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222
Chapter 22
(b)
Total flux:
22
25
(0.081 0.135)(N/C) m
0.054 N m /C.
Φ=Φ +Φ =
−
⋅
=−
⋅
Therefore,
13
0
4.78 10
C.
q
−
=Φ
=
−
×
P
EVALUATE:
Flux is positive when
E
G
is directed out of the volume and negative when it is directed into the
volume.
22.5.
IDENTIFY:
The flux through the curved upper half of the hemisphere is the same as the flux through the flat circle
defined by the bottom of the hemisphere because every electric field line that passes through the flat circle also must
pass through the curved surface of the hemisphere.
SET UP:
The electric field is perpendicular to the flat circle, so the flux is simply the product of
E
and the area of
the flat circle of radius
r
.
EXECUTE:
Φ
E
= EA = E
(
2
r
π
) =
2
r
E
EVALUATE:
The flux would be the same if the hemisphere were replaced by any other surface bounded by the flat
circle.
22.6.
IDENTIFY:
Use Eq.(22.3) to calculate the flux for each surface.
SET UP:
ˆ
cos
where
E
AA
φ
Φ= ⋅ =
E
=
n
GG
G
.
EXECUTE:
(a)
1
ˆ
ˆ
(left)
S
−
n= j
.
1
32
2
(4 10 N/C)(0.10 m) cos(90
36 9 )
24 N m /C.
S
.
Φ=
−×
−
°=
−
⋅
°
2
S
ˆ
ˆ
(top)
+
n=k
.
2
(4 10 N/C)(0.10 m) cos90
0
S
°
=
.
3
ˆ
ˆ
(right)
S
+
.
3
2
(4 10 N/C)(0.10 m) cos(90
36.9 )
24 N m /C
S
+×
°
−
+
⋅
.
4
ˆ
ˆ
(bottom)
S
−
.
4
(4 10 N/C)(0.10 m) cos90
0
S
Φ= ×
°
=
.
5
ˆ
ˆ
(front)
S
+
n=i
.
5
2
(4 10 N/C)(0.10 m) cos36.9
32 N m /C
S
°
=
⋅
.
6
ˆ
ˆ
(back)
S
−
.
6
2
(4 10 N/C)(0.10 m) cos36.9
32 N m /C
S
°
=
−
⋅
.
EVALUATE:
(b)
The total flux through the cube must be zero; any flux entering the cube must also leave it, since
the field is uniform. Our calculation gives the result; the sum of the fluxes calculated in part (a) is zero.
22.7.
(a) IDENTIFY:
Use Eq.(22.5) to calculate the flux through the surface of the cylinder.
SET UP:
The line of charge and the cylinder are sketched in Figure 22.7.
Figure 22.7
EXECUTE:
The area of the curved part of the cylinder is
2
.
Ar
l
=
The electric field is parallel to the end caps of the cylinder, so
0
⋅
=
EA
G
G
for the ends and the flux through the
cylinder end caps is zero.
The electric field is normal to the curved surface of the cylinder and has the same magnitude
0
/2
E
r
λ
=
P
at all
points on this surface. Thus
0
=°
and
()
(
)
( )
6
52
0
12
2
2
0
6.00 10 C/m 0.400 m
cos
/ 2
2
2.71 10 N m /C
8.854 10
C / N m
E
l
EA
EA
r
rl
φλ
−
−
×
= =
=
×
⋅
×⋅
P
P
(b)
In the calculation in part (a) the radius
r
of the cylinder divided out, so the flux remains the same,
2.71 10 N m /C.
E
×
⋅
(c)
( )
6
12
2
2
0
6.00 10 C/m 0.800 m
5.42 10 N m /C
8.854 10
C / N m
E
l
−
−
×
Φ= =
=
×
⋅
P
(twice the flux calculated in parts (b) and (c)).
EVALUATE:
The flux depends on the number of field lines that pass through the surface of the cylinder.
22.8.
IDENTIFY:
Apply Gauss’s law to each surface.
SET UP:
encl
Q
is the algebraic sum of the charges enclosed by each surface. Flux out of the volume is positive and
flux into the enclosed volume is negative.
EXECUTE:
(a)
1
92
10
0
/
(4.00 10 C)/
452 N m /C.
S
q
−
=
×
=
⋅
PP
(b)
2
20
0
/
( 7.80 10 C)/
881 N m /C.
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This note was uploaded on 07/16/2011 for the course PHY 2053 taught by Professor Buchler during the Spring '06 term at University of Florida.
 Spring '06
 Buchler
 Physics

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