Online Homework 8 Solution
Current Sheet
Consider an infinite sheet of parallel wires.
The sheet lies in the xy plane. A current I
runs in the -y direction through each wire.
There are N/a wires per unit length in the
x direction.
r
Part A
Write an expression for B(d ) , the magnetic field a distance d above
the xy plane of the sheet. Use µ 0 for the permeability of free space. Express
the magnetic field as a vector in terms of any or all of the following: d, I, N, a,
µ 0 , and the unit vectors x̂ , ŷ , and/or ẑ .
Sol:)
Use Ampere’s Law
r r
B
∫ • d l = µ 0 I encl
I encl = NI
在單一導線上方的磁場並無 y 方向分量
本題為 current sheet
如下圖，磁場的 z 分量被抵消
因此，total 磁場方向為 − x̂
r r
B
∫ • d l = 2 Ba
⇒ 2 Ba = µ 0 NI
v
µ NI
B=− 0
xˆ
2a
Note：
：
比較 E =
σ
，
2ε 0
若令 current per unit length λ J =
則B =
NI
，
a
µ0λJ
2
1
Problem 28.31
A coaxial cable consists of a solid inner conductor of radius R1 surrounded
by a concentric cylindrical tube of inner radius R2 and outer radius R3 (the
figure). The conductors carry equal and opposite currents I0 distributed
uniformly across their cross sections.
Part A Determine the magnetic field at a distance R from the axis for R<R1.
Express your answer in terms of some or all of the variables R, R1, R2, R3, I0,
and appropriate constants.
Sol:)
I0
πR12
r r
B
∫ • d l = µ 0 I encl
截面的電流密度 J inner =
Use Ampere’s Law
(
B ⋅ (2πR ) = µ 0 J inner ⋅ πR 2
B=
µ0 I 0 R
2πR12
)
Part B Determine the magnetic field at a distance R from the axis for R1 < R2.
Express your answer in terms of some or all of the variables R, R1, R2, R3, I0,
and appropriate constants.
Sol:)
同理，Use Ampere’s Law
r r
B
∫ • d l = µ 0 I encl
B ⋅ (2πR ) = µ 0 I 0
B=
µ0 I 0
2πR
2
Part C Determine the magnetic field at a distance R from the axis for R2 < R<
R3. Express your answer in terms of some or all of the variables R, R1, R2, R3, I0,
and appropriate constants.
Sol:)
I0
π R − R22
r r
B
∫ • d l = µ 0 I encl
截面的電流密度 J outer = −
Use Ampere’s Law
(
)
2
3
(
(
)
)

π R 2 − R22 
B ⋅ (2πR ) = µ 0 I 0 + J outer π R 2 − R22 = µ 0  I 0 − I 0

π R32 − R22 

µ I R2 − R2
B = 0 0 32
2πR R3 − R22
[
(
(
)]
(
)
)
Part D Determine the magnetic field at a distance R from the axis for R>R3.
Express your answer in terms of some or all of the variables R, R1, R2, R3, I0,
and appropriate constants.
Sol:)
r
Use Ampere’s Law
r
∫ B • dl = µ
I
0 encl
B ⋅ (2πR ) = 0
B=0
Part E Let I0 = 1.50 A, R1 = 1.00 cm, R2 = 2.00 cm, and R3 = 2.50 cm. Graph
B from R = 0 to R = 3.00 cm.
Sol:)
3
Problem 28.24
A triangular loop of side length a carries a current I (see the figure).
Part A If this loop is placed a distance d away from a very long straight wire
carrying a current I, determine the force on the loop.
Sol:).
將三角形線圈分成三段，將三段的力加起來即為所求。
第一段：
F1 =
µ0 II ′
a
2πd
第二段：
將導線分為無限多段，每段長 dx。令第二段導線的最左端為 x=0，
則長 dx 的導線與下方直導線之距離 r = d + 3 x 。
F2 = ∫
a/2
0
µ 0 II ′
(
2π d + 3 x
dx =
)
µ 0 II ′ 
3a 

ln1 +
2d 
2π 3 
第二段與第三段受的力大小相等，且第二、三段與下方長直導線電流方向
相反，受排斥力。
F = F1 − 2 F2 =
µ 0 II ′
µ II ′ 
3a  µ 0 II ′  a
3 
3a 
=
1 +

a − 2 0 ln1 +
−
ln

2πd
2d 
π  2d 3 
2d 
2π 3 
4
Force between an Infinitely Long Wire and a Square Loop
A square loop of wire with side length a
carries a current I1. The center of the loop is
located a distance d from an infinite wire
carrying a current I2. The infinite wire and
loop are in the same plane; two sides of the
square loop are parallel to the wire and two
are perpendicular as shown.
Part A What is the magnitude, F, of the net force on the loop? Express your
answer in terms of I1, I2, a, d, and µ 0 .
Sol:)
在 wire loop 中，四段導線受力的總和即為 wire loop 受的淨力。
v
v v
由 F = I l × B ，wire loop 的四邊受力方向如 Fig. 1
µ I
wire loop 在磁場 B = 0 中受力大小如 Fig. 2
2πr
其中，B 可由 Ampere’s Law 求得。
µ 0 I1
F1 = I 2 a
a

2π  d − 
2

F = F1 − F2 = I 2 a
=
Fig. 1
µ 0 I1
a

2π  d − 
2

− I 2a
µ 0 I1
a

2π  d + 
2

µ 0 I1 I 2 a 2

a2
2π  d 2 +
4




Fig. 2
r
Part B The magnetic moment m of a current loop is defined as the vector
whose magnitude equals the area of the loop times the magnitude of the current
flowing in it (m = IA), and whose direction is perpendicular to the plane in
which the current flows. Find the magnitude, F, of the force on the loop from
Part A in terms of the magnitude of its magnetic moment. Express F in terms of
m, I2, a, d, and µ 0 .
Sol:)
F=
µ 0 I1 I 2 a 2

a2 

2π  d 2 +
4 

=
µ0 I 2m

a2 

2π  d 2 +
4 

5
Problem 28.37
A wire is formed into the shape of two half
circles connected by equal-length straight
sections as shown in the figure. A current I flows
in the circuit clockwise as shown.
Part A Determine the magnitude of the magnetic field at the center, C.
Express your answer in terms of the variables I, R1, R2, and appropriate
constants.
Sol:)
For upper half circle segment
dBupper
r
d
µ I l × rˆ µ 0 I dl µ 0 I R1 dθ µ 0 I dθ
= 0
=
=
=
4π r 2
4π r 2
4π R12
4π R1
Bupper = ∫ dBupper = ∫
π
0
µ 0 I dθ µ 0 I
=
4π R1 4 R1
同理，For lower half circle segment
µ0 I
Blower =
4R2
B = Bupper + Blower =
µ0 I
4 R1
+
µ0 I
4 R2
=
µ0 I  1
1 
 +

4  R1 R2 
Part B Determine the direction of the magnetic field at the center, C.
Sol:)
The direction of the magnetic field is into the screen.
Part C Determine the magnitude of the magnetic dipole moment of the
circuit. Express your answer in terms of the variables I, R1, R2, and appropriate
constants.
Sol:)
v
πR 2
πR 2 Iπ (R12 + R22 )
v
µ =I A =I 1 +I 2 =
2
2
2
Part D Determine the direction of the magnetic dipole moment of the circuit.
Sol:)
The direction of the magnetic dipole moment of the circuit is into the screen.
6