271
M
AGNETIC
F
IELD AND
M
AGNETIC
F
ORCES
27.1.
I
DENTIFY
and
S
ET
U
P
:
Apply Eq.(27.2) to calculate
.
F
!
Use the cross products of unit vectors from Section 1.10.
E
XECUTE
:
(
)
(
)
4
4
°
°
4.19
10
m/s
3.85
10
m/s
= +
×
+ −
×
v
i
j
!
(a)
(
)
°
1.40 T
=
B
i
!
(
)
(
)
(
)
(
)
8
4
4
°
°
°
°
1.24
10
C
1.40 T
4.19
10
m/s
3.85
10
m/s
q
−
⎡
⎤
=
×
= −
×
×
×
−
×
×
⎣
⎦
F
v
B
i
i
j
i
!
!
!
°
°
°
°
°
0,
×
=
×
= −
i
i
j
i
k
(
)
(
)
(
)
(
)
(
)
8
4
4
°
°
1.24
10
C
1.40 T
3.85
10
m/s
6.68
10
N
−
−
= −
×
−
×
−
= −
×
F
k
k
!
E
VALUATE
:
The directions of
and
v
B
!
!
are shown in Figure 27.1a.
The righthand rule gives that
×
v
B
!
!
is directed
out of the paper (+
z
direction). The charge is
negative so
F
!
is opposite to
;
×
v
B
!
!
Figure 27.1a
F
!
is in the

z
−
direction. This agrees with the direction calculated with unit vectors.
(b) E
XECUTE
:
(
)
°
1.40 T
=
B
k
!
(
)
(
)
(
)
(
)
8
4
4
°
°
°
°
1.24
10
C
1.40 T
4.19
10
m/s
3.85
10
m/s
q
−
⎡
⎤
=
×
= −
×
+
×
×
−
×
×
⎣
⎦
F
v
B
i
k
j
k
!
!
!
°
°
° °
°
°
,
×
= −
×
=
i
k
j j
k
i
(
)
(
)
(
)
(
)
(
)
4
4
4
4
°
°
°
°
7.27
10
N
6.68
10
N
6.68
10
N
7.27
10
N
−
−
−
−
⎡
⎤
= −
×
−
+
×
=
×
+
×
⎣
⎦
F
j
i
i
j
!
E
VALUATE
:
The directions of
and
v
B
!
!
are shown in Figure 27.1b.
The direction of
F
!
is opposite to
×
v
B
!
!
since
q
is negative. The direction of
F
!
computed
from the righthand rule agrees qualitatively
with the direction calculated with unit vectors.
Figure 27.1b
27.2.
I
DENTIFY
:
The net force must be zero, so the magnetic and gravity forces must be equal in magnitude and
opposite in direction.
S
ET
U
P
:
The gravity
force is downward so the force from the magnetic field must be upward. The charge±s
velocity and the forces are shown in Figure 27.2. Since the charge is negative, the magnetic force is opposite to the
righthand rule direction. The minimum magnetic field is when the field is perpendicular to
v
!
. The force is also
perpendicular to
B
!
, so
B
!
is either eastward or westward.
E
XECUTE
:
If
B
!
is eastward, the righthand rule direction is into the page and
B
F
!
is out of the page, as required.
Therefore,
B
!
is eastward.
sin
mg
q vB
φ
=
.
90
φ
=
°
and
3
2
4
8
(0.195
10
kg)(9.80 m/s
)
1.91 T
(4.00
10
m/s)(2.50
10
C)
mg
B
v q
−
−
×
=
=
=
×
×
.
27
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272
Chapter 27
E
VALUATE
:
The magnetic field could also have a component along the northsouth direction, that would not
contribute to the force, but then the field wouldn±t have minimum magnitude.
Figure 27.2
27.3.
I
DENTIFY
:
The force
F
!
on the particle is in the direction of the deflection of the particle. Apply the righthand
rule to the directions of
v
!
and
B
!
. See if your thumb is in the direction of
F
!
, or opposite to that direction. Use
sin
F
q vB
φ
=
with
90
φ
=
°
to calculate
F
.
S
ET
U
P
:
The directions of
v
!
,
B
!
and
F
!
are shown in Figure 27.3.
E
XECUTE
:
(a)
When you apply the righthand rule to
v
!
and
B
!
, your thumb points east.
F
!
is in this direction,
so the charge is positive.
(b)
6
3
sin
(8.50
10
C)(4.75
10
m/s)(1.25 T)sin90
0.0505 N
F
q vB
φ
−
=
=
×
×
=
°
E
VALUATE
:
If the particle had negative charge and
v
!
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 Spring '08
 Jabbour
 Materials Science And Engineering

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