276 Chapter 30 · 2015. 8. 12. · 276 Chapter 30 . Calculating the magnetic field due to a current...
Transcript of 276 Chapter 30 · 2015. 8. 12. · 276 Chapter 30 . Calculating the magnetic field due to a current...
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Chapter 30
Calculating the magnetic field due to a current
In vector form
rrdqEd 3
041πε
=
Element ds
In vector form
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Two long parallel wires a distance 2d apart carry equal currents i in opposite directions, as shown in Fig30-10a. Derive an expression for B(x), the magnitude of the resultant magnetic field for points at a distance x from the midpoint of a line joining the wires. x a p b d d
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Sol:
B(x)=Ba(x)+Bb(x)=)()(2)(2 22
000
xdid
xdi
xdi
−=
−+
+ πμ
πμ
πμ
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ienc=i1-i2 Note: the right-hand rule
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Application:
Case 1 B outside a long straight line with current
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Case 2 uniformly distributed current i
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Magnitude: 2.0X10-5(T) Direction: counterclockwise
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Only (n: the number of turns per unit length)
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If Z>>R , B(Z)~ 3
20
2 ZiRμ
B(Z)= 30
2 ZNiA
πμ
(N turns A: the area of the loop)
In vector form 30
2)(
ZZB μ
πμ
=
Exercises:37,39,45,67
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Chapter 31
In section29-8,we saw that if we put a closed conducting loop in a B and then
send current through the loop, forces due to the magnetic field create a torque
to turn the loop current + magnetic field torque
With i=0, torque + magnetic field current?
Let us consider two experiments.
First experiment
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We note that
The current: an induced current The work per unit charge in producing that current: an induced emf The process: induction
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We note that
Switch on a current
Switch off a current
Only when there is a change in the current
Faraday’s law of induction:
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To oppose the magnetic field increase being caused by the approaching
magnet.
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(a)
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Direction: Clockwise(to oppose B)
(b)
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Direction: counterclockwise
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The same A small increase in the temperature of the loop.
Not a single loop!
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New form of faraday’s law
E: induced electric field
*Induced electric fields are produced not by static charges but by a changing
magnetic flux. Although electric fields produced in either way exert forces on
charged particles, there is an important difference between them.
Induced electric field: field lines form closed loops.
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0 or 5V?
In figure 31-14b, take R=8.5cm and dB/dt=0.13 T/S (a) Find a expression for the magnitude E of the induced electric field at
points within the magnetic field, at radius r from the center of the magnetic field. Evaluate the expression for r=5.2cm
(b) Find an expression for the magnitude E of the induced electric field at points that are outside the magnetic field. Evaluate the expression r=12.5cm
Copper ring removed
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(a)
∫∫∫Φ
===•=•dt
drEdSEdSESdE B)2( π
mmVEcmrdtdBrE
dtdBrrE
rBBAB
/4.32.52
)2(
)(
2
2
=⇒=
=
=
==Φ
ππ
π
(b)
)!0(/8.35.12
2
)(2
2
rtransformeEmmVEcmr
dtdB
rRE
RBBAB
⇒≠=⇒=
=
==Φ π
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: can be used to produce a desired magnetic field.
Def inductance
SI unit:
Inductance-like capacitance-depends only on the geometry of the device.
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Figure 31-16 shows a cross section, in the plane of the page, of a toroid of N
turns like that in fig.30-21a but of rectangular cross section; its dimensions are as indicated.
(a) What is its inductance L? (b) The toroid shown in fig.31-16 has N=1250turns, a =52mm, b=95mm,and
h=13mm what is the inductance?
(a)
B= riNπ
μ2
0
hdrr
iNBhdrAdBb
a
b
a πμ2
0∫∫∫ ==•=Φ
)/ln(2
12
00 abiNh
hdrr
iNh b
a πμ
πμ
==Φ ∫
)/ln(2
/2
0 abhN
iNπ
μ=Φ L=
(b)
=2.45X10-3H~2.5mH L
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( ) NΦ=Li
dtdiL− =
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(Like a RC circuit)
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S b
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(a)
i(0)=0
Inductors: broken wire
(b)
Long after the switch has been closed
Equilibrium
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Inductors: connecting wire
3R in parallel
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and are both in units of power (work (energy)/time )
A 3.56H inductor is placed in series with a 12.8Ω resistor, and an emf of
3.24V is then suddenly applied across the RL combination.
(a)At 0.278s (which is one inductive time constant) after the emf is applied,
what is the rat P at which energy is being delivered by the battery?
(b)At 0.278s, at what rate PR is energy appearing as thermal energy in the
resistor?
