Download - Ch5. 靜磁學 (Magnetostatics)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
Ch5. 靜磁學 (Magnetostatics)
5.1 羅倫茲力定律 (The Lorentz Force Law)
5.2 必歐沙伐定律 (The Biot-Savart Law)
5.3 磁場的散度與旋度 (The Divergence and Curl of B)
5.4 磁向量位勢 (Magnetic Vector Potential)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.1 The Lorentz Force Law-磁場 (Magnetic Fields)
Bar magnet Conducting wire
Parallel wires carrying currents in the same direction attract each other
Parallel wires carrying currents in opposite direction repel each other
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.2.1 磁力 (Magnetic Forces)
BvqEqFFFBE
The Lorentz force
Additional speed parallel to B
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.2.2 磁力 (Magnetic Forces) z
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.2.3 磁力 (Magnetic Forces)
0)0(z)0(y But the particle started from rest ( ), at the origin ( );0)0(z)0(y
EtB
EtsinAtcosB)t(y
0
0
CtsinBtcosA)t(z
EtB
EtsinAtcosB)t(y
0
0 0
0
B
EtcosAtsinB)t(y
0EB)0(y 0B
EA)0(y
0
0
CtsinBtcosA)t(z tcosBtsinA)t(z
0CA)0(z 0B)0(z
0BE CB
EA
0
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.2.4 磁力 (Magnetic Forces)
)tsint(B
EEt
B
EtsinAtcosB)t(y
0
0
0
0
)tcos1(B
ECtsinBtcosA)t(z
0
0
0
0
B
ER
tsinRtR)t(y
tcosRR)t(z
tsinR]tR)t(y[ 222
tcosR]R)t(z[ 222
222 R]R)t(z[]tR)t(y[
This is the formula for a circle, of radius R, whose center (0, Rt, R) travels in the y-direction at a constant speed
0
0
B
ERv
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.3.1 電流 (Currents)
Example 5.3 : A rectangular loop of wire, supporting a mass m, hangs vertically with one end in a uniform magnetic field B, which points into the page in the shaded region. For what current I, in the loop, would the magnetic force upward exactly balance the gravitational force downward?
mgIBaFmag
Ba
mgI
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.3.2 電流 (Currents)
What happens if we now increase the current?
mgIBaFmag The loop rises a height of h
?0IBahhFW magmag
u
w
vmagF qwB
quB
IaBawBqwBFvert Rise the loop
auBquBFhoriz
IBahuwdtaBauBwdtwdtFW horizagent
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.3.3 電流 (Currents)
Example 5.4
a) A current I is uniformly distributed over a wire of circular cross section, with radius a. Find the volume current density J.
2a
IJ
b) Suppose the current density in the wire is proportional to the distance from the axis, J = ks. Find the total current in the wire.
The current in the shaded path dsdks)ds)(sd(JJda 2
3
ka2dssk2dsdksJdaI
3a
0
22
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.1.3.4 電流 (Currents)
SS
adJJdaI
The current crossing a surface S can be written as
The total charge per unit time leaving a volume V is
VS
d)J(adJ
Because charge is conserved
VVV
d)t
(ddt
dd)J(
tJ
Continuity equation
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.1 穩定電流 (Steady Currents)
0t
J
Continuity equation
Actually, it is not necessary that the charges be stationary, but only that the charge density at each point be constant ( be independent with time).
