Magnetic Fields Chapter 29 (continued)

1
Magnetic FieldsChapter 29(continued)
2
Force on a Charge in aMagnetic Field
F
v
q
m
B
(Use Right-Hand Rule to determine direction of
F)
3
Trajectory of Charged Particlesin a Magnetic
Field
(B field points into plane of paper.)
v
B
B


v
F
F
Magnetic Force is a centripetal force
4
Radius of a Charged ParticleOrbit in a Magnetic
Field
Centripetal Magnetic Force
Force

v
B

F
r
Note as , the magnetic force does
no work.
5
Exercise
electron
B
v
v

• In what direction does the magnetic field
  point?
• Which is bigger, v or v ?

6
Exercise answer
electron
B
v
v
F

• In what direction does the magnetic field point
  ?
• Into the page F -e v x B
• Which is bigger, v or v ?
• v v B does no work on the electron, F?v

7
Trajectory of Charged Particlesin a Magnetic
Field
What if the charged particle has a velocity
component along B?
v
8
Trajectory of Charged Particlesin a Magnetic
Field
What if the charged particle has a velocity
component along B?
v
Fz0 so vzconstant
The force is in the xy plane. It acts exactly as
described before, creating circular motion in the
xy plane.
Result is a helix
9
Trajectory of Charged Particlesin a Magnetic
Field
What if the charged particle has a velocity
component along B?
v
Fz0 so vzconstant
The force is in the xy plane. It acts exactly as
described before, creating circular motion in the
xy plane.
Result is a helix
10
The Electromagnetic Force
If a magnetic field and an electric field are
simultaneously present, their forces obey the
superposition principle and must be added
vectorially
The Lorentz force


q
11
The Electromagnetic Force
If a magnetic field and an electric field are
simultaneously present, their forces obey the
superposition principle and must be added
vectorially
The Lorentz force
FB


FE
q
12
The Electromagnetic Force
When B, E, and v are mutually perpendicular, as
pictured here, FE and FB point in opposite
directions.


FB
FE
q
The magnitudes do not have to be equal, of
course. But by adjusting E or B you can set this
up so the net force is zero.
qE qvB
Set FE qE equal to FB qvB
Hence with the pictured orientation of fields and
velocity, the particle will travel in a straight
line if v E / B.
13
The Hall Effect
vd
I

• Consider a conducting bar, carrying a current,
  with a perpendicular magnetic field into the
  picture.

14
The Hall Effect
vd
I

• Consider a conducting bar, carrying a current,
  with a perpendicular magnetic field into the
  picture.
• The electrons drifting to the right tend to move
  down because of the magnetic force.

15
The Hall Effect


vd
I
- - - - - - - - -
- - -

• Consider a conducting bar, carrying a current,
  with a perpendicular magnetic field into the
  picture.
• The electrons drifting to the right tend to move
  down because of the magnetic force.
• Thus you get a charge separation a net negative
  charge along the bottom edge, and positive along
  the upper.

16
The Hall Effect


E
vd
I
- - - - - - - - -
- - -

• Consider a conducting bar, carrying a current,
  with a perpendicular magnetic field into the
  picture.
• The electrons drifting to the right tend to move
  down because of the magnetic force.
• Thus you get a charge separation a net negative
  charge along the bottom edge, and positive along
  the upper.
• This charge separation sets up an electric field,
  top to bottom, which pulls electrons up
   opposing the magnetic force.

17
The Hall Effect


E
vd
I
- - - - - - - - -
- - -

• Consider a conducting bar, carrying a current,
  with a perpendicular magnetic field into the
  picture.
• The electrons drifting to the right tend to move
  down because of the magnetic force.
• Thus you get a charge separation a net negative
  charge along the bottom edge, and positive along
  the upper.
• This charge separation sets up an electric field,
  top to bottom, which pulls electrons up
   opposing the magnetic force.
• The charge separation builds up until the two
  forces are equal
• eEevdB

18
The Hall Effect


d
vd
I
- - - - - - - - -
- - -

• The charge separation builds up until the two
  forces are equal
• eEevdB

19
The Hall Effect


d
vd
I
- - - - - - - - -
- - -

• The charge separation builds up until the two
  forces are equal
• eEevdB
• This means an electric potential difference
  develops between the two edges VHEdvdBd -the
  Hall voltage

20
The Hall Effect


d
vd
I
- - - - - - - - -
- - -

• The charge separation builds up until the two
  forces are equal
• eEevdB
• This means an electric potential difference
  develops between the two edges VHEdvdBd -the
  Hall voltage
• This means that measuring the Hall voltage lets
  you work out the drift velocity.

