p212c29: 1 chapter 29: sources of magnetic field sources are moving electric charges single charged...
TRANSCRIPT
p212c29: 1
2
2
2
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comparer
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o
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Chapter 29: Sources of Magnetic Field
Sources are moving electric charges
single charged particle:
v
field point
Brr
^
field lines circulate around “straight line trajectory”
r^
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from the definition of a Coulomb and other standards,
o= 4x10-7 N s2/C2 = 4x10-7 T m/A
with o , the speed of light as a fundamental constant can be determined (in later chapters) from
c2 = 1/(o o)
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Simple geometry (all 90 degree angles)
Fmagnetic = k’|q2v2 q1v1|/ (r12)2
= k’ e2v2/r2
Felectric = k e2/r2
Fmagnetic /Felectric = k’ v2 /k = (o/4v2/ (1/4o
= v2 /c2 but... depends upon frame?
Consider: Two protons move in the x direction at a speed v.
Calculate all the forces each exerts on the other.
magnetic: F1 on 2 = q2v2xBat2due to 1
= q2v2x (k’ (q1v1xr12)/(r12)2)
electric F1 on 2 = (k q1q2)/(r12)2 r12
B
v1q1
r12^
^
r12
v2
F1 on 2
^
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Gauss’s Law for magnetism:
Magnetic field lines encircle currents/moving charges. Field lines do not end or begin on any “charges”.
for any closed surface.
B dA 0
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Current elements as magnetic field sources
2
2
2
2
2
ˆ4
ˆ4
ˆ)(
4
ˆ4
ˆ
4
rrlId
B
rrlId
Bd
rrvq
nAdl
rrvq
N
r
rvqB
o
o
do
do
i
iiio
I dl
^
field point
Brr
superposition of contributions of all charge carriers
+ large number of carriers + small current element
^r
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Field of a long straight wire
x
I
dl
r̂
dB
y
rI
B
zx
IB
zdyyx
xIB
zdyyx
xI
zr
Idyr
rlIdBd
o
o
o
o
o
o
2
)ˆ(2
)ˆ(4
)ˆ(4
page) (into )ˆ(sin
4
ˆ4
2322
2322
2
2
r x y
y r
2 2
sin /
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Example: A long straight wire carries a current of 100 A. At what distance will the magnetic field due to the wire be approximately as strong as the earth’s field (10-4 T)?
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Force between two long parallel current carrying wires
(consider, for this example, currents in the same direction)I1 I2
Bdue to I1Fdue to I1
F = I2 Bdue to I1 L
= I2 (oI1/(2r))L
F/L = oI1 I2 /(2r)
force on I1 is towards I2
force is attractive (force is repulsive for currents in opposite directions!)
Example: Two 1m wires separated by 1cm each carry 10 A in the same direction. What is the force one wire exerts on the other
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Magnetic Field of a Circular Current Loop
I dl r̂
ra
x
dB
dBx
arc) of(segment
'
)90cos(||
sin)90cos(
ˆ'
|ˆ|'||
2222
22
22
2
2
dsdlax
aax
Idlk
BddB
ax
a
axr
rldrIdl
k
rrldI
kBd
x
2322
2322
22
02322
)(2
'
)(2
')(
'
axM
kB
axIa
kax
aIdsk
dBB
x
a
xx
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ax
xM
k
x
IaB
a
IB
ax
IaB
ox
ox
ox
3
3
2
2322
2
2'
2
centerat 2
)(2
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Example: A coil consisting of 100 circular loops .2 m in radius carries a current of 5 A. What is the magnetic field strength at the center?
N loops => N x magnetic field of 1 loop.
At what distance will the field strength be half that at the center?
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Ampere’s Law
•Equivalent to Biot-Savart Law
•Useful in areas of high symmetry
•Analogous to Gauss’s Law for Electric Fields
Formulated in terms of: B dl B dl ||
For simplicity, consider single long straight wire (source) and paths for the integral confined to a plane perpendicular to the wire.
IB
dl
dBIdl r
ro
4 2
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B dl Bdl
Brd
I
rrd
Id
B dlI
d
Id
I
o
o
o
o
o
cos
2
2
2
2
0
path encloses current
path does not
enclose current
B
dl
d
r
B
dl
d
r
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B dl I
I dl
B dl J dA
o enclosed
o
Right Hand Rule
relates conventions for directions of and
In general
.
•Amperian Loop analogous to Gaussian surface
•Use paths with B parallel/perpendicular to path
•Use paths which reflect symmetry
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Application of Ampere’s Law: field of a long straight wire
Cylindrical Symmetry, field lines circulate around wire.
r
I
r
IrB
I
rrB
dlrBldrB
o
o
2)(
2)(
)()(
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Field inside a long conductor
I
Jr
22
encl
encl
2)(
)()(
rRI
JAI
I
rrB
dlrBldrB
o
outside] 2
)([
2)(
2)(
2
22
r
IrB
RrI
rB
rRI
rrB
o
o
o
R
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B/Bmax
rr=R
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I
I
Homework: Coaxial cable
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Magnetic Field in a SolenoidClose packed stacks of coils form cylinder
Fields tend to cancel in region right between wires.
Field Lines continue down center of cylinder
Field is negligible directly outside of the cylinder
I
B
I
B
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B
L I
solenoid) finitefor (
length)unit per turns(
ILN
B
nIB
LnIBL
nLnII
BLldB
o
o
o
enc
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Example: what field is produced in an air core solenoid with 20 turns per cm carrying a current of 5A?
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Toroidal Solenoid
r
NIB
NIrB
NII
rBldB
o
o
enc
2
2
2
torusofcenter along Field
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Magnetic MaterialsMicroscopic current loops:
electron “orbits”
electron “spin”
L
nh
nL
Lme
mmvr
e
rr
veIA
rv
eI
2
22
2
2
2
B
BU
TJ
me
B
/10274.9
MagnetonBohr 2
24
Quantum Effects: quantized L, Pauli Exclusion Principle important in macroscopic magnetic behavior.
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Magnetic Materials: Microscopic magnetic moments interact with an external (applied) magnetic field Bo and each other, producing additional contributions to the net magnetic field B.
Magnetization M = tot/V B= Bo + o M
linear approximation: M proportional to Bo
o => = Km o = permeability m = Km-1 magnetic Susceptibility
Types of MaterialsDiamagnetic: Magnetic field decreases in strength.Paramagnetic: Magnetic field increases in strength.Ferromagnetic: Magnetic field increases in strength!Diamagnetic and Paramagnetic are often approximately linear with m
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Ferromagnetism:
Greatly increases field
Highly nonlinear, with Hysteresis:
Hysteresis= magnetic record
Magnetization forms in Magnetic Domains
SaturationPermanent Magnetization
M
Bo
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Displacement Current“Generalizing” displacement current for Ampere’s Law
conduction current creates magnetic field
iC
Amperian loop with surfaceIenc = iC
Parallel PlateCapacitor
iC
Parallel PlateCapacitor
Amperian loop with surfaceIenc = 0???
p212c29: 27
Define Displacement Current between plates so that iD = iC
Fields! Magnetic createcan Fields Electric changing
)(
dtEd
j
AdEdtd
i
iildB
idt
ddtdq
i
EAEddA
Cvq
D
oCo
DCo
DE
C
E