L111

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L11-1

• Inductance

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Inductance
The magnetic flux FB through a circuit is
proportional to the current.
The proportionality constant is called the
inductance, L L FB/I
The SI units of inductance are Wb/A H
(henry). L is also called self inductance.
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Using L FB/I along with Faradays law Vind -d
FB/dt, we have Vind -d(LI)/dt -L dI/dt.
The current-voltage relation for an inductor is V
- L dI/dt.
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Note differences with other elements
-
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Example Solenoid
What is the inductance of a solenoid?
The inductance of a solenoid is L µ0N 2A/ l
(N turns, A cross-section, l length)
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Example Solenoid
By definition L FB/I. For a tightly packed coil
with N turns, FB is N times the flux through one
turn.
The field inside a solenoid of length l and is B
µ0IN/l, so the flux through one turn is µ0INA/l
where A is the cross-sectional area of the
solenoid. Put it all together
The inductance of a solenoid is L µ0N 2A/l
(N turns, A cross-section, l length)
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inductor obeys ?V - L dI/dt.

• The current in a 90.0 mH inductor changes with
  time as
• I t 2 - 6.00t (in SI units).
• What is the magnitude of the induced emf
  (voltage) at time t ?

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The potential at a is higher than the potential
at b. Which of the following statements about the
inductor current I could be true?
1. I flows from a to b and is steady. 2. I
flows from a to b and is decreasing. 3. I flows
from b to a and is steady. 4. I flows from b to
a and is increasing. 5. I flows from a to b and
is increasing.
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We can vary the current through the circuits by
varying the resistance. What is the graph of ?B
through the lower loop as a function of the
current I flowing in the upper loop?
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We can vary the current through the circuit loop
below by varying the resistance. What is the
graph of ?B through the loop as a function of the
current I flowing in the loop itself?
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We can vary the current through a solenoid in the
circuit below by varying the resistance. What is
the graph of ?B through the solenoid as a
function of current I flowing in the solenoid?
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A resistor and an inductor are in a series
circuit together. At the instant that the switch
is closed, across which element is the potential
difference equal to that across the battery? 1.
The resistor 2. The inductor 3. Both equally
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A resistor and an inductor are in a series
circuit together. After a very long time after
the switch is closed, across which element is
the potential difference equal to that across
the battery? 1. The resistor 2. The inductor 3.
Both equally
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RL circuit
What happens to an inductor in a circuit?
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Result Charging
t L/R
-time constant
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Discharging
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The switch S below is initially unconnected. At
time tA the switch is connected to point A. What
is the equation from KLR for this circuit?
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The RL Circuit
Just as for an RC circuit, the voltage and
current in an RL circuit decay exponentially
towards their long-time values, ?I, ?V e-t/?
For an RC circuit, the time constant was RC
For an RL circuit, the time constant is ? L/R
Units ohm-farad sec henry/ohm
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• The current in a certain RL circuit decays to
  1 of its initial value in 100 ms. If the
  resistance in the circuit is doubled, the time
  for the current to decay to 1 will become
• 1. 25 ms
• 2. 50 ms
• 3. 200 ms
• 4. (100 ms)/ln(2)
• 5. (100 ms)ln(2)

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• Class Question. The current in a certain RL
  circuit decays to 1 of its initial value in 100
  ms. If the resistance in the circuit is doubled,
  the time for the current to decay to 1 will
  become
• A. 25 ms
• B. 50 ms
• C. 200 ms
• D. (100 ms)/ln(2)
• E. (100 ms)ln(2)

Answer (a) (Easy way) the time constant ? L/R
is halved, so the required time is also halved to
50 ms. (b) (Worked out in more detail) in both
cases e-t/? 1 0.01, so -t/? ln(0.01) ?t
-ln(0.01)L/R (4.61)L/R. Thus the time t is
inversely proportional to the resistance R.
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Rank in order, from largest to smallest, the time
constants of these three
circuits.
1. t1 gt t2 gt t3 2. t2 gt t1 gt t3 3. t2 gt t3 gt
t1 4. t3 gt t1 gt t2 5. t3 gt t2 gt t1
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Both resistors below are identical. Just after
the switch S is closed, _______ a long time
later, _______.
I1
I2
R
e
L
R

• 5) I1 gt I2 I1 I2
• 6) I1 lt I2 I1 I2
• 7) I1 I2 I1 gt I2
• 8) I1 I2 I1 lt I2

• I1 gt I2 I1 gt I2
• I1 gt I2 I1 lt I2
• I1 lt I2 I1 lt I2
• I1 lt I2 I1 gt I2

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LC circuits - new phenomena resonance - no
resistance here
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Results - oscillations
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Analogy -oscillations in springs
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An LC circuit oscillates with a frequency of
2000Hz. What will the frequency be if the
inductance is quadrupled. 1) 500 Hz 2) 1000
Hz 3) 2000 Hz 4) 4000 Hz 5) 8000 Hz
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Briefly LRC circuit
resistor dissipates energy -decaying oscillations
Formulas in book
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Energy Stored in an Inductor
The power flowing into an inductor is P IV I
(L dI/dt) d(LI 2/2)/dt
Given this power, what is the energy stored?

