physics 4 inductance prepared by vince zaccone for campus learning assistance services at ucsb
TRANSCRIPT
Physics 4
Inductance
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Inductance
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Mutual Inductance of two coils:
Some of the magnetic flux through one coil also passes through the other coil, inducing a voltage.
Inductance is magnetic flux/current.
Self-Inductance
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Changing current through the wires in a coil will induce a voltage that opposes the CHANGE in the current.
Magnetic Field Energy
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When an inductor has a steady current, it stores potential energy.
This leads to a general formula for potential energy stored in any magnetic field:
π’= π΅2
2π0
This formula is for magnetic energy density,
Which is energy per unit volume.
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Example: A solenoid 25.0cm long and with a cross-sectional area of 0.500cm2 contains 400 turns of wire and carries a current of 80.0A.
Calculate:
(a) the magnetic field in the solenoid.
(b) the energy density in the magnetic field if the solenoid is air-filled.
(c) the total energy contained in the coilβs magnetic field (assume uniform field).
(d) The inductance of the solenoid.
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Example: A solenoid 25.0cm long and with a cross-sectional area of 0.500cm2 contains 400 turns of wire and carries a current of 80.0A.
Calculate:
(a) the magnetic field in the solenoid.
(b) the energy density in the magnetic field if the solenoid is air-filled.
(c) the total energy contained in the coilβs magnetic field (assume uniform field).
(d) The inductance of the solenoid.
Solution:
π΅π πππππππ=π0ππΌ=(4π β10β7 )( 400π‘π’πππ 0.25π ) (80 π΄ )=0.16π
π’= π΅2
2π0=ΒΏΒΏ ΒΏ
ππππ=π’β (π£πππ’ππ )=(10,294 π½π3 ) (0.25π ) (0.5ππ2 )( 1π
100ππ )2
=0.129 π½
For part d) we can use the formula for energy in an inductor:
π=12πΏπΌ 2βπΏ=2π
πΌ 2=2(0.129 π½)
(80 π΄)2=4 β10β5π»=400ππ»
R-L Circuit
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When connected in a circuit with a resistor, an inductor will have the effect of slowing down changes in the current through the resistor.
When the current is steady (the switch has been closed for a long time), the inductor has no effect, but there is potential energy stored in the inductor.
R-L Circuit
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If switch S1 is closed in the circuit, current will begin to flow through the resistor and inductor as shown. This increasing current will induce current to flow the opposite direction, slowing the growth of the current.
We can write down a formula for the current as a function of time:
π (π‘ )= ππ
(1βπβ( π πΏ )π‘ )
The quantity is called the βtime constantβ for this exponential decay.
R-L Circuit
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Once the current reaches a steady value we can flip the switches, opening S1 and closing S2. Then current will keep flowing for while as the inductor opposes this decreasing current.
A similar formula describes this decaying current as a function of time:
π (π‘ )=πΌ 0(πβ( π πΏ )π‘)
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Example: A 35.0V battery with negligible internal resistance, a 50.0Ξ© resistor and a 1.25mH inductor are connected in series with an open switch. The switch is suddenly closed.
(a) How long after closing the switch will the current through the inductor reach half of its maximum value?
(b) How long after closing the switch will the energy stored in the inductor reach half its maximum value?
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Example: A 35.0V battery with negligible internal resistance, a 50.0Ξ© resistor and a 1.25mH inductor are connected in series with an open switch. The switch is suddenly closed.
(a) How long after closing the switch will the current through the inductor reach half of its maximum value?
(b) How long after closing the switch will the energy stored in the inductor reach half its maximum value?
π (π‘ )= ππ
(1βπβ( π πΏ )π‘ )
As soon as the switch is closed, current begins to flow around the circuit, increasing toward a maximum value given by Ohmβs Law. Here is the formula:
We want to find the time when the current is half of the maximum.
π (π‘ )=12 ( ππ )= π
π (1βπβ(π πΏ )π‘ )β 1
2=(1βπβ(π πΏ )π‘)βπβ(π πΏ )π‘
=12
β(π πΏ )π‘=ln( 12 )βπ‘=β( πΏπ )β ln( 12 )=β( 1.25β10β 3π»
50Ξ© ) β ln( 12 )=1.73 β10β5 π =17.3ππ For part b) we want the energy to be half of its maximum, so use the energy formula:
π=12πΏπ2=1
2 ( 12 πΏπΌ 2)β π(π‘)= πΌβ2
Using the formula for current again: 1
β2=(1βπβ( π πΏ )π‘ )β π‘=30.7ππ
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L-C CircuitA circuit containing a capacitor and an inductor will exhibit an oscillating current, with potential energy transferring back and forth.
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L-C CircuitThe oscillations in an L-C circuit should look familiar. This situation is directly analogous to an undamped mass-spring system that we saw previously.
All of the formulas we developed for that case are repeated here, with charge, q, taking the place of displacement, x. The capacitor is related to the spring constant, and the inductance is like mass.
To add in the damping, we just include a resistor in the circuitβ¦
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Example: In an L-C circuit, L=85.0mH and C=3.20ΞΌF. During the oscillations the maximum current in the inductor is 0.850mA.
(a) What is the maximum charge on the capacitor?
(b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500mA?
(c) How long does it take for the capacitor to go from maximum charge to zero charge?
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Example: In an L-C circuit, L=85.0mH and C=3.20ΞΌF. During the oscillations the maximum current in the inductor is 0.850mA.
(a) What is the maximum charge on the capacitor?
(b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500mA?
(c) How long does it take for the capacitor to go from maximum charge to zero charge?
(a) What is the maximum charge on the capacitor?
