Download - Electromagnetism
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Electromagnetism
Zhu Jiongming
Department of Physics
Shanghai Teachers University
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Electromagnetism
Chapter 1 Electric Field
Chapter 2 Conductors
Chapter 3 Dielectrics
Chapter 4 Direct-Current Circuits
Chapter 5 Magnetic Field
Chapter 6 Electromagnetic Induction
Chapter 7 Magnetic Materials
Chapter 8 Alternating Current
Chapter 9 Electromagnetic Waves
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Chapter 5 Magnetic Field
§1. Introduction to Basic Magnetic Phenomena
§2. The Law of Biot and Savart
§3. Magnetic Flux
§4. Ampere’s Law
§5. Charged Particles Moving in a Magnetic Field
§6. Magnetic Force on a Current-Carrying Conductor
§7. Magnetic Field of a of a Current Loop
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§1. Basic Magnetic Phenomena
Comparing with Electric Fields :E : charge electric field charge
( produce ) ( force )M :
Permanent Magnets Magnetic Effect of Electric Currents Molecular Current
Movingcharge
Movingcharge
magnetic field
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Permanent Magnets
Two kinds of Magnets : natural 、 manmade Two Magnetic Poles : south S 、 north N Force on each other : repel ( N-N, S-S ) attract ( N-S ) Magnetic Monopole ?
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Magnetic Effect of Electric Currents
Experiments Show Straight Line Current
I
S
N
I
N
S
Molecular Current—— Ampere’s Assumption
Two Parallel Lines Circular Current Solenoid and Magnetic Bar
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Magnetic Field B
Experiment : Helmholts coils in a
hydrogen bulb , an electron gun
I I
M M’ Conclusion : moving charge
F = q v B ( Definition of B ) ( Electric Field : F = qE ) Unit : Tesla )Gauss10(
m/s1C
1NTesla 1 4
Magnetic Field Lines :( curve with a direction ) Tangent at any point on a line is in the direction of t
he magnetic field at that point
● Number of field lines through unit area perpendicular to B equals the magnitude of B
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§2. The Law of Biot and Savart
1. The Law of Biot and Savart
2. Magnetic Field of a Long Straight Line Current
3. Magnetic Field of a Circular Current Loop
4. Magnetic Field on the Axis of a Solenoid
5. Examples
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1. The Law of Biot and Savart
The field of a current element Idl
dB Idl , 1/r2 , sin r : Idl P : angle between Idl and r Proportionality constant : 0/4 = 10-7
20 sind
4d
r
lIB
20 ˆd
4d
r
I rlB
2
0 ˆd
4d
r
I rlBB
dB
Idl
r
P
rEE ˆd
4
1d
20
r
q
Direction : dBIdl , dB r
Integral :
Compare with :
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2. Field of a Long Straight Line Current
current I , distance a
all dB in same direction
20 ˆd
4 r
I rlB
2
0 sind
4 r
lIB
2
1
dsin1
40
a
I
)cos(cos4 21
0
a
I
r
2
1
a PO
I
dllsin = a/r
ctg = - l/a
r = a/sin l = - a ctg
dl= ad/sin2Infinite long : 1= 0 , 2= ,
a
IB
2
0 Direction : right hand rule
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3. Magnetic Field of a Circular Current
current I , radius R ,P on axis , distance a
20 ˆd
4d
r
I rlB
zoR
r
a
dB
P
dl
I
cos
sind
4 20
r
lIB
= 90o
cos = R/r
r2 = R2 + a2
lr
RId
4 30
2/322
20
)(2 aR
IR
Rl 2d
R
IBa
20,Oat)1( 0
:3
20
2,far)2(
a
RIBRa
:
component dB||= dBcossymmetry , dB cancel , B = 0
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4. Field on the Axis of a Solenoid
current I, radius R, Length L,
n turns per unit length
dB at P on axis caused by nIdl
( as circular current )
I
PR l
dl
L
2/322
20
)(
d
2d
lR
lnIRB
d)sin(
20 nI
1
2
d)sin(20
nIB
)cos(cos2 210
nI
ctg = l/R
l = R ctg
dl= - Rd /sin2R2+ l2 = R2/sin2
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Field on the Axis of a Solenoid
Direction :right hand rule
)cos(cos2 210
nI
B
PR
L
2
1
B
I
B
O L
(1) center ( or R << L ) 1= 0 , 2= , B = 0nI
(2) ends ( Ex. : left ) 1= 0 , 2= /2 , B = 0nI /2
(3) outside, cos1 、 cos2 same sign, minus, B small
inside, opposite sign, plus , B large
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Example ( p.345/5 - 3 -11)Uniform ring with current , find B at the center.Sol. :
I
I
O
B
C
12
I1
I2
20 ˆd
4d
r
I rlB
0ˆd rlI
R
IB
20
2
2
210
1
R
IB
22
202 R
IB B1= B2
opposite direction
B = 0
2
2
1
R
R
21 )2( II
Straight lines : ( circular current : )
arc 1 :
arc 2 :
parallel : I1R1 = I2R2
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Exercises
p.212 / 5-2- 3, 8, 12, 13, 16
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§3. Magnetic Flux
1. Magnetic Flux
2. Magnetic Flux on Closed Surface
3. Magnetic Flux through Closed Path