The energy stored in an inductor is U LI 2/2
Similar to formula for energy stored in a
capacitor, U CV 2/2.
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Energy Stored in an Inductor
The power flowing into an inductor is P IV I
(L dI/dt) d(LI 2/2)/dt
The total energy that flows into an inductor
going from current zero up to current I is U
?Pdt LI 2/2 so
The energy stored in an inductor is U LI 2/2
Similar to formula for energy stored in a
capacitor, U CV 2/2.
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Example Energy in LC circuit
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• An LC circuit oscillates at a frequency of 1.0
  kHz. At time t 0, the energy stored in the
  inductor is 1.0 J and the energy stored in the
  capacitor is zero. After one half of an
  oscillation cycle, how much energy is stored in
  the inductor?
• 1. 1.0 J
• 2. zero
• 3. -1.0 J
• 4. 0.707 J
• 5. 1.414 J

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Energy Density for the Magnetic Field
Energy density in the ELECTRIC field is u e0 E2
/2 What is the corresponding result for B field?
Consider a solenoid (N turns, length l , area
A).
The energy density (energy per unit volume) in
the magnetic field is u B2/2 µ 0. The stored
energy is U u(Vol) where Vol is the volume over
which the field B exists.
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Energy Density for the Magnetic Field
Consider a solenoid (N turns, length l, area A).
We know Field inside B µ0IN/l Volume over
which field exists v Al Inductance L µ0N 2A/l
Stored energy U LI 2/2 B2Al/2µ0
(B2/2µ0)v
The energy density (energy per unit volume) in
the magnetic field is u B2/2 µ0. The stored
energy is U uv where v is the volume over which
the field B exists.
Similar to energy density in the ELECTRIC field,
u ?0E 2/2
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energy density u B2/2 µ 0
solenoid B µ 0IN/l

• The linear dimensions of a solenoid (length
  and diameter) are doubled while the current and
  number of turns per meter remain constant. As a
  result the stored energy is multiplied by
• 1. 1/2
• 2. 1
• 3. 2
• 4. 4
• 5. 8

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Generators are explained by Faradays law
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Transformers
Key Physics Changing current in one loop creates
current in second nearby loop
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A transformer is an arrangement of two closely
spaced coils (often with an iron core) used to
change the voltage of electrical signals or power.
A transformer does not generate power(P IV) -
if the voltage goes up, the current must go down
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The math of transformers
Two solenoids
Induced voltage
the other current
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AC transformer
AC current input in primary
If all flux also runs through secondary
Then voltage change
No power added
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The primary coil of a transformer has 100 turns
and provides 120 V. If the output coil has 400
turns, what is the output voltage? 1) 30 V 2) 60
V 3) 120 V 4) 240 V 5) 480 V
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B-field throughout iron uniform, proportional to
current flowing around it
Iron core (electromagnet)
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An EMF has angular frequency w. The average
power dissipated in the inductor L and in the
light bulb of resistance R, respectively, is
R

• zero zero
• zero erms2 / R
• zero erms2 R / (R2w2L2)
• erms2 L erms2 R
• erms2 R / (R2w2L2) each

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The light bulb has a resistance R, and the emf
drives the circuit with a frequency w. The light
bulb glows most brightly at

• very low frequencies
• very high frequencies
• the frequency w 1/vLC
• the frequency w 1/LC

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If the current through the inductor is increasing
at a rate di/dt the magnitude and direction of
the induced EMF is
a
b

• Li to the left
• Li to the right
• Ldi/dt to the left
• Ldi/dt to the right
• 0 (the inductor prevents the current from
  increasing)

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A charge Dq flows from point a to point b while
the current is increasing at a rate di/dt. The
charges potential energy
b
a

• increases by i2R
• decreases by i2R
• increases by i2L
• decreases by i2L
• increases by Dq Ldi/dt
• decreases by Dq Ldi/dt

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i
a,
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Chapter 29, 30 Summary
Faradays Law of Induction is Vind e -d FB/dt.
Magnetic flux is computed as FB BA cos?
Faradays Law can also be expressed Eds d
FB/dt
Lenzs Law gives the direction of induced
current - opposing change
Faraday Lenz give rise to eddy currents drag
force on conductors moved in a magnetic field.
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The inductance of a circuit is L F B/I. The
SI unit of inductance is the henry (H) .
The inductance of a solenoid is L µ 0N2 A/ (N
turns, A cross-section, length)
The current-voltage relation for an inductor is V
L dI/dt.
For an RL circuit, the time constant is t
L/R. For an RL circuit with no battery and
current starting at I0 I I0e-t/t . For an RL
circuit with a battery and current starting at
zero I (VB/R)(1 e-t/t) .
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The energy stored in an inductor is U LI 2/2
The energy density in the magnetic field is u
B2/2 µ0. The stored energy is U u (Vol)
Transformers