AC Circuits II Physics 2415 Lecture 23 Michael Fowler, UVa.

Slides:



Advertisements
Similar presentations
Oscillations in an LC Circuit
Advertisements

Chapter 32 Inductance.
Inductance Self-Inductance RL Circuits Energy in a Magnetic Field
Dale E. Gary Wenda Cao NJIT Physics Department
Copyright © 2012 Pearson Education Inc. PowerPoint ® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures.
Chapter 30.
Chapter 31B - Transient Currents and Inductance
Lecture 7 Circuits Ch. 27 Cartoon -Kirchhoff's Laws Topics –Direct Current Circuits –Kirchhoff's Two Rules –Analysis of Circuits Examples –Ammeter and.
The current through the inductor can be considered a sum of the current in the circuit and the induced current. The current in the circuit will be constant,
Physics 1402: Lecture 21 Today’s Agenda Announcements: –Induction, RL circuits Homework 06: due next MondayHomework 06: due next Monday Induction / AC.
Lesson 6 Capacitors and Capacitance
Physics 1502: Lecture 22 Today’s Agenda Announcements: –RL - RV - RLC circuits Homework 06: due next Wednesday …Homework 06: due next Wednesday … Induction.
Ben Gurion University of the Negev Week 9. Inductance – Self-inductance RL circuits Energy in a magnetic field mutual inductance.
Physics 4 Inductance Prepared by Vince Zaccone
AC Circuits III Physics 2415 Lecture 24 Michael Fowler, UVa.
Physics 2415 Lecture 22 Michael Fowler, UVa
Ch. 30 Inductance AP Physics. Mutual Inductance According to Faraday’s law, an emf is induced in a stationary circuit whenever the magnetic flux varies.
Physics 2102 Inductors, RL circuits, LC circuits Physics 2102 Gabriela González.
-Self Inductance -Inductance of a Solenoid -RL Circuit -Energy Stored in an Inductor AP Physics C Mrs. Coyle.
1 W12D2 RC, LR, and Undriven RLC Circuits; Experiment 4 Today’s Reading Course Notes: Sections , 11.10, ; Expt. 4: Undriven RLC Circuits.
Fall 2008Physics 231Lecture 10-1 Chapter 30 Inductance.
Inductance Self-Inductance A
Chapter 32 Inductance. Joseph Henry 1797 – 1878 American physicist First director of the Smithsonian Improved design of electromagnet Constructed one.
1 Faraday’s Law Chapter Ampere’s law Magnetic field is produced by time variation of electric field.
Inductance and AC Circuits. Mutual Inductance Self-Inductance Energy Stored in a Magnetic Field LR Circuits LC Circuits and Electromagnetic Oscillations.
© 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their.
Chapter 32 Inductance. Self-inductance  A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying.
Chapter 32 Inductance. Introduction In this chapter we will look at applications of induced currents, including: – Self Inductance of a circuit – Inductors.
Wednesday, Nov. 16, 2005PHYS , Fall 2005 Dr. Jaehoon Yu 1 PHYS 1444 – Section 003 Lecture #20 Wednesday, Nov. 16, 2005 Dr. Jaehoon Yu Self Inductance.
Copyright © 2009 Pearson Education, Inc. Chapter 33 Inductance, Electromagnetic Oscillations, and AC Circuits.
Thursday, Dec. 1, 2011PHYS , Fall 2011 Dr. Jaehoon Yu 1 PHYS 1444 – Section 003 Lecture #23 Thursday, Dec. 1, 2011 Dr. Jaehoon Yu LR circuit LC.
Self-Inductance, RL Circuits
Physics 2112 Unit 18 Today’s Concepts: A) Induction B) RL Circuits Electricity & Magnetism Lecture 18, Slide 1.
Lecture 18-1 Ways to Change Magnetic Flux Changing the magnitude of the field within a conducting loop (or coil). Changing the area of the loop (or coil)
Class 34 Today we will: learn about inductors and inductance
Chapter 32 Inductance L and the stored magnetic energy RL and LC circuits RLC circuit.
Exam review Inductors, EM oscillations
IV–3 Energy of Magnetic Field Main Topics Transformers Energy of Magnetic Field Energy Density of Magnetic Field An RC Circuit.
Physics 2 for Electrical Engineering Ben Gurion University of the Negev
Chapter 32 Inductance. Joseph Henry 1797 – 1878 American physicist First director of the Smithsonian Improved design of electromagnet Constructed one.
Chapter 32 Inductance. Self-inductance Some terminology first: Use emf and current when they are caused by batteries or other sources Use induced emf.
Today Course overview and information 09/16/2010 © 2010 NTUST.
Copyright © 2009 Pearson Education, Inc. Chapter 32: Inductance, Electromagnetic Oscillations, and AC Circuits.
L C LC Circuits 0 0 t V V C L t t U B U E Today... Oscillating voltage and current Transformers Qualitative descriptions: LC circuits (ideal inductor)
Inductance and AC Circuits. Mutual Inductance Self-Inductance Energy Stored in a Magnetic Field LR Circuits LC Circuits and Electromagnetic Oscillations.
My Chapter 20 Lecture Outline.
PHYSICS 222 EXAM 2 REVIEW SI LEADER: ROSALIE DUBBERKE.
Monday, Apr. 16, 2012PHYS , Spring 2012 Dr. Jaehoon Yu 1 PHYS 1444 – Section 004 Lecture #20 Monday, April 16, 2012 Dr. Jaehoon Yu Today’s homework.
Tuesday, April 26, PHYS Dr. Andrew Brandt PHYS 1444 – Section 02 Lecture #20 Tuesday April 26, 2011 Dr. Andrew Brandt AC Circuits Maxwell.
Lesson 10 Calculation of Inductance LR circuits
Monday, April 23, PHYS , Spring 2007 Dr. Andrew Brandt PHYS 1444 – Section 004 Lecture #19 Monday, April 23, 2007 Dr. Andrew Brandt Inductance.
Self Inductance and RL Circuits
Wednesday, Apr. 19, 2006PHYS , Spring 2006 Dr. Jaehoon Yu 1 PHYS 1444 – Section 501 Lecture #21 Wednesday, Apr. 19, 2006 Dr. Jaehoon Yu Energy.
Thursday August 2, PHYS 1444 Ian Howley PHYS 1444 Lecture #15 Thursday August 2, 2012 Ian Howley Dr. B will assign final (?) HW today(?) It is due.
Copyright © 2009 Pearson Education, Inc. Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits.
Last time Ampere's Law Faraday’s law 1. Faraday’s Law of Induction (More Quantitative) The magnitude of the induced EMF in conducting loop is equal to.
Physics 6B Inductors and AC circuits Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.
For vacuum and material with constant susceptibility M 21 is a constant and given by Inductance We know already: changing magnetic flux creates an emf.
Electromagnetic Induction
Mutual Inductance Mutual inductance: a changing current in one coil will induce a current in a second coil: And vice versa; note that the constant M, known.
Coils sharing the same magnetic flux, BA
Concept Questions with Answers 8.02 W12D2
Eddy Current A current induced in a solid conducting object, due to motion of the object in an external magnetic field. The presence of eddy current in.
Electromagnetic Induction
Induction -->Inductors
Electromagnetic Induction
PHYS 1444 – Section 04 Lecture #22
AC circuits Physics /27/2018 Lecture IX.
University Physics Chapter 14 INDUCTANCE.
Chapter 31B - Transient Currents and Inductance
Presentation transcript:

