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Self-Inductance When the switch is closed, the current does not immediately reach its maximum value Faraday’s law can be used to describe the effect

Self-Inductance

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Self-Inductance. When the switch is closed, the current does not immediately reach its maximum value Faraday’s law can be used to describe the effect. Self-induced emf. A current in the coil produces a magnetic field directed toward the left (a) - PowerPoint PPT Presentation

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Page 1: Self-Inductance

Self-Inductance

• When the switch is closed, the current does not immediately reach its maximum value

• Faraday’s law can be used to describe the effect

Page 2: Self-Inductance

Self-induced emf

• A current in the coil produces a magnetic field directed toward the left (a)

• If the current increases, the increasing flux creates an induced emf of the polarity shown (b)

• The polarity of the induced emf reverses if the current decreases (c)

Page 3: Self-Inductance

Self Inductance

oBo

d nIAd dI NBA dIN N N nAdt dt dt I dt

oB nI

BN dI dILI dt dt

BNLI

Define: Self Inductance

Page 4: Self-Inductance

Inductance of a Solenoid

• The magnetic flux through each turn is

• Therefore, the inductance is

• This shows that L depends on the geometry of the object

IB oNBA μ A

2

IB oN μ N AL

Page 5: Self-Inductance

Inductance Units

BNLI

dILdt

VL s Henry HA / s

Page 6: Self-Inductance

3. A 2.00-H inductor carries a steady current of 0.500 A. When the switch in the circuit is opened, the current is effectively zero after 10.0 ms. What is the average induced emf in the inductor during this time?

5. A 10.0-mH inductor carries a current I = Imax sin ωt, with Imax = 5.00 A and ω/2π = 60.0 Hz. What is the back emf as a function of time?

7. An inductor in the form of a solenoid contains 420 turns, is 16.0 cm in length, and has a cross-sectional area of 3.00 cm2. What uniform rate of decrease of current through the inductor induces an emf of 175 μV?

Page 7: Self-Inductance

LR Circuits

Charging

Kirchhoff Loop Equation:

odIV RI L 0dt

Solution:

t /oVI 1 eR

LR

Page 8: Self-Inductance

LR Circuits

Discharging

Kirchhoff Loop Equation:

dIRI L 0dt

Solution:

t /oI I e

LR

Page 9: Self-Inductance

Active Figure 32.3

(SLIDESHOW MODE ONLY)

Page 10: Self-Inductance

14. Calculate the resistance in an RL circuit in which L = 2.50 H and the current increases to 90.0% of its final value in 3.00 s.

20. A 12.0-V battery is connected in series with a resistor and an inductor. The circuit has a time constant of 500 μs, and the maximum current is 200 mA. What is the value of the inductance?

24. A series RL circuit with L = 3.00 H and a series RC circuit with C = 3.00 μF have equal time constants. If the two circuits contain the same resistance R, (a) what is the value of R and (b) what is the time constant?

Page 11: Self-Inductance

Energy in a coil

dIP VI L Idt

21U LI2

PE in an Inductor

PE in an Capacitor21U CV

2

Page 12: Self-Inductance

Energy Density in a coil21U LI

2PE in an Inductor

oB N NI / AN NBALI I I

2

o 2

o o

N N A1 B 1U B A2 N 2

o

BIN

2

o

1u B2

2o

1u E2

Page 13: Self-Inductance

31. An air-core solenoid with 68 turns is 8.00 cm long and has a diameter of 1.20 cm. How much energy is stored in its magnetic field when it carries a current of 0.770 A?

33. On a clear day at a certain location, a 100-V/m vertical electric field exists near the Earth’s surface. At the same place, the Earth’s magnetic field has a magnitude of 0.500 × 10–4 T. Compute the energy densities of the two fields.

36. A 10.0-V battery, a 5.00-Ω resistor, and a 10.0-H inductor are connected in series. After the current in the circuit has reached its maximum value, calculate (a) the power being supplied by the battery, (b) the power being delivered to the resistor, (c) the power being delivered to the inductor, and (d) the energy stored in the magnetic field of the inductor.

Page 14: Self-Inductance

Example 32-5: The Coaxial Cable

• Calculate L for the cable• The total flux is

• Therefore, L is

• The total energy is

ln2 2

I Ibo o

B a

μ μ bB dA drπr π a

ln2I

B oμ bLπ a

221 ln

2 4II oμ bU Lπ a

Page 15: Self-Inductance

Mutual Inductance

2 12 1N I

2 12 12 1N M I

12 1

212 12 2 2 12

M IdNd dIN N M

dt dt dt

2

1 21dIMdt

Page 16: Self-Inductance

Mutual Inductance example

11 0 1

NB I

112 1 0 1

NB A I A

10 1

21 1 22 2 2 0

Nd I Ad N N dIN N A

dt dt dt

1 212 0

N NM A M

Page 17: Self-Inductance

LC Circuits

Q dIL 0C dt

2

2

d Q Q 0dt LC

Kirchhoff Loop Equation: Solution:

maxQ Q cos t

1LC

I t 0 0 maxQ(t 0) Q

Page 18: Self-Inductance

Energy in an LC circuit

22

2maxE

Q1 QU cos t2 C 2C

maxQ Q cos t 1LC

2 2 2

2 2 2max maxB

L Q Q1U LI sin t sin t2 2 2C

maxdQI Q sin tdt

2 2 2

2 2max max maxE B

Q Q QU U cos t sin t2C 2C 2C

Page 19: Self-Inductance

Active Figure 32.17

(SLIDESHOW MODE ONLY)

Page 20: Self-Inductance

LRC Circuits

Q dIRI L 0C dt

Kirchhoff Loop Equation:

Solution:

2

2

d Q dQ QL R 0dt dt C

tmax dQ Q e cos t

2

d 2

1 RLC 4L

R2L

Page 21: Self-Inductance

Damped RLC Circuit

• The maximum value of Q decreases after each oscillation– R < RC

• This is analogous to the amplitude of a damped spring-mass system

Page 22: Self-Inductance

Active Figure 32.21

(SLIDESHOW MODE ONLY)

Page 23: Self-Inductance

LRC Circuits

• Underdamped

• Critically Damped

• Overdamped

R t2L

oQ Q e cos ' t

2

2

1 R'LC 4L

2

2

1 RLC 4L

2

2

1 RLC 4L

2

2

1 RLC 4L

24L RC

24L RC

24L RC

Page 24: Self-Inductance
Page 25: Self-Inductance

41. An emf of 96.0 mV is induced in the windings of a coil when the current in a nearby coil is increasing at the rate of 1.20 A/s. What is the mutual inductance of the two coils?

49. A fixed inductance L = 1.05 μH is used in series with a variable capacitor in the tuning section of a radiotelephone on a ship. What capacitance tunes the circuit to the signal from a transmitter broadcasting at 6.30 MHz?

55. Consider an LC circuit in which L = 500 mH and C = 0.100 μF. (a) What is the resonance frequency ω0? (b) If a resistance of 1.00 kΩ is introduced into this circuit, what is the frequency of the (damped) oscillations? (c) What is the percent difference between the two frequencies?

Page 26: Self-Inductance

LC Demo

R = 10 C = 2.5 FL = 850 mH

1. Calculate period2. What if we change

C = 10 F3. Underdamped?4. How can we

change damping?