(c)At 0.278s, at what rate PB is energy being stored in the magnetic field?
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(a)
i=32.4/12.8(1-e-1)=0.16A
P=ξi=0.5184W~518mW
(b)
PR=i2R=(0.16)2(12.8)=0.3277W~328mW
(c)
PB=dUB/dt=Li(di/dt)
di/dt= L
t
LRt
eR
eLR
Rτξξ −−
=)(
PB=0.1907W~191mW
P= PB+ PR (energy conservation)
A solenoid: length l and cross-sectional area A
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AnlL 2
0μ= (31-33)
( )
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(a)
(b)
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Mutual induction: two coils
Self induction: one coil
Def M21 of coil 2 with respect to coil 1
L (self inductance)
(Not simple!)
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(a)
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R1>>R2
We may take B1 to be the magnetic field at all points within the boundary of
the smaller coil
(b)
What about ?
Exercises: 5,63,91,98
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Chapter 32
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Surface 1: N pole
Surface 2: No magnetic dipole
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M lies along
Magnetic materials are magnetic because of the electrons within them. We have already seen one way in which electrons can generate a magnetic field: send them through a wire as an electric current, and their motion produces a magnetic field around the wire.
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Two important issues:
Protons and neutrons ( 1/1000 !)
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Classical analysis
Net component: upward
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Electron atom material If the combined magnetic dipole moments produce a magnetic field, then the material is magnetic. *A diamagnetic material placed in an external magnetic field Bext develops a magnetic dipole moment directed opposite Bext . If the field is non-uniform, the diamagnetic material is repelled from a region of greater magnetic field toward a region of lesser field. * A paramagnetic material placed in an external magnetic field Bext develops a magnetic dipole moment in the direction of Bext. If the field is non-uniform, the paramagnetic material is attracted toward a region of greater magnetic field from a region of lesser field. Def. Magnetization M: the vector quantity, the magnetic dipole moment per unit volume
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If T>TC, ferromagnetic materials paramagnetic
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Rowland ring: thin toroidal core of circular cross section.
(Iron core)
T<TC, strong alignment of adjacent atomic dipoles in a ferromagnetic
material.
Q: An iron nail, a naturally strong magnet?
A: No. There are magnetic domains.
Regions of the crystal throughout which the alignment of the atomic dipoles is
essentially perfect. For the crystal as a whole, however, the domains are so
oriented that they largely cancel each other as far as their external magnetic
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effects are concerned.
=?
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Recall ampere’s law
(a)
ienc=0
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(For r≦R) r=0 B=0
(b)
(c)
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(a)
(b)
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Exercise :17,33,39,51
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Chapter 33
New physics – old mathematics
LC oscillations
See fig.33-1. Electric energy magnetic energy
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In contrast to RC and RL circuits, the total energy (UB+UE) in a LC circuit is
conserved.
Q=CV
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q x 1/c k i v L m
mk
=ω (Block-spring system) LC1
=ω (LC circuit)
(U is a constant)
(U is a constant)
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UE+UB=C
Q2
2
(a) In an oscillating LC circuit, what charge q, expressed in terms of the
maximum charge Q, is present on the capacitor when the energy is shared equally between the electric and magnetic fields? Assume that L=12mH
and C=1.7μF.
(b) When does this condition occur if the capacitor has its maximum charge at time t=0?
Sol:
(a) UE=1/2UE,max
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UE =C
q2
2
and UE,max= CQ2
2
Cq2
2
=C
Q22
1 2
QQq 707.0~2
=⇒
(b)
0.707Q=Q tωcos
ωt=45。=4π
t=ωπ4
= ssXLC μπ 110~1012.14
4−=
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LRte
CQ /
2
2−U= (decay)
(a)
(b)
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An external emf device supplies enough energy to make up for the energy
dissipated as thermal energy in the resistance R.
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(v and i in phase)
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(v and i 900 out of phase )
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In an RLC circuit, Let R=160Ω, c=15.0μF,L=230mH, fd=60.0Hz and
ξm=36.0V
(a) What is the current amplitude?
(b) What is the phase constant Φ?
Sol:
(a)
Ω=−+= 184)( 22CL XXRZ
I= AZm 196.0=
ξ
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(b)
radR
XX CL
513.0)564.0(tan
564.0tan
1 −=−=
−=−
=
−φ
φ
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(d) With that change in capacitance, what would Pav be ?
(a)
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(b)
(c)
(d) ZI rms
rmsξ
= )(0.72)0cos( 0 WIP rmsrmsav == ξ
I2R: Ohmic losses
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Exercises:27,53,75,87