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Flux on area element dS
dB = B · dS = B dS cos Flux on surface S ( integral ) if B and dS in same direction ( = 0 ), write dS
= Magnitude of B
Unit : 1 Web = 1 T · m2
define number of B lines through dS = B · dS = dB
then line density =
1. Magnetic Flux
dS
B
SB B
d
dB : Flux per unit area perpendicular to B
BS
B
d
d
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Show : (1) dB of current element Idl
B lines are concentric circles
these circles and the surface S either not intercross ( no contribution to flux ) or intercross 2 times ( in/out , flux +/- )
2. Magnetic Flux on Closed Surface
)surfaceclosedany(0d SS
SB
dB
11111 dddddin SBSB:
22222 dddddout SBSB:21 dd BB
21 dd SS
21 dd 0dd 21 0d
S
Idl
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Magnetic Flux on Closed Surface
Show : (2) magnetic field of any currents
superposition : B = B1 + B2 + …
0dddd 21 SSSSSB
B lines are continual , closed , or
—— called The field without sources
Compare with : E lines from +q or , into -q or
—— called The field with sources
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Turn the normal vector of S1
opposite, same as that of S2
then
3. Magnetic Flux through Closed Path
Any surfaces bounded by the closed path L have the same fluxShow :
L
S1 S2
n
n
0ddd21
SSSSBSBSB
21
ddSS
SBSB
21
ddSS
SBSB
—— called Magnetic Flux through Closed Path L
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Exercises
p.214 / 5-3- 1, 3
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§4. Ampere’s Law
1. Ampere’s Law
2. Magnetic Field of a Uniform Long Cylinder
3. Magnetic Field of a Long Solenoid
4. Magnetic Field of a Toroidal Solenoid
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1. Ampere’s Law
Ampere’s Law : L : any closed loop I : net current enclosed by L
Three steps to show the law : L encloses a Long Straight Current I L encloses no Currents L encloses Several Currents
IL 0d lB
I
L
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L Encloses a Long Straight Current I
Field of a long line current I : I
La
IB
2
0
sBlB dcosdd lB
d
20 aa
I
d2
0I
II
LL 00 d
2d
lBI
ds dld
LB
( direction: tangent )
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L Encloses no Currents
Current I is outside L
21
dddLLL
lBlBlBI L2L1
)dd(2 21
0 LL
I
0)(2
0
I
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L Encloses Several Currents
L encloses several currents
Principle of superposition : B = B1 + B2 + …
LL
lBBlB d)(d 21
I is algebraic sum of the currents enclosed by L direction of Ii with direction of L ( integral ):
right hand rule , take positive sign
)( 210 II I0
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I0
2. Field of a Uniform Long Cylinder
radius R , current I ( outgoing ),find B at P a distance r from the axis
concentric circle L with radius r ,symmetry : same magnitude of B on L ,
direction: tangent
P
L
rBL
2d lB
)(2
0 Rrr
IB
rBL
2d lB
)(2 2
0 RrR
IrB
0 r
B
R
outside :
inside : 2
2
0 R
rI
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radius R , current I ( outgoing ),field B at P a distance r from the axis
symmetry : B in direction of tangent
Direction is along the Tangent
P
B
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3. Magnetic Field of a Long Solenoid
Field inside is along axis
Show : turn 180 o round zz’ : B B’
I opposite : B’ B’’
B’’ should coincide with B
a
d
d
c
c
b
b
aLlBlBlBlBlB ddddd
nIllBlB cdab 000 nIBab 0
0 cdB
nIllBlBab 0outin 00axison not if :nIB 0in
B’B
B’’
z’
z
ab
dc
direction :right hand rule
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4. Field of a Toroidal Solenoid
Symmetry : B on the circle L
magnitude : same
direction : tangent
( L >> r , N turns )in :
out :
NILBL 0ind lB
nIL
NIB 0
0in
0d out LBL
lB
0out B
direction : right hand rule
if L , becomes a long solenoid
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Surface current ( width l , thickness d )
5. Field of a Uniform Large Plane
dB
l
Jdl
Jld
l
I
llBlB zzL012d lB z
20
12
zz BB
012 2
nn EE比较电场:
Direction : parallel opposite on two sides
( right hand rule )
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Exercises
p.215 / 5-4- 2, 3, 4, 5
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§5. Charged Particles Moving in B
1. Motion of Charged Particles in a Magnetic Field
2. Magnetic Converging
3. Cyclotrons
4. Thomson’s e/m Experiment ( skip )5. The Hall Effect
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1. Motion of Charged ParticlesLorents Force : F = q ( E + v B )if E = 0 , F = q v B
if v B , q moves in a circle
with constant speed
Centripetal force :qvB
R
mvF
2
C
OR
v
F
m, q=- e
