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