Mass Spectrometer

• Ion source (bio-molecules)
• Velocity selector v
• Semicircular orbit in magnetic field.
• Radius r mv/qB measures mass m, but must make
  an assumption about charge q e, 2e,,

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Mass Spectrometer
Magnetic Field ? to drawing
?
?
?
?
?
?
?
?
?
?
E?B Velocity Selector
?
?
- -

Position Sensitive Detector
Ion source
sample
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Mass Spectrometer
Measures m/q of ions
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Uranium-235, -238 Separation

• U.S. Manhattan Project Calutron
• Iraq pre-1990 project

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Helical Motion

• Resolve velocity into components parallel and
  perpendicular to magnetic field.
• FB is ? to B
• Acceleration a?? 0
• Acceleration a? q v? B / m
• Charged particle spirals along magnetic field
  lines.
• v?? is constant

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Helical motion in space

• Solar flares (left)
• Aurora Borealis (center and right)
• X-Ray Pulsars

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Force on parallel wires

• Each of two parallel wires with current I,
  experiences an attractive magnetic force that
  diminishes as one over the distance separating
  the wires F ? I1 I2 L / d.
• L length
• We use this proportionality to define the unit of
  current
• The force on wire 2 is equal to current in wire 2
  times magnetic field from wire 1 times length of
  wire 2.
• Magnetic field generated by a current diminishes
  as one over distance from wire (1/d)

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Force on perpendicular wires
I1
d
I2

• Two infinitely long perpendicular wires with
  currents I1 and I2 , experience NO (net)
  magnetic force, independent of distance d.
• Magnetic force on wire 2 involves three vectors.
• The displacement vector from wire 1 to wire 2
• The direction of the current in wire 2
• The direction of the current in wire 1

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Definition of Ampere

• The ampere (A) is defined such that two parallel
  wires separated by 1.0m and each carrying 1.00 A
  of current experience a force of attraction of
  210-7 N on each 1.00 m length of wire.

• This defines m0 4p x 10-7 N / A2
• m0 Permeability of free space.
• Positive magnetic force is attractive
• Definition of Coulomb
• 1.00 Coulomb (1.00 Amp) (1.00 sec)

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B Field Outside a Wire

• Earlier we said that magnetic fields are created
  by moving charges. A current in a wire,
  therefore, must create a magnetic field.
• Unlike the Electric field from a line of charges,
  the magnetic field generated by a current in a
  straight wire cannot be radial (outward).
• There is no mathematically consistent way for the
  sign of the current to define whether B is
  radially inward or outward.
• There is no mathematically consistent way for a
  radial B-field to explain that parallel currents
  attract, anti-parallel currents repel, and
  infinitely long perpendicular currents have no
  mutual force.

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Magnetic Field from a Wire

• The magnetic field lines from a current form
  circles around a straight wire with the direction
  given by another right hand rule (thumb in
  direction of current, finger curl around current
  indicating direction of magnetic field).
• Derive strength of magnetic field from equation
  for force per unit length from current I1 on
  current I2

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Quiz 1 Magnetic Field Direction

• What is the direction of the magnetic field at
  the point a) created by the current I?
• Into the screen
• Up on the screen
• Out of the screen
• Down on the screen.

a)
I
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Magnetic Field Superposition

• To compute the force of Current I1 on I2, we
  first compute the magnetic field generated by I1
  at the location of I2.
• To compute the force on a third current I3, we
  must use the superposition principle to calculate
  the magnetic field B1 from I1 and B2 from I2 at
  the location of I3. The total magnetic field
  acting on I3 is BB1B2

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Quiz 2 Superposition

• At the position a) the total magnetic field has a
  contribution from Current 1) and from Current 2)
  B B1 B2
• At a), B1 and B2 are both in same direction
• At a) B1 and B2 are in opposite directions

a)
I1
I2
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Quiz 3 Superposition

• At the position b) the total magnetic field has a
  contribution from Current 1) and from Current 2)
  B B1 B2
• At b), B1 and B2 are both in same direction
• At b) B1 and B2 are in opposite directions

I1
b)
I2
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No Self-Forces

• In Newtonian mechanics, an object cannot exert a
  force on itself.
• What happens if we try to calculate the force on
  current I1 from its own magnetic field?
• At r0 the B-field has no direction, there is no
  net force on the wire from its own magnetic field.

I1
r
B
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Magnetic Torque on current loop

• In a uniform magnetic field, the net force on a
  current loop (independent of geometry) is 0.
• However, there can be a torque t Sr?F
• Each segment of loop experiences a torque rF sinq
• r distance from center of rotation to loop
  segment
• F magnetic force on segment
• q angle between vector r and vector F (tails
  drawn together).
• Torque is a rotational force If you point your
  right-hand thumb in the direction of the torque,
  the torque creates an angular acceleration in the
  direction of your fingers.