Radius : R = mv/qB Period : T = 2R/v = 2 m/qB Frequency : f = 1/ T = qB/ 2 m Ratio of charge to mass : q/ m= v/BR = /B
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2. Magnetic Converging
v making an angle with B : v || = v cos
v = v sin
Helical pathradius : sin
qB
mv
qB
mvR
cos
2|| qB
mvTvh
)( small ~sin vv
)small (~cos vv
R
h
P P’
pitch :
Magnetic Converging : different R , same h from P to P’ distance h
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Take period of emf same as that of qaccelerated 2 times per revolution
v r ( ), T not changed
3. Cyclotrons
Principle : uniform field , outward
2 Dees , alternating emf
q accelerated as crossing the gap
qB
mT
2 ( not depend on v, r )
qB
mvr
Application : accelerating proton 、 etc. to slam into a solid target to learn it’s structure
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Ex. : deuteron q/ m ~ 10 7 , B ~ 2 , R ~ 0.5
need U ~ 10 7 ( volts ) frequency of emf f = qB/ 2m ~ B magnetic field relativity : v m f varying frequency —— Synchrotrons
Cyclotrons Compare with straight line accelerator
Str. :Cyc. :
qUmv 2
2
1
m
RBq
m
qBRmmv
2)(
2
1
2
1 22222
m
RqBU
2
22
22
0
/1 cv
mm
To gain the same v , need
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Exp. carrier q , force fL = q v B
q > 0 , v - positive x , fL - positive z
q < 0 , v - opposite x , fL - positive z direction
A’
5. The Hall Effect A conducting strip of width l, thickness d x - current , y - magnetic field z - voltage UAA’
x
yz
I
BA
l
d
fL
fe
for q > 0 , positive charges pile up on side A , negative on A’ produce an electric field Et
fe = qEt opposite to fL slow down qEt = qvB
stop piling q moves along x ( as without B ) the Hall potential difference : UAA’ = Et l = vBl
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The Hall Constant
I = q n ( vld )
v = I/qnld
UAA’ = IB/qnd
write : UAA’ = K IB/d
proportional to IB/d ( macroscopic )Hall constant : K = 1/qn ( microscopic )
determined by q 、 n q > 0 , K > 0 UAA’ > 0
q < 0 , K < 0 UAA’ < 0
( A - negative charges , A’ - positive )
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Exercises
p.216 / 5-5- 1, 3, 4, 5, 6
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§6. Magnetic Force on a Conductor
1. Ampere Force
2. Rectangular Current Loop in a Uniform
Magnetic Field
3. The Principle of a Galvanometer
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1. Ampere Force
current carriers magnetic force on conductor
electron : f = - ev B
current : j = - env
force on current element Idl : dF = N (- ev B )
= n dS dl (- ev B )
= dS dl ( j B )
= Idl B
Ampere force :
L
I0
d BlF
I
B
dldS
dl and j in same direction
j and dS in same direction
I = j dS = j dS
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2. Rectangular Loop in Magnetic Field
Normal vector n and current I
------ right-hand ruleu① :
d③ :l ② :r ④ :
II
B
n
①
②③
④
l2
l1
)90sin(d o
01
1 l
lIBF
cos1IBl ( up )cos13 IBlF ( down )
2
02 dl
lIBF 2IBl ( ⊙ )
24 IBlF ( )F1 , F3 cancel out
B
n
l1
F2
F4
F2 , F4 produce a net torque : T = F2l1sin = IBl2l1sin = ISBsin
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The Magnetic Dipole Moment
Torque on a current carrying rectangular loop :T = ISBsin ( direction : n B )
Definition :Magnetic Dipole Moment of a
current carrying rectangular loop
pm = IS n
then the torque
T = pm B
B
pm
I
T
( Comparison : in an electric field
p = ql , T = p E )
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Magnetic Moment of Any Loop
Divided into many small rectangular loops
outline ~ the loop , inner lines cancel out
dT = dpm B = IdS n B
all dT in the same direction
T = dT = IdSnB = InBdS
= IS n B = pm B
Definition : Magnetic Dipole Moment of Any Loop pm = IS n
no matter what shape
( same form as that of a rectangular loop )
n
BI
S
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Magnetic Dipole Moment of Any Loop
pm making an angle with B
maximum T for = /2 T = 0 for = 0
equilibrium, stable
lowest energy T = 0 for =
equilibrium, unstable
highest energy
n
BI
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3. The Principle of a Galvanometer
n turns : T = nISB
countertorque by springs
T’ = kwhen in balance
= nISB/ k I
( = 0 for I = 0 )
NS
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Exercises
p.217 / 5-6- 1, 5, 8
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§7. Field of a Current Loop
Circular loop of radius R , current I , on axis
o
R
a PI2/322
20
)(2 aR
IRB
3
20
2for
a
RIBRa
:
ISpRS m2
3m0
2 a
pB
)2
1 withcompare(
30 a
pE
:
pm = IS n is important
• torque exerted by magnetic field• produce magnetic field
B
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Exercises
p.219 / 5-6- 11