HW8 Solution
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Transcript of HW8 Solution
GENERAL PHYSICS (I I) 2015 HOMEWORK 8 – SOLUTION SET
Problem 28-32
Suppose the current in the coaxial cable of Problem 31, Fig. 28-42, is not uniformly
distributed, but instead the current density j varies linearly with distance from the
center: j1 = C1R for the inner conductor and j2 = C2R for the outer conductor. Each
conductor still carries the same total current I0 , in opposite directions. Determine the
magnetic field in terms of I0 in the same four regions of space as in Problem 31.
Sol: 1: Iknown j dA
0 2 : ' encAmpere s lawknow dn B l I
1( )a R R3
2 1 01 0 0 1 1 0
0 0
2' 2 ' ' 2 ' '
3
R R
enc
C RB dl I C R R dR C R dR
1
ˆ2 B R r2
1 0
2
0 01 3
1
ˆ 3
ˆ 2
C RB r
I Rr
R
13
01 10 1 1 30
1
32' 2 ' '
3 2
R IC RI C R R dR C
R
3
2
3 3
2 3 2 00 2 2 3 3
3 2
2 ( ) 3' 2 ' '
3 2 ( )
R
R
C R R II C R R dR C
R R
Problem 28-32
1 2( )b R R R
3( )d R R
2 3( )c R R R
2 0 0 0encB dl I I
2
ˆ2 B R r
0 02
ˆ 2
IB r
R
2
3 0 0 0 2 ' 2 ' 'R
encR
B dl I I C R R dR
3
ˆ2 B R r
3
3 3
0 0 3
3 3
3 2
3
2 20 0
1
2 ( )
3ˆ
( )ˆ
2 ( )2
C R RI
B rR
I R Rr
R R R
4 0 0encB dl I
4
ˆ2 B R r4 0B
2
2
0 0 22 ' 'R
RI C R dR
3 3
2 20 0
2 ( )
3
C R RI
Problem 28-57 A very large flat conducting sheet of thickness t carries a uniform current density throughout (Fig. 28-56) . Determine the magnetic field (magnitude and direction) at a distance y above the plane. (Assume the plane is infinitely long and wide).
j
:sol
y t
’
.
Use with a rectangular loop
that extend
Amper
s a distance y above and below the curr
e s l
ent sh
a
eet
w
0 0 0encB dl I j A j xt
ˆ ˆ ( )top downB x z B x z
x
y
z
0 ˆ2
P
jtB z
j
P
A
x
2y t
x
/ / / /
B x B x
’ Ampere s loop
Problem 3
As shown in Fig. 2, an infinite plate of thickness 4R carries a electric current in the +z-axis
direction (out of page) with uniform current density J (A/m2). There is a infinitely long
cylindrical hollow (空心) region of radius R in the middle of the plate. Find:
(a) The magnitude and the direction of the magnetic field for points along the x-axis at 0 < x < R,
R < x < 2R, and 2R < x.
(b) The x-, y- and z-component of the magnetic field at point P in the x-y plane.
(If you use Ampere’s law, you need to draw the path of integral for integration.)
B dl:sol
0 0 : encB dk l I J dAnown
( )0a x R 2R x R 2R x
J J
Problem 3(a)
(a) The magnitude and the direction of the magnetic field for points along the x-axis at 0 < x < R,
R < x < 2R, and 2R < x.
:sol0 0 : encB dk l I J dAnown
0for x R 1 0 encB dl I
2 0 encB dl I
JJ
1 2 2
2 02 ( ) B x J x
02 ˆ
2
xJB y
0
ˆ ˆ ˆ2 2 ( )
ˆ ( ) ( 2 )
right up left
down
B x x B l y B x x
B l y J xl
xx
yy
2x
l
x
infinite
infinite
1 0 2 ( 2 )B l J xl
1 0 ˆB xJ y
0 ˆ2
tot
xJB y
& infinite long opposite direction
1loop 2loop
2for R x R
1 0 encB dl I
2 0 encB dl I
2
2 02 ( ) B x J R
2
02 ˆ
2
R JB y
x
J J
1 2
xx
yy
2x
l
x
1 0 2 ( 2 )B l J xl
1 0 ˆB xJ y
2
0ˆ( )
2tot
RB J x y
x
1loop2loop
2for R x
1 0 encB dl I
2 0 encB dl I
2
2 02 ( ) B x J R
2
02 ˆ
2
R JB y
x
JJ
1 2
xx
yy
2x
l
x
1 0 2 ( 4 )B l J Rl
1 0ˆ2 B RJ y
0ˆ(2 )
2tot
RB J R y
x
1loop
2loop
Problem 3(b)
(b) The x-, y- and z-component of the magnetic field at point P in the x-y plane.
:sol
J
J
(3 ,3 ,0)P R R(3 ,3 ,0)P R R
1 0 encB dl I
1 0 2 ( 4 )B l J Rl
1 0ˆ2 B RJ y
2 0 encB dl I 2
2 02 ( ) B x J R
02
2ˆ ˆ( sin cos )
12 4
4
RJB x y
0 ˆ ˆ( 23 )12
tot
RJB x y
1 2
l
6R
2 2(3 ) (3 )
3 2
x R R
R
02
ˆ ˆ( )12
RJ
B x y
4
4
1loop2loop