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Galvanometer

• Current in coil is finite, due to non-zero
  resistance of coil
• Magnetic field produces torque on current in
  coil.
• Needle swings until magnetic torque is balanced
  by torsion of spring

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Magnetic Force on Current loop

• A free current loop will rotate from the magnetic
  torque until it is ? to magnetic field.
• The loop is then attracted to a region of
  stronger magnetic field.
• This is why iron filings are attracted to the
  poles of a magnet

F
?I
S
N
?I
F
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Walker, Problem 22-37
The loop contains 10 turns with a current 0.22
A. B0.050 T (in horizontal plane) a) Find force
on each segment b) Find net force c) Find
torque d) If the loop rotates freely about the
vertical axis (with a small friction), what is
its equilibrium orientation?
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Problem 22-37a,b

• F1N I L1 B sina 10(0.22 A)(0.08 m) (0.050 T)
  sin90
• F1 0.0088 N perpendicular to I and B
• F2 N I L2 B sin(90-q) 10 (0.22A)(0.15m)(0.050
  T) sin 25
• F2 0.0070 N

F2

• F3 N I L3 B sina
• 10 (0.22 A)(0.08 m) (0.050T) 1
• 0.0088 N
• F4 N I L2 B sin(90q)
• 10 (0.22A)(0.15m)(0.050 T) sin 155
• 0.0070 N

F1
F3
F4
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Problem 37 c,d Torque

• F1 and F3 both tend to rotate loop clockwize, as
  viewed from above.
• F2 and F4 exert no torque about vertical axis.
• t1 F1 (0.15m/2) sin90
• t3 F3(0.15 m/2) sin90
• t t1 t3 (0.0088 N) (0.15 m)
• t 0.00132 Nm
• Equilibrium when q0,

F2
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B Fields of Current Distributions

• By winding wires in various geometries, we can
  produce different magnetic fields.
• For example, a current loop
• (? to plane, radius a, current emerging from
  plane at top of loop)

Magnetic field at center of loop B m0 I /
(2a) Magnetic field far from loop B ? I(Area
of loop) / r3
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Ampères Theorem

• Consider any closed loop in space
• Doesnt have to be a circle, or lie flat.
• The sum (over all segments of the loop) of the
  product of the component of Magnetic field
  parallel to the loop times the length of the loop
  segment is equal to the product of m0 times the
  current enclosed by the loop.

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Amperes Law and a straight wire

• We already argued that the B-field generated by a
  wire has to form circles around the wire.
• Apply Amperes law

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Solenoids

• If we stack several current loops together we end
  up with a solenoid
• In the limit of a very long solenoid, the
  magnetic field inside is very uniform
• Bm0 n I
• n number of windings per unit length,
• I current in windings
• B ? 0 outside windings

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Solenoid Amperes Law
n turns per unit length nL total of windings
LB 0 00 m0 (nL)I Bm0 n I
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Ferromagnetism

• The magnetic torque of one current loop on
  another tends to cause the microscopic magnetism
  of atoms of Fe to align in mesoscopic domains.
• At high temperature, however, the microscopic
  magnets remain randomly oriented (thermal energy
  gt magnetic energy).

• As the solid material cools below the Curie
  temperature (770 C for Fe) magnetic domains form,
  but they remain randomly oriented.

• If a Fe-ore cools in the presence of an external
  magnetic field (Earths field) below the Curie
  temperature, magnetic domains form with a net
  alignment along the Earths field.

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Continental Drift (several cm/year)

• The Atlantic sea-floor on either side of the
  mid-Atlantic ridge forms a mirror-image geologic
  record of the reversals of the Earths magnetic
  field over the past 100 million years.

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Forces on wire, and Current loop.

• Find net force on loop from 14 A current in
  straight wire.
• In which direction will the loop tend to rotate?

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Magnetic field from wire

• Looking in same direction as current flow (14 A)
  , Magnetic Field lines (from that current) form
  loops in clockwise direction. (I am not showing
  the Field generated by 2.5 A current)

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Forces on Loop
F1 I L1 B(r10.2m) (note B is
perpendicular to I) F1 (2.5 A) (1.0m) (2.8e-6
Tm)/ (0.2m) 3.5e-5(Tm A) 3.5e-5 N F2
?? F4 - F2 F4 F2 0 F3 I L3 B(r11.2m) F3
F1 /6 F3 5.83e-6 N No Torque
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Permittivity, Permeability, and the speed of What?

• e0 , m0 defined from electrostatics and
  magnetic forces.

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Permittivity, Permeability, and the resistance of
what?

• e0 , m0 defined from electrostatics and
  magnetic forces.

• Coax cable, 50 W
• Twisted pair, 100 W
• Ratio of inductance to capacitance, in wave
  propagation

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Quiz 2, Feb 21, 2004Name
a)
b)
I1
c)
I2
d)