19'3 Magnetic Fields

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19.3 Magnetic Fields

• Stationary charged particles do NOT interact with
  a magnetic field.
• Charge moving through a magnetic field experience
  a magnetic force.
• Value of the force is maximum when the charge
  moves perpendicularly to the field lines.
• Value of the force is zero when the charge moves
  parallel to the field lines.

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Magnetic Fields in analogy with Electric Fields

• Electric Field
• Distribution of charge creates an electric field
  E(r) in the surrounding space.
• Field exerts a force Fq E(r) on a charge q at r
• Magnetic Field
• Moving charge or current creates a magnetic field
  B(r) in the surrounding space.
• Field exerts a force F on a charge moving q at r

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Strength of the Magnetic Field

• Define the magnetic field, B, at a given point in
  space in terms of the magnetic force imparted on
  a moving charge at that point.
• Observations show that the force is proportional
  to
• The field
• The charge
• The velocity of the particle
• The sine of the angle between the field and the
  direction of the particles motion.

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Strength and direction of the Magnetic Force on a
charge in motion
F
B
q
v
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Magnetic Field Magnitude
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Magnetic Field Units

• F newton
• v m/s
• q C
• B tesla (T).
• Also called weber (Wb) per square meter.
• 1 T 1 Wb/m2.
• 1 T 1 N s m-1 C-1.
• 1 T 1 N A-1 m-1.
• CGS unit is the Gauss (G)
• 1 T 104 G.

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Right Hand Rule

• Provides a convenient trick to remember the
  spatial relationship between F, v, and B.
• Consider the motion of positive charge
• Direction of force reversed if negative charge.

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Example Proton traveling in Earths magnetic
field.

• A proton moves with a speed of 1.0 x 105 m/s
  through the Earths magnetic field which has a
  value of 55 mT a particular location. When the
  proton moves eastward, the magnetic force acting
  on it is a maximum, and when it moves northward,
  no magnetic force acts on it. What is the
  strength of the magnetic force? And what is the
  direction of the magnetic field?

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19.4 Magnetic Force on Current-carrying conductor.

• A magnetic force is exerted on a single charge in
  motion through a magnetic field.
• That implies a force should also be exerted on a
  collection of charges in motion through a
  conductor I.e. a current.
• And it does!!!
• The force on a current is the sum of all
  elementary forces exerted on all charge carriers
  in motion.

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19.4 Magnetic Force on Current
x x x x x x x x x x x x x
x x x x x x x x x x x

• If B is directed into the page we use blue
  crosses representing the tail of arrows
  indicating the direction of the field,
• If B is directed out of the page, we use dots.
• If B is in the page, we use lines with arrow
  heads.

. . . . . . . . . . . . .
. . . . . . . . . . .
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• Force on a wire carrying current in a magnetic
  field.

Bin
x x x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x x x
x x x x x x x x x x x
Bin
Bin
I
I 0
I
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Force on a wire carrying current in a magnetic
field.
x x x x x x x x x x x x
x x x x x x x x x x x x x
x x x x x x x x x x x x x
x x x x
vd
q
A
Magnetic Field and Current at right angle from
each other.
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Force on a wire carrying current in a magnetic
field.

• General Case field at angle q relative to
  current.

B
B sin q
q
I
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Voice Coil
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Mini-Quiz

• In a lightning strike, there is a rapid flow of
  negative charges from a cloud to the ground. In
  what direction is a lightning strike deflected by
  the Earths magnetic field?

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Example Wire in Earths B Field

• A wire carries a current of 22 A from east to
  west. Assume that at this location the magnetic
  field of the earth is horizontal and directed
  from south to north, and has a magnitude of 0.50
  x 10-4 T. Find the magnetic force on a 36-m
  length of wire. What happens if the direction of
  the current is reversed?

B0.50 x 10-4 T. I 22 A l 36 m Fmax BIl
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19.5 Torque on a Current Loop

• Imagine a current loop in a magnetic field as
  follows

19
I
F
B
B
F
a/2
b
F
F
a
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In a motor, one has N loops of current
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Example Torque on a circular loop in a magnetic
field
30.0o

• A circular loop of radius 50.0 cm is oriented at
  an angle of 30.0o to a magnetic field of 0.50 T.
  The current in the loop is 2.0 A. Find the
  magnitude of the torque.

B
r 0.500 m q 30o B 0.50 T I 2.0 A N 1
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19.6 Galvanometer/Applications

• Device used in the construction of ammeters and
  voltmeters.

Scale
Current loop or coil
Magnet
Spring
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Galvanometer used as Ammeter

• Typical galvanometer have an internal resistance
  of the order of 60 W - that could significantly
  disturb (reduce) a current measurement.
• Built to have full scale for small current 1 mA
  or less.
• Must therefore be mounted in parallel with a
  small resistor or shunt resistor.

60 W
Galvanometer
Rp
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60 W
Galvanometer
Rp

• Lets convert a 60 W, 1 mA full scale
  galvanometer to an ammeter that can measure up to
  2 A current.
• Rp must be selected such that when 2 A passes
  through the ammeter, only 0.001 A goes through
  the galvanometer.

• Rp is rather small!
• The equivalent resistance of the circuit is also
  small!

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Galvanometer used as Voltmeter

• Finite internal resistance of a galvanometer must
  also addressed if one wishes to use it as
  voltmeter.
• Must mounted a large resistor in series to limit
  the current going though the voltmeter to 1 mA.
• Must also have a large resistance to avoid
  disturbing circuit when measured in parallel.

Rs
60 W
Galvanometer
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Maximum voltage across galvanometer
Suppose one wish to have a voltmeter that can
measure voltage difference up to 100 V
Large resistance
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19.7 Motion of Charged Particle in magnetic field
Bin

• Consider positively charge particle moving in a
  uniform magnetic field.
• Suppose the initial velocity of the particle is
  perpendicular to the direction of the field.
• Then a magnetic force will be exerted on the
  particle and make follow a circular path.

r
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The magnetic force produces a centripetal
acceleration.
The particle travels on a circular trajectory
with a radius
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Example Proton moving in uniform magnetic field

• A proton is moving in a circular orbit of radius
  14 cm in a uniform magnetic field of magnitude
  0.35 T, directed perpendicular to the velocity of
  the proton. Find the orbital speed of the proton.

r 0.14 m B 0.35 T m 1.67x10-27 kg q 1.6
x 10-19 C
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Example 2
Consider the mass spectrometer. The electric
field between the plates of the velocity selector
is 950 V/m, and the magnetic fields in both the
velocity selector and the deflection chamber have
magnitudes of 0.930 T. Calculate the radius of
the path in the system for a singly charged ion
with mass m2.1810-26 kg.