Download Magnetic Fields, Chapter 29

Magnetism
Magnetism
• Our most familiar experience of magnetism is through
permanent magnets.
• These are made of materials which exhibit a property we
call “ferromagnetism” - i.e., they can be magnetized.
• Depending on how we position two magnets, they will
attract or repel, i.e. they exert forces on each other.
• Thus, a magnet must have an associated field:
a magnetic field.
What Do We Know About
Permanent Magnets?
• We have not been able, so far, to isolate a magnetic
monopole (the equivalent of an electric charge).
• We describe magnets as having two magnetic poles:
North (N) and South (S).
• Like poles repel, opposite poles attract.
Field of a Permanent Magnet

B
N
S
Magnetic field lines exit from the north pole
of the magnet, and enter at the south pole
Field of a Permanent Magnet

B
N
N
S
S
What happens when a small bar magnet or compass
is placed in a magnetic field ?
Field of a Permanent Magnet

B
N
N
S
S
The south pole of the small bar magnet is attracted towards the north
pole of the big magnet.
Also, the small bar magnet (a magnetic dipole) wants to align with the
B-field.
The field attracts and exerts a torque on the small magnet.
Field of a Permanent Magnet

B
N
N
S
S
The small bar magnet or compass aligns with the B field.
If free to move, it will move towards the big magnet.
The south pole of the small magnet is attracted towards the
north pole of the big magnet.
Field of a Permanent Magnet
The earth has a magnetic field whose origin is not clear yet.
Note: geographic north is magnetic south
Magnetism
• The origin of magnetism lies in moving electric charges.
Moving (or rotating) charges generate magnetic fields.
• An electric current generates a magnetic field.
• A magnetic field will exert a force on a moving charge,
and therefore on a conductor that carries an electric current
• Two conductors that carry electric currents will exert
forces on each other.
• We will study the relation and interaction between
moving charges, currents, and magnetic fields.
What Force Does a Magnetic Field
Exert on Charges?

B
q
• NONE!, If the charge is
not moving with respect to the
field (or if the charge moves
parallel to the field).
What Force Does a Magnetic Field
Exert on Charges?

B
q

v

q B
• NONE!, If the charge is
not moving with respect to the
field (or if the charge moves
parallel to the field).
• If the charge is moving, there
is a force on the charge,
perpendicular to both v and B.
F=qvxB
Force on a Charge in a Magnetic Field
• As we saw, force is perpendicular to both v and B.
• The force is also largest for v perpendicular to B,
and zero for v parallel to B.
This can be summarized as:

 
F  qv  B
F
v
or:
F  qvBsin 
B
mq
Force on a Charge in a Magnetic Field
Force points out of the page
Force on a Charge in a Magnetic Field
Units of Magnetic Field
As
F  qvB sin
F
B
qv sin 
Therefore, the units of magnetic
N
field are:
 Ns / Cm
Cm / s
...or: Tesla ,( T )
1T  1Ns / Cm
(Note: 1 Tesla = 10,000 Gauss)
The Magnetic Force is Different
From the Electric Force.
Whereas the electric force
acts in the same direction as
the field:
The magnetic force acts in a
direction orthogonal to the
field:


F  qE



F  qv  B
(Use “Right-Hand” Rule to
determine direction of F)
And --- the charge must be moving !!
Trajectory of Charged Particles
in a Magnetic Field
(B field points into plane of paper.)
+
B
+
+
+
+
+
+
+
+ F
+
+
+
+
+
+
+
+
+
+
v
+
Trajectory of Charged Particles
in a Magnetic Field
(B field points into plane of paper.)
+
+B
+
v+
+
+
+
+
+
+
+
+ F
+
+
+
+ F +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
B
+
+
+
+
+
v
Magnetic Force is a centripetal force
Trajectory of Charged Particles
in a Magnetic Field
Magnetic Force is a centripetal force
Rotational Motion

 = s / r  s =  r  ds/dt = d/dt r  v =  r
s
r
 = angle,  = angular speed,  = angular acceleration

at
ar
at = r 
tangential acceleration
ar = v2 / r radial acceleration
The radial acceleration changes the direction of motion,
while the tangential acceleration changes the speed.
Uniform Circular Motion
ar
 = constant  v and ar constant but direction changes

v
ar = v2/r = 2 r
KE = ½ mv2 = ½ mw2r2
F = mar = mv2/r = m2r
Radius of a Charged Particle
Orbit in a Magnetic Field
+B
+
+
v+
+
+
+
+
+
+
+
+
+
r
+
+
+
+
+
+
F
+
Centripetal
Force
=
Magnetic
Force
mv 2

 qvB
r

mv
r
qB
 
Note: as Fv , the magnetic
force does no work!
Cyclotron Frequency
+B
+
v+
+
+
+
+
+
+
+
+
+
+
r
+
+
+
+
+
+
F
+
The time taken to complete one
orbit is:
2r
T
v
2 mv

v qB
Hence the orbit frequency, f
1
qB
f  
T 2 m
known as the “cyclotron frequency”
T = 2/ = 1/ƒ  ƒ = /2
The Electromagnetic Force
If a magnetic field and an electric field are simultaneously
present, their forces obey the superposition principle and
may be added vectorially:


 
F  qE  qv  B
The Electromagnetic Force
If a magnetic field and an electric field are simultaneously
present, their forces obey the superposition principle and
may be added vectorially:


 
F  qE  qv  B

B+

v
+ +
+
+
+
+
+ +
+q +
+ +
+ +

E
What is the direction
of the net force?
Exercise
electron
v
B
v’
• In what direction does the magnetic field point?
• Which is bigger, v or v’ ?
What is the orbital radius of a charged particle (charge q, mass m)
having kinetic energy K, and moving at right angles to a magnetic
field B, as shown below?.
x x x
B
x x x
K
q• m
What is the orbital radius of a charged particle (charge q, mass m)
having kinetic energy K, and moving at right angles to a magnetic
field B, as shown below?.
F = q v x B = m a and a = v2 / r
q v B = m v2 / r
x x x
B
x x x
qB=mv/rrqB=mv
r
K = ½ mv2
q• m
r = m v / (q B)
r2 = m2 v2 / (q B)2  (1/2m) r2 = ½ m v2 / (q B)2
(1/2m) r2 = K / (q B)2  r = [2mK]1/2 / (q B)
What is the relation between the intensities of the electric and
magnetic fields for the particle to move in a straight line ?.
x x x B
E
x x x
v
q• m
Trajectory of Charged Particles
in a Magnetic Field
What if the charged particle has a velocity component along B?

Vz

B

Vz
unchanged
Circular motion
in xy plane.
x
z
The particle follows a helical path
y