Electromagnetic

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Chapter 12
Electromagnetic Induction
???? ????
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12-1 Nonelectrostatic Force, Source
Electromotive Force ???? ?? ???
1. Nonelectrostatic Force
When S is turned off,
the charges on A and B are constant,
?the E-field between A and B is constant.
When S is turned on, there is a current in circle,
the charges on A and B decrease, ? E-field ?
?
q0, E0, I0
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Nonelectrostatic Field
---nonelectrostatic force on per unit positive
charge.
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2. Electromotive Force ???
The work done by nonelectrostatic force for
moving per unit positive charge from B to A
within the source is defined as
---emf
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The emf in a complete circuit equals to the line
integral of the nonelectrostatic force round the
circuit .
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3. Complete circuit Ohm law ???????
Source emf ? , internal resistance r
In a time interval dt, emf moves dq from B to A
dq idt
So the work done by emf is
dW?dq ? idt
Meanwhile, the resistances R and r consume energy
Pdt (i2R i2r) dt
thermal energy
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According to the principle of conservation of
energy,
dW Pdt
i.e.
?idt (i2R i2r) dt
?
Vab iR
Vab ? - ir
---Complete circuit Ohm law
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12-2 Faradays Law of Induction ??????
1. Induction phenomena
Ii
Ii
Ii
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2. Lenzs Law
Determine the direction of an induced current.
The induced emf and current are in such a
direction as to oppose the change that produces
them.(???????????????????????????)
I
I
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3. Faradays law of induction
For SI
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Induced current
The induced charge passing through the loop
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Note
for a coil of N turns
---total flux
--number of flux linkages
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Example
Alternating current (AC) generator
A coil of area S,N turns rotating in a uniform
M-field. Angular velocity is ?,and ?0 when t0.
Find ??
Let
alternating emf
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Example A long, straight wire carries a
variable current I (t). Find the induced emf in
the rectangular loop.
?
Directioninverse-clockwise
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12?3 Motional electromotive force ?????
----motional emf
----induced emf
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The charge q in the rod
Motional emf
q suffers Lorentz force
--nonelectrostatic force
?
nonelectrostatic field is
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Notes
This formula can be used to calculate the
motional emf induced by any shape of moving
conductor in any M-field.
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Question
is the work done by Lorentz force to positive
unit charge.
From another formula ,
We get .
Lorentz force does not do any work for q.
How can we explain their conflict?
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q has two motions in the rod
The total work done by resultant force is
?
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solution
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Another solution
using
Make a subsidiary line,
then the M-flux through the fan area is
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A metal disk with radius R is rotating around its
center in the uniform M-field.
The motional emf from O to any point P of its
edge is
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Solution
Uagt Ub
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12-4 Induced electric field
1. Maxwell s hypothesis
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2. Induced emf
3 . The relation between induced emf and
changing M-field
S the area surrounded by L.
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3 . The differences between electrostatic field
and induced electric field
? Electrostatic field is set up by static charges.
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--non-vortex field (???)
?
E-potential can be introduced to describe
electrostatic field.
--vortex field (???)
Potential can not be introduced to describe
induced electric field.
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rltR
Choose a closed circle L passing point P
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use
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L
rgtR
?P
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12-5 Applications of induced electric field
1. Betatron ?????
A device that can accelerate electrons to high
nenrgies by induced electric field.
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When an electron enters the changing M-field,
,
Tangent acceleration
speed
when B(t) ?, v(t) ? .
( r keeps a constant. )
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2.Vortex Current ???
(1) formation
(2) application
Using the hot effect of current, vortex current
can be used on electromagnetic stove or induced
stove to produce heat energy.
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The energy loss of vortex current should be
decreased as less as possible on the transformer
systems.
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12-5 Self-induction Mutual-induction ?????
1. Self-induction
Induced emf
I
M-field
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--Self inductance of the circle. It is determined
by the size, the shape, the number of turns, and
the magnetic properties of the material around
the circuit. It has nothing with I.
The SI unit of L is Henry(H)??
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Self-induced emf when I is changing,
Self inductance
or
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Solution
Assume the solenoid carries current I,
then
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--the volume of the solenoid.
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ExampleFind L? for the unit length of coaxial
cable. (a,b,?0 are known)
Solution assume it carries I ,
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The self inductance for unit length cable is
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2. Mutual induction
I1
I2
2
1
Mutual induced emf
when the currents I1, I2 change,
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Circle 1
the mutual inductance of loop 1 relative to 2.
It is determined by the sizes, the shapes, the
number of turns, and the magnetic properties of
the material around the loops, the relative
position of the two loops. It has nothing with
I2.
The SI unit of M is Henry(H)
?
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Circle 2
the mutual inductance of loop 2 relative to 1.
?
It was proved that
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Calculate M
or
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ExampleA long and thin straight solenoid with
and two coils . Find their M?
Solution
N2
Assume the coil 1carries current I1,
S
then the M-field produced by I1 is
N1
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The number of magnetic flux linkages through the
coil 2 is
If the coils 2 carries current I2,M?
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12-7 Energy of the M-field
Turn on the switch K , the current in the circle
0? i(t),
At any time,
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When t??,
(constant)
During the current changes from 0 to ?,M-field
changes from 0 to ?.
The work done by the battery
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i.e. the part of work is transferred in the
energy of M-field.
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Energy of M-field is
The energy density of M-field (the density of
magnetic energy)is
For long, thin straight solenoid,
,
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The density of magnetic energy is
The energy of M-field is
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ExampleFind Wm? for the unit length of
coaxial cable. (I,a,b,?0 are known)
a
b
choose a section of the cable with h1,
I
I
on the region of rlta and rgtb ,
we have
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altrltb
We have known that