Download Chapter 29

Chapter 29
Magnetic Fields
Magnets
• In each magnet there are two poles present (the ends
where objects are most strongly attracted): north and
south
• Like (unlike) poles repel (attract) each other (similar to
electric charges), and the force between two poles
varies as the inverse square of the distance between
them
• Magnetic poles cannot be isolated – if a permanent
magnetic is cut in half, you will still have a north and a
south pole (unlike electric charges)
• There is some theoretical basis for monopoles, but
none have been detected
Magnets
• The poles received their names due to the way a
magnet behaves in the Earth’s magnetic field
• If a bar magnet is suspended so that it can move
freely, it will rotate
• The magnetic north pole points toward the Earth’s
north geographic pole
• This means the Earth’s north geographic pole is a
magnetic south pole
• Similarly, the Earth’s south geographic pole is a
magnetic north pole
Magnets
• An unmagnetized piece of iron can be magnetized by
stroking it with a magnet (like stroking an object to
charge an object)
• Magnetism can be induced – if a piece of iron, for
example, is placed near a strong permanent magnet, it
will become magnetized
• Soft magnetic materials (such as iron) are easily
magnetized and also tend to lose their magnetism
easily
• Hard magnetic materials (such as cobalt and nickel)
are difficult to magnetize and they tend to retain their
magnetism
Magnetic Fields
• The region of space surrounding a moving charge
includes a magnetic field (the charge will also be
surrounded by an electric field)
• A magnetic field surrounds a properly magnetized
magnetic material
• A magnetic field is a vector quantity symbolized by B
• Its direction is given by the direction a north pole of a
compass needle pointing in that location
• Magnetic field lines can be used to show how the field
lines, as traced out by a compass, would look
Magnetic Field Lines
• A compass can be used to show the direction of
the magnetic field lines
Magnetic Field Lines
• Iron filings can also be used
to show the pattern of the
magnetic field lines
• The direction of the field is
the direction a north pole
would point
• Unlike poles (compare to the
electric field produced by an
electric dipole)
Magnetic Field Lines
• Iron filings can also be used
to show the pattern of the
magnetic field lines
• The direction of the field is
the direction a north pole
would point
• Unlike poles (compare to the
electric field produced by an
electric dipole)
• Like poles (compare to the
electric field produced by like
charges)
Magnetic Fields
• When moving through a magnetic field, a charged
Nikola Tesla
particle experiences a magnetic force
1856 – 1943
• This force has a maximum (zero) value when the
charge moves perpendicularly to (along) the magnetic
field lines
• Magnetic field is defined in terms of the magnetic force
exerted on a test charge moving in the field with
velocity v
F
• The SI unit: Tesla (T)
N
T
A m
B
N
T
qv sin 
C  (m / s)
F  qvB sin 
Magnetic Fields
• Conventional laboratory magnets: ~ 2.5 T
• Superconducting magnets ~ 30 T
• Earth’s magnetic field ~ 5 x 10-5 T
Direction of Magnetic Force
• Experiments show that the direction of
the magnetic force is always
perpendicular to both v and B
• Fmax occurs when v is perpendicular to
B and F = 0 when v is parallel to B
FB  qv  B
• Right Hand Rule #1 (for a + charge):
Place your fingers in the direction of v
and curl the fingers in the direction of B
– your thumb points in the direction of F
• If the charge is negative, the force
points in the opposite direction
Direction of Magnetic Force
• The x’s indicate the magnetic field when it is directed
into the page (the x represents the tail of the arrow)
• The dots would be used to represent the field directed
out of the page (the • represents the head of the arrow)
Differences Between Electric and
Magnetic Fields
• The electric force acts along the direction of the
electric field, whereas the magnetic force acts
perpendicular to the magnetic field
• The electric force acts on a charged particle regardless
of whether the particle is moving, while the magnetic
force acts on a charged particle only when the particle
is in motion
• The electric force does work in displacing a charged
particle, whereas the magnetic force associated with a
steady magnetic field does no work when a particle is
displaced (because the force is perpendicular to the
displacement)
Force on a Charged Particle in a
Magnetic Field
• Consider a particle moving in an
external magnetic field so that its
velocity is perpendicular to the field
• The force is always directed toward
the center of the circular path
• The magnetic force causes a
centripetal acceleration, changing
the direction of the velocity of the
particle
mv
F  qvB sin   qvB 
r
2
mv
r
qB
Force on a Charged Particle in a
Magnetic Field
• This expression is known as the
cyclotron equation
• r is proportional to the momentum
of the particle and inversely
proportional to the magnetic field
• If the particle’s velocity is not
perpendicular to the field, the path
followed by the particle is a spiral
(helix)
v qB
 