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.2 穩定電流的磁場 (The Magnetic Field of a Steady Current)
Example 5.5
Find the magnetic field a distance s from a long straight wire carrying a steady current I
I
sr
ℓ dℓ
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.3 穩定電流的磁場 (The Magnetic Field of a Steady Current)
20
r
rd
4
IB
I
sr
ℓ dℓ
dcos
scosdsindrd
tans d
cos
sd 2
2
2
2 s
cos
r
1
2
1
2
1
dcoss4
I
s
cosd
cos
s
4
I
r
rd
4
IB 0
2
20
20
)sin(sins4
I12
0
s2
I)]1(1[
s4
I 00
2
,2 21
1
2
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.4 穩定電流的磁場 (The Magnetic Field of a Steady Current)
The field at 2 due to 1 is
d2
IB 10
into the page
BdIF
The force on 2 due to 1 is d)d2
I(IF 10
2
The force per unit lengthd2
IIF 210
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.5 穩定電流的磁場 (The Magnetic Field of a Steady Current)
Example 5.6 Magnetic Field on the Axis of a Circular Current Loop
)Rx(
ds
4
I
r
rsd
4
IdB
22
0
2
0
)Rx(
cosds
4
IcosdBB
22
0
x
2/122 )Rx(
Rcos
2/322
2
0
2/322
0
x )Rx(2
IRds
)Rx(4
IRB
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.6 穩定電流的磁場 (The Magnetic Field of a Steady Current)
For surface and volume currents, the Biot-Savart Law becomes
'daˆ)'r(K
4)r(B 2
0
'dˆ)'r(J
4)r(B 2
0
Problem 5.8
a) Find the magnetic field at the center of a square loop, which carries a steady current I. Let R be the distance from center to side.
R R4
I2)
2
2
2
2(
R4
I)sin(sin
s4
IB 00
120
for 10
2 45,Rs
Four sidesR
I2
R4
I24B 00
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.7 穩定電流的磁場 (The Magnetic Field of a Steady Current)
b) Find the field at the center of a rectangular n-sided polygon, carring a steady current I. Let R be the distance from center to any side.
nsin
R2
I)
nsin
n(sin
R4
I)sin(sin
s4
IB 00
120
for 12 n,Rs
n sidesn
sinR2
InB 0
c) Check that your formula reduces to the field at the center of a circular loop, in the limit n ∞.
n ∞nn
sinsinsin 12
R2
I
nR2
In
nsin
R2
InB 000
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.2.8 穩定電流的磁場 (The Magnetic Field of a Steady Current)Problem 5.9
Find the magnetic field at point P for each of the steady current configurations.
P
I b
a
The vertical and horizontal lines produce no field at P.
The two quarter-circles
)b
1
a
1(
8
I)
b2
I
a2
I(
4
1B 000
R I
IP
The two half-lines are the same as one infinite line
R2
IB 0
The half-circle: R4
IB 0
Total:
)2
1(R4
IB 0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.1.1 磁場的散度與旋度– 直線電流 (Straight-Line Currents)
The magnetic field of an infinite straight wire
Ids2
Id
s2
IdB 0
00
Notice that the answer is independent of s; that’s because B decreases at the same rate as the circumference increases.
In fact, it doesn’t have to be a circle; any old loop that encloses the wire would give the same answer. For if we use cylindrical coordinates (s,,z), with this current flowing along the z axis,
ˆ
s2
IB 0
zdzˆsdsdsd
Id2
Isd
s
1
2
IdB 0
2
0
00
This assumes the loop encircles the wire exactly once.
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.1.2 磁場的散度與旋度– 直線電流 (Straight-Line Currents)
Suppose we have a bundle of straight wires. Each wire that posses through our loop contributes 0I, and those outside contribute nothing.
I1
I2
I3
I4
enc0IdB
I enc : total current enclosed by the integration path
If the flow of charge is represented by a volume current density J, the enclosed current is
adJIenc
Applying Stokes’ theorem
adJIad)B(dB 0enc0
JB 0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.2.1 磁場的散度與旋度 (The Divergence and Curl of B)
(x,y,z)r
(x’,y’,z’)
d’
The Biot-Savart law for the general case of a volume current reads
'd]ˆ)'r(J
[4
)r(B 20
)B(A)A(B)BA(
)ˆ
()'r(J)]'r(J[ˆ
]ˆ
)'r(J[ 222
0)'r(J
Because J doesn’t depend on the unprimed variables (x,y,z)
0ˆ2
0]ˆ
)'r(J[ 2
0)r(B
The divergence of the magnetic field is zero!