21
The Hall Effect


d
vd
I
- - - - - - - - -
- - -

• The charge separation builds up until the two
  forces are equal
• eEevdB
• This means an electric potential difference
  develops between the two edges VHEdvdBd -the
  Hall voltage
• This means that measuring the Hall voltage lets
  you work out the drift velocity.
• Moreover, using Jnevd and IJA (with A the
  slabs cross-sectional area) gives vdI/(Ane) and
  so VHIBd/Ane . Measuring the Hall voltage lets
  you find the density of conduction electrons.

22
The Hall Effect
VH

• The Hall effect also lets you find the sign of
  the charge carriers that make up the current.
  Above is the picture for electrons.
• But if the charge carriers actually had a
  positive charge, the picture would look like this

- - - - - - - - -
- - -
VH
vd
I

• The carriers would move to the bottom edge still,
  and the Hall voltage would point in the opposite
  direction.

23
Magnetic Force on a Current
A

• Force on one charge
• F q vd x B
• Forces on all charges in a length L of a
  conductor
• F n A L q vd x B
• Use I n q vd A and define a vector L whose
  length is L, and has the same direction as the
  current I. Then

L
I n q vd A
F
I

F I L x B
L
F points out of the page
24
Magnetic Force on a Current
Example A current, I10 A, flows through a
wire, of length L20 cm, between the poles of a
1000 Gauss magnet. The wire is at q 900 to
the field as shown. What is the force on the wire?
L
N
S
25
Magnetic Force on a Current
Example A current, I10 A, flows through a
wire, of length L20 cm, between the poles of a
1000 Gauss magnet. The wire is at q 900 to
the field as shown. What is the force on the wire?
L
N
S
(up)
26
Magnetic Force on a Current
Example A current, I10 A, flows through a
wire, of length L20 cm, between the poles of a
1000 Gauss magnet. The wire is at q 900 to
the field as shown. What is the force on the wire?
L
N
S
(up)
(up)
27
Magnetic Force on a Current Loop
A current loop is placed in a uniform magnetic
field as shown below. What will happen?
I
28
Magnetic Force on a Current Loop
No net force but a torque is imposed.
FBIL
F
q
L
B
I
F
FBIL
29
Magnetic Torque on a Current Loop
Simplified view
30
Magnetic Torque on a Current Loop
Simplified view
31
Magnetic Torque on a Current Loop
for a current loop
Avector with magnitude ALd and direction given
by a RH rule.
32
Magnetic Force on a Current Loop Torque
Magnetic Dipole
By analogy with electric dipoles, for
which The expression, implies the a current
loop acts as a magnetic dipole! Here
is the magnetic dipole moment, and
(Torque on a current loop)
33
Potential Energy of a Magnetic Dipole
By further analogy with electric dipoles So
for a magnetic dipole (a current loop)
The potential energy is due to the fact that the
magnetic field tends to align the current loop
perpendicular to the field.
34
Nonuniform Fields and Curved Conductors

• So far, we have considered only uniform fields
  and straight current paths.
• If this is not the case, we must build up using
  calculus.

Consider a small length, dL, of current
path. The force on dL is dF dL I x B
35
Nonuniform Fields and Curved Conductors
For a conductor of length L F L I x B For a
bit of length dL dF dL I x B Then, for the
total length of the curved conductor in
a non-uniform magnetic field F ? dF ? dL
I x B
To find the force exerted by a non-uniform
magnetic field on a curved current we divide the
conductor in small sections dL and add
(integrate) the forces exerted on every section
dL.
36
Nonuniform Fields and Curved Conductors Example

• What is the force on the current-carrying
  conductor shown?

R
I
L
L
37
Nonuniform Fields and Curved Conductors Example
R
I
q
L
L
38
Nonuniform Fields and Curved Conductors Example
q
R
I
q
L
L
39
Nonuniform Fields and Curved Conductors Example
40
Nonuniform Fields and Curved Conductors Example
41
Nonuniform Fields and Curved Conductors Example
42
Nonuniform Fields and Curved Conductors Example
43
Nonuniform Fields and Curved Conductors Example
44
Nonuniform Fields and Curved Conductors Example
Equal to the force we would find for a straight
wire of length 2(RL)