r m
2m

T
 qB
2
mv
r
qB
Chapter 29
Problem 3
A proton moves perpendicularly to a uniform magnetic
field at 1.0 × 107 m/s and exhibits an acceleration of 2.0 ×
1013 m/s2 in the +x-direction when its velocity is in the +zdirection. Determine the magnitude and direction of the
field.
Chapter 29
Problem 15
A cosmic-ray proton in interstellar space has an energy of
10.0 MeV and executes a circular orbit having a radius
equal to that of Mercury’s orbit around the Sun (5.80 × 1010
m). What is the magnetic field in that region of space?
Particle in a Nonuniform Magnetic Field
• The motion is complex
Charged Particles Moving in Electric and
Magnetic Fields
• In many applications, charged particles move in the
presence of both magnetic and electric fields
• In that case, the total force is the sum of the forces due
to the individual fields:
F  qE  qv  B
Magnetic Force on a Current Carrying Wire
• The current is a collection of many charged
particles in motion
• The magnetic force is exerted on each
moving charge in the wire
• The total force is the sum of all the
magnetic forces on all the individual
charges producing the current
• Therefore a force is exerted on a currentcarrying wire placed in a magnetic field:





 
F  L I B


 
 
F  qvd  B # carriers  qvd  B  nAL

Magnetic Force on a Current Carrying Wire
• The direction of the force is given by right hand rule #1,
placing your fingers in the direction of I instead of v
Chapter 29
Problem 25
A wire having a mass per unit length of 0.500 g/cm carries a 2.00-A
current horizontally to the south. What are the direction and
magnitude of the minimum magnetic field needed to lift this wire
vertically upward?
Magnetic Force on a Current Carrying
Wire of an Arbitrary Shape
• For a small segment of the wire, the force exerted on
this segment is
dFB  I ds  B
• The total force is
b
FB  I  ds  B
a
Torque on a Current Loop
F2  F4  BIa  max
b
b
 F2  F4
2
2
b
b
 BIa  BIa  BIab  BIA
2
2
  BIA sin 
  NBIA sin 
Torque on a Current Loop
• Applies to any shape loop
• Torque has a maximum value when  =
90°
• Torque is zero when the field is
perpendicular to the plane of the loop
  NBIA sin 
Magnetic Moment
• The vector  is called the magnetic dipole
moment of the coil
• Its magnitude is given by
μ = IAN
• The vector always points perpendicular to the plane of
the loop(s)
• The equation for the magnetic torque can be written as
τ = BIAN sinθ = μB sinθ
   B
• The angle is between the moment and the field
Potential Energy
• The potential energy of the system of a magnetic
dipole in a magnetic field depends on the orientation of
the dipole in the magnetic field
 
U    B
• Umin = – μB and occurs when the dipole moment is in
the same direction as the field
• Umax = + μB and occurs when the dipole moment is in
the direction opposite the field
Chapter 29
Problem 38
A wire is formed into a circle having a diameter of 10.0 cm and placed
in a uniform magnetic field of 3.00 mT. The wire carries a current of
5.00 A. Find (a) the maximum torque on the wire and (b) the range of
potential energies of the wire-field system for different orientations of
the circle.
Answers to Even Numbered Problems
Chapter 29:
Problem 4
(a) 86.7 fN
(b) 51.9 Tm/s2
Answers to Even Numbered Problems
Chapter 29:
Problem 46
39.2 mT