'dˆ)'r(J
4)r(B 2
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.2.2 磁場的散度與旋度 (The Divergence and Curl of B)
'd]ˆ)'r(J
[4
)r(B 20
)A(B)B(AB)A(A)B()BA(
)]'r(J[ˆ
)ˆ
)('r(Jˆ
])'r(J[)'r(J)ˆ
(]ˆ
)'r(J[ 22222
)'rr(4)(4)ˆ
( 332
= 0 = 0
)ˆ
)('r(Jˆ
])'r(J[]ˆ
)'r(J[ 222
'dˆ)'r(J
4)r(B 2
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.2.3 磁場的散度與旋度 (The Divergence and Curl of B)
22
ˆ]')'r(J[
ˆ])'r(J[
)'xx(f'x
)'xx(fx
The x component )'xx
](')'r(J[ 3
)f(A)A(f)Af(
)A(f)Af()f(A
)]'r(J')['xx
()]'r(J'xx
[')'xx
](')'r(J[ 333
For steady currents the divergence of J is zero 0)'r(J'
)]'r(J'xx
[')'xx
](')'r(J[ 33
This contribution to the integral can be written
0'ad)'r(J'xx
'd)]'r(J'xx
['S
3V
3
On the boundary J = 0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.2.4 磁場的散度與旋度 (The Divergence and Curl of B)
'd]ˆ)'r(J
[4
)r(B 20
)'rr()'r(J4)()'r(J4)ˆ
)('r(J]ˆ
)'r(J[ 3322
'd)'rr()'r(J44
'd]ˆ)'r(J
[4
)r(B 302
0
)r(J0
)r(J)r(B 0
Ampère’s law in differential form
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.3.1 安培定律的應用 (Applications of Ampère’s Law)
)r(J)r(B 0
Ampère’s law in differential form
enc00 IadJdBad)B(
Ampère’s law in integral form
Example 5.8
Find the magnetic field of an infinite uniform surface current , flowing over the xy plane
xKK
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.3.2 安培定律的應用 (Applications of Ampère’s Law)
What is the direction of B?
KB
From the Biot-Savart law
Could it have a z-component ? no (symmetry)
The magnetic field points to the left above the plane and to the right below it
KIB2dB 0enc0 2
KB 0
0zfory2
K
0zfory2
K
B0
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.3.3 安培定律的應用 (Applications of Ampère’s Law)
Example 5.9
Find the magnetic field of a very long solenoid, consisting of n closely wound turns per unit length on a cylinder of radius R and carrying a steady current I.
BdsBsdBsdB
1side1side
NIBsdB0
Where N is the number of turns in the length nII
NB
00
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.3.4 安培定律的應用 (Applications of Ampère’s Law)
Example 5.10
Find the magnetic field of a toroidal coil, consisting of a circular ring around which a long wire is wrapped.
NI)r2(BdsBsdB0
r2
NIB 0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.3.5 安培定律的應用 (Applications of Ampère’s Law)
Problem 5.13
A steady current I flows down a long cylindrical wire of radius a. Find the magnetic field, both inside and outside the wire, if(a). The current is uniformly distributed over the outside surface of the wire.(b). The current is distributed in such a way that J is proportional to s, the distance from the axis.
Ia
(a)
enc0IsB2dB
asforˆs2
Iasfor0
B0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.3.5 安培定律的應用 (Applications of Ampère’s Law)
(b)
enc0IsB2dB
ksJ 3
ka2ds)s2(ksJdaI
3a
0
3a2
I3k
3a2
Is3ksJ
For s < a
3
303
30
s
030enc0 a
Iss
3
1
a
I3sds2
a2
Is3IsB2dB
3
20
a2
IsB
For s > a
Ia3
1
a
I3sds2
a2
Is3IsB2dB 0
330
a
030enc0
s2
IB 0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.3.4. 靜磁學與靜電學之比較 (Comparison of Magnetostatics and Electrostatics)
The divergence and curl of the electrostatic field are
0E
1E
0
Gauss’s law
The divergence and curl of the magnetostatic field are
JB
0B
0
Ampère’s law
“The electric force is stronger than the magnetic force. Only when both the source charge and the test charge are moving at velocities comparable to the speed of light, the magnetic force approaches the electric force.”
