chapter 30 inductance - physics and astronomy at...
TRANSCRIPT
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CHAPTER 30
INDUCTANCE
BASIC CONCEPTS
Mutual Inductance
Self Inductance
Magnetic Field Energy
Circuits with Inductors
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Mutual Inductance
I1 #1
If I1 changes with time t
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Put second coil at P
I1 #1 #2
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If I1 changes with time
For example
Then
And
B
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Or
And
I1 #1 #2
I2
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be the flux through
coil #2 due to B of coil #1.
Let be the number of turns
of coil #2.
Then depends on the
magnitude of B.
Magnitude of B depends on
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Therefore
depends on
In fact
Call constant of proportionality
Then
is the Mutual Inductance
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Solve for
Faraday’s Law
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is a constant depending only on
geometry‐ position, size and shape.
By Symmetry
And
is Mutual Inductance.
Units are Henry (H).
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Example
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From Chapter 28
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Only depends on geometry, number of
turns, area and length.
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Self Inductance
If the current is changing then there will be
a changing magnetic flux through the coil.
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A changing magnetic flux will induce an
emf.
Therefore we define in analogy to the
mutual inductance
This is Self‐Inductance
Solve for
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Faraday’s Law
Self‐Induced emf
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What is the inductance, , of the black coil?
B
Area
A
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Magnetic Field Energy
Consider the coil
Current starts at zero and increases to .
Or
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And
Power is work per unit time.
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How much work to bring current through
inductor from zero to ?
Integrate.
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Apply to solenoid
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So
Put into equation for
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And
So
Energy stored in magnetic field .
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R‐L Circuit
Consider this circuit
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First consider that switch is closed and
calculate how the current will increase with
time.
Second, then while there is a current in the
circuit consider opening and closing
and calculate how the current will decay.
Close
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At time close switch.
Kirchkoff’s loop starting at the switch and
going ccw.
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Integrate
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Raise to the power of each side
Solve for i
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Let to get
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The L‐C Circuit
The capacitor has been charged to an initial
charge and then placed across the
inductor.
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Kirchhoff’s Law
A solution of this differential equation is
Where is the initial charge on C.
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Differentiate to get current
Remember energy in
Capacitor
And
Inductor
Then study
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