We can use energy for this if we want to. When all the energy is in the inductor it will have maximum current. When all the energy is in the capacitor it will have maximum charge.
ππππ=12πΏπΌ 2=
12(85 β10β3π» )ΒΏ
πππππ=12π2
πΆ=0.307ππ½βπ=44.3ππΆ
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Example: In an L-C circuit, L=85.0mH and C=3.20ΞΌF. During the oscillations the maximum current in the inductor is 0.850mA.
(a) What is the maximum charge on the capacitor?
(b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500mA?
(c) How long does it take for the capacitor to go from maximum charge to zero charge?
b) We can use energy again, or we can use the formula for the charge as a function of time.
πΈπ‘ππ‘ππ=12πΏ π2+ 1
2π2
πΆ
Total energy can be found from max current or max charge. Should be the same either way. We can solve for the charge when the current is as given:
πΈπ‘ππ‘ππ=3.07 β10β8 π½=1
2(85 β10β 3π» ) (0.5 β10β 3π΄ )2+ 1
2π2
3.2 β10β6πΆβπ=3.58 β10β7πΆ=358ππΆ
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Example: In an L-C circuit, L=85.0mH and C=3.20ΞΌF. During the oscillations the maximum current in the inductor is 0.850mA.
(a) What is the maximum charge on the capacitor?
(b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500mA?
(c) How long does it take for the capacitor to go from maximum charge to zero charge?
To go from no charge to fully charged is a quarter of a cycle, so we need to find the period of the oscillation. We have a formula for angular frequency:
π=β 1πΏπΆ
=1917ππππ
Rearrange this to get the period, then divide by 4:
π=2ππ
=0.00328π β14π=0.82ππ
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L-R-C Series Circuit
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L-R-C Series CircuitFormulas for this case are developed in the same way as the L-C circuit, we just include an extra term involving resistance:
π2πππ‘2
+ π πΏππππ‘
+ 1πΏπΆ
π=0
Solving this differential equation gives a general solution:
π=π΄πβ( π 2πΏ )π‘
πππ (β 1πΏπΆ
βπ 2
4πΏ2π‘+π)
This solution is for the underdamped case: R2<4L/C
The angular frequency in this case is:
π β²=β 1πΏπΆ
βπ 2
4πΏ2Notice this is less than the frequency in the undamped L-C circuit β the resistor slows down the oscillations.
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Example: Assume the switch has been in the position shown in the figure for a long time (so the capacitor is fully charged and no current is flowing). When the switch is moved (to connect points a and d in the figure), find the following:
a) The initial charge on the capacitor, and initial total energy in this system.
b) The frequency of oscillation for this circuit.
c) The maximum current through the inductor, and the time when that current is first achieved.
Assume the following values: Ξ΅=10.0V, R=1kΞ©, C=1ΞΌF, L=2H.
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a) At the beginning, the capacitor is fully charged, so the voltage matches the battery.
π=πΆ βπ=(1ππΉ ) (10π )=10ππΆ
πππππ=12πΆπ 2=
12
(1ππΉ ) (10π )2=50π π½
Example: Assume the switch has been in the position shown in the figure for a long time (so the capacitor is fully charged and no current is flowing). When the switch is moved (to connect points a and d in the figure), find the following:
a) The initial charge on the capacitor, and initial total energy in this system.
b) The frequency of oscillation for this circuit.
c) The maximum current through the inductor, and the time when that current is first achieved.
Assume the following values: Ξ΅=10.0V, R=1kΞ©, C=1ΞΌF, L=2H.
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a) At the beginning, the capacitor is fully charged, so the voltage matches the battery.
π=πΆ βπ=(1ππΉ ) (10π )=10ππΆ
πππππ=12πΆπ 2=
12
(1ππΉ ) (10π )2=50π π½
b) Frequency for an underdamped system:
π β²=β 1πΏπΆ
βπ 2
4πΏ2=β 1
(2π» )(10β 6πΉ )β
(1000Ξ©)2
4 (2π» )2=661 πππ
π
Example: Assume the switch has been in the position shown in the figure for a long time (so the capacitor is fully charged and no current is flowing). When the switch is moved (to connect points a and d in the figure), find the following:
a) The initial charge on the capacitor, and initial total energy in this system.
b) The frequency of oscillation for this circuit.
c) The maximum current through the inductor, and the time when that current is first achieved.
Assume the following values: Ξ΅=10.0V, R=1kΞ©, C=1ΞΌF, L=2H.
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c) Maximum current in inductor happens when the capacitor discharges β this is ΒΌ cycle.
π=ππβ( π 2πΏ )π‘
πππ (π β² π‘ )
π β²=661ππππ
=2ππβπ=0.0095π =9.5ππ
Imax at t= ΒΌ (9.5ms)=2.4ms
We will need to put this time into the formula for current, which is the derivative of the formula for charge on the capacitor.
π=β( π 2πΏ )ππβ( π 2πΏ )π‘πππ (πβ²π‘ )βπ β²ππ
β( π 2πΏ )π‘π ππ (πβ² π‘ )
π (2.4ππ )=β(661 ππππ ) (10ππΆ )πβ(1000Ξ©2 (2π» ) ) (2.4ππ )
=3.63ππ΄
Example: Assume the switch has been in the position shown in the figure for a long time (so the capacitor is fully charged and no current is flowing). When the switch is moved (to connect points a and d in the figure), find the following:
a) The initial charge on the capacitor, and initial total energy in this system.
b) The frequency of oscillation for this circuit.
c) The maximum current through the inductor, and the time when that current is first achieved.
Assume the following values: Ξ΅=10.0V, R=1kΞ©, C=1ΞΌF, L=2H.