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.1. 磁向量位勢 (Magnetic Vector Potential)
0E
VE
V: electric scalar potential
You can add to V any function whose gradient is zero
VfV)fV(
0B
AB
: magnetic vector potentialA
You can add to any function whose curl is zeroA
AA)A(
AB
JA)A()A(B 02
0A
We will prove that :
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.2. 磁向量位勢 (Magnetic Vector Potential)
Suppose that our original vector potential is not divergenceless0A
0A0
Because we can add to any function whose curl is zero0A
0AA
200 AAA
If a function can be found that satisfies
02 A
0A
Mathematically identical to Poisson’s equation0
2V
In particular, if goes to zero at infinity, then the solution is
'd4
1V
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.3. 磁向量位勢 (Magnetic Vector Potential)
By the same token, if goes to zero at infinity, then 0A
'dA
4
1 0
It is always possible to make the vector potential divergenceless
0A
so JAA)A(B 022
This again is a Poisson’s equation
Assuming goes to zero at infinity, then J
'd)'r(J
4)r(A 0
For line and surface currents
'd)'r(I
4)r(A 0
'da)'r(K
4)r(A 0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.4. 磁向量位勢 (Magnetic Vector Potential)
Example 5.11
A spherical shell, of radius R, carrying a uniform surface charge , is set spinning at angular velocity . Find the vector potential it produces at point . r
The integration is easier if we let lie on the z axis, so that is titled at an angle .
r
r
'r
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.5. 磁向量位勢 (Magnetic Vector Potential)
'da)'r(K
4)r(A 0
where vK
'cosRr2rR 22
'cosR'sin'sinR'cos'sinR
cos0sin
zyx
'rv
]z)'sin'sin(siny)'cossin'cos'sin(cosx)'sin'sin(cos[R
Because 0'd'cos'd'sin2
0
2
0
We just consider y'cossinRv
y)'d'cosRr2rR
'cos'sin(
2
sinR
y'd'd'sinR'cosRr2rR
)'cossinR(
4)r(A
022
30
2
22
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.6. 磁向量位勢 (Magnetic Vector Potential)
'cosu
)]rR)(RrrR(rR)RrrR[(rR3
1
]Rru2rRrR3
)RrurR([
])Rru2rR(rR3
1Rru2rR
Rr
u[
duRru2rR
u'd
'cosRr2rR
'cos'sinI
222222
1
1
2222
22
1
1
2/32222
22
1
122
022
Letting , the integral becomes
If the point lies inside the sphere, Then R > r. r
2R3
r2I
If the point lies outside the sphere, Then R < r. r
2r3
R2I
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.7. 磁向量位勢 (Magnetic Vector Potential)
yI2
sinRy)'d
'cosRr2rR
'cos'sin(
2
sinR)r(A
30
022
30
If the point lies inside the sphere, Then R > r. r
2R3
r2I
If the point lies outside the sphere, Then R < r. r
2r3
R2I
Noting that ysinr)r(
For the point inside the sphere
)r(3
Ry
R3
r2
2
sinRyI
2
sinR)r(A 0
2
30
30
For the point outside the sphere
)r(r3
Ry
r3
R2
2
sinRyI
2
sinR)r(A 3
40
2
30
30
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.8. 磁向量位勢 (Magnetic Vector Potential)
We revert to the original coordinates, in which coincides with the z axis and the point is at (r,,)
r
For the point inside the sphere
ˆsinr
3
R)r(
3
R)r(A 00
For the point outside the sphere
ˆsin
r3
R)r(
r3
R)r(A 2
40
3
40
z3
R2)ˆsinr(cos
3
R2)r(AB 00
The magnetic field inside this spherical shell is uniform!
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.1.9. 磁向量位勢 (Magnetic Vector Potential)
Example 5.12
Find the vector potential of an infinite solenoid with n turns per unit length, radius R, and current I
We cannot use because the current itself extends to infinity.
