phy202_7

1
Lesson 7
Magnetic Fields

• Magnetic Force on moving charge
• Motion of charge in Magnetic Field
• Magnetic Force on current carrying conductor
• Torque on current carrying loop
• Hall Effect
• Magnetic Devices

2
Magnetic Force

• Some metallic objects cause other metallic
  objects to accelerate
• Thus must be producing a force
• This force is different to Gravitational Force
  and Electric Force
• Can plot the directions of lines field
• By alignment of magnetic dipoles
• MAGNETIC FORCE

3
Magnetic Field Lines
4
Properties of Magnetic Fields

• Properties
• Moving charge (current) is effected
• Force proportional
• Q and v
• FB 0 if v parallel to Magnetic Field B
• FB perpendicular to v and B
• FB on positive charge opposite to that on
  negative charge
• FB proportional to Sin of the angle v makes
  with B

5
Mathematical Expression
6
Vector Product I
Vector Product



A
i
j
k
A
A
A
x
y
z



B
i
j
k
B
B
B
x
y
z
i
j
k
A
A
A
A
A
A
y
z
x
y
x
z



-

A
B
i
j
k
A
A
A
x
y
z
B
B
B
B
B
B
y
z
x
y
x
z
B
B
B
x
y
z
(
)
(
)
(
)

-
-
-

-
i
j
k
A
B
A
B
A
B
A
B
A
B
A
B
y
z
z
y
x
z
z
x
x
y
y
x
7
Vector Product II
8
Unit vector products
9
Work done by Magnetic Field
Work Done by Magnetic Force on charge

10
Work done by Electric Field
Compare to Electric Force
11
Implications

• Thus Kinetic Energy of charge is not changed by
  FB
• Potential Energy is not changed

1
1
-

2

2
DK
m
m
vf
0
vi
2
2
Þ

vi
vf
B
i
.
e
.
speed is constant under influence of
v
but
can change direction
12
SI units
13
Magnetic Flux
Magnetic Flux
F
B
For constant magnetic field over flat area
B
with area vector
A

F


B
AABcosq
B
in general
ò
F


B
A
d
B
surface


F

2

Tm
Wb
(
Weber
)
B
14
Right Hand Rule for Magnetic Force
15
Magnetic Force on Moving Charge
FB
B
v
Q
k
j
i



-

F
v
B
Q
j
k
Q
v
B
B
y
z
i
j
k

-


-

-
j
k
i
Qv
B
Qv
B
0
1
0
Qv
B
y
z
y
z
y
z
0
0
1
16
Mathematical expression
17
Uniform Circular Motion of Charge
Uniform Circular Motion
18
Mathematical analysis
2
F
v
QvB



a
c
r
m
m
2
mv
Þ

QvB
r
mv

r
radius of motion
QB
w
v

r
BQ
rad
\
w

angular speed

angular freqency
(
)
s
m
w
BQ


f
frequency
(
cps

Hz
)
p
p
2
2
m
p
1
2
m


T
period of the motion
.
f
BQ

19
Magnetic Bottles
magnetic bottles
20
Magnetic Force on Current
Force on current carrying conductor in
a uniform constant magnetic field

ò
b


I

F
s

l
B
I
d
B
B
a
(
)

-
l
r
r
where
b
a
é
ù
æ
ö
ds
dQ
(
)




v
s
ç

s
dQ
dQ
ˆ
ds
ˆ
I ds
ˆ
I d
ê
ú
s
s
è
ø
ë
û
dt
dt

21
Vector Integral I
ds
b
rb
B is constant
vector integral
ra
a
Current I
Want to find total force on charge as it flows
from a to b
22
Vector Integral II
ds
b
rb
B
rb- ra
ra
a
I
23
Vector Integral III
Sum
ds
24
Magnetic Force on Current Loop I
Force on Current Loop
I
B
k
j
i
25
Magnetic Force on Current Loop II
Total Force on Loop
26
Torque on Current Loop I
Torque on Current Loop
??
I
B
??
k
z
??
j
i
?
??
y
27
Torque on Current Loop II
28
Torque on Current Loop III
29
Magnetic Moment
Torque on Current Loop in yz plane in
a uniform magnetic field in positive y direction
t
k

IAB
In general the torque on a current loop in
magnetic field is
t

M
B


M
A
I
,
where
is the area vector of the loop
,
A
with the orientation given by the current flow
.


30
Oriented Area Vector
AAi Area vector with orientation
I
k
A
j
i
31
Hall Effect
I
Hall Effect
A
t
d
vd
FE
FB
B
V
E
-

32
Mathematical analysis
At Equilibrium

F
F
E
B
V

E
d



F
F
eE
v
eB
E
B
d
E
\

v
d
B
J
I


Remember
v
d
ne
neA
I
E
BI
BI
BI
\

Þ



n
V
neA
B
EAe
Vte
dte
d
33
Measurement
can measure vd by measuring the PD V.
34
Lorentz Force