'd)'r(I
4)r(A 0
Notice that adBad)A(dA
Since the magnetic field is uniform inside the solenoid : znIB 0
)s(nIadB)s2(AdA 20
ˆ2
nIsA 0
For s < R
For an amperian outside the solenoid
)R(nIadB)s2(AdA 20
ˆ
s2
nIRA
20
For s > R
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.2.1. 總結 ;靜磁邊界條件 (Summary; Magnetostatic Boundary Conditions)
B
J
A
'd)'r(J
4)r(A 0
JA 02
'dˆ)'r(J
4)r(B 2
0
JB 0
BA
?
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.2.2. 總結 ;靜磁邊界條件 (Summary; Magnetostatic Boundary Conditions)
0B
0adBd)B(V
belowabove BB
KI)BB(dB 0enc0//below
//above KBB 0
//below
//above
For an amperian loop running perpendicular to the current
For an amperian loop running parallel to the current
0I)BB(dB enc0//below
//above
Perpendicular to the current
//below
//above BB
Parallel to the current
)nK(BB 0belowabove
Where is a unit vector perpendicular to the surface, pointing “upward”.n
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.2.3. 總結 ;靜磁邊界條件 (Summary; Magnetostatic Boundary Conditions)
The vector potential is continuous across any boundary
danBdan)A(dA
For an amperian loop of vanishing thickness 0
belowabove AA
KBB 0//below
//above B
z
A
00A
z/y/x/
zyx
)A(
y
y
Kn
A
n
A0
belowabove
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.3.1. 向量位勢的多極展開 (Multipole Expansion of the Vector Potential)
I
r
'r
'd'rd
'
A multipole expansion which is an approximate formula and valid at distant points) for the vector potential of a localized current distribution.
0n
nn
22)'(cosP)
r
'r(
r
1
'cos'rr2)'r(r
11
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.3.2. 向量位勢的多極展開 (Multipole Expansion of the Vector Potential)
0nn
n1n
00 'd)'(cosP)'r(r
1
4
I'd
1
4
I)r(A
....]'d)2
1'cos
2
3()'r(
r
1'd'cos'r
r
1'd
r
1[
4
I)r(A 22
320
monopole dipole quadrupole
The magnetic monopole tern is always zero0'd
'd)'rr(r4
I'd'cos'r
r4
I)r(A 2
02
0dip
Let , where is a constant vector.Tcv
c
area
'da'n)v'('dv
area
'da'n)Tc'('dTc
0B
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.3.3. 向量位勢的多極展開 (Multipole Expansion of the Vector Potential)
T'cT'cT)c'(cT'T)c'()Tc('
areaarea
'da)'nT'(c'da)T'c('n'dTc
area
'da)'nT'('Td
Let 'rrT
area
'da]'n)'rr('['d)'rr(
ar'da'nr'da)'nr('d)'rr(areaarea
'da'n'd'r2
1a
)'d'r
2
1(r'd)'rr(
20
dip r
rm
4)r(A
where aIadIm
)ra(r4
I)ar(
r4
I'd)'rr(
r4
I)r(A 2
02
02
0dip
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.3.4. 向量位勢的多極展開 (Multipole Expansion of the Vector Potential)
Example 5.13
Find the magnetic dipole moment of the “Bookend-shaped” loop shown in Figure below. All sides have length w, and it carries a current I.
w
w
w
The wire could be considered the superposition of two plane square loops shown in Fig. 5.53 of text book.
The combined (net) magnetic dipole moment is
zIwyIwm 22
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University
of Kaohsiung
5.4.3.5. 向量位勢的多極展開 (Multipole Expansion of the Vector Potential)
The magnetic dipole moment is independent of the choice of origin.
The magnetic field of a (pure) dipole is easier to calculate if we put the dipole moment at the origin and let it point in the z-direction.
20
dip r
rm
4)r(A
ˆ
r
sinm
4)r(A 2
0dip
)ˆsinrcos2(r4
m
)r4
sinm(sinr00
//r/
ˆsinrˆrr
sinr
1)r(A)r(B
30
20
2dipole
This is identical in structure to the field of an electric dipole!