AC Circuits II Physics 2415 Lecture 23 Michael Fowler, UVa

Today’s Topics Review self and mutual induction LR Circuits LC Circuits

Definition of Self Inductance For any shape conductor, when the current changes there is an induced emf E opposing the change, and E is proportional to the rate of change of current. The self inductance L is defined by: and symbolized by: Unit: for E in volts, I in amps L is in henrys (H).

Mutual Inductance We’ve already met mutual inductance: when the current I 1 in coil 1 changes, it gives rise to an emf E 2 in coil 2. The mutual inductance M 21 is defined by: where is the magnetic flux through a single loop of coil 2 from current I 1 in coil 1.. Coil 1: N 1 loops Coil 1 Coil 2: N 2 loops Coil 2

Mutual Inductance Symmetry Suppose we have two coils close to each other. A changing current in coil 1 gives an emf in coil 2: Evidently we will also find: Remarkably, it turns out that M 12 = M 21 This is by no means obvious, and in fact quite difficult to prove.

Mutual Inductance and Self Inductance For a system of two coils, such as a transformer, the mutual inductance is written as M. Remember that for such a system, emf in one coil will be generated by changing currents in both coils:

Energy Stored in an Inductance If an increasing current I is flowing through an inductance L, the emf LdI / dt is opposing the current, so the source supplying the current is doing work at a rate ILdI / dt, so to raise the current from zero to I takes total work This energy is stored in the inductor exactly as is stored in a capacitor.

Energy is Stored in Fields When a capacitor is charged, an electric field is created. The capacitor’s energy is stored in the field with energy density. When a current flows through an inductor, a magnetic field is created. The inductor’s energy is stored in the field with energy density.

LR Circuits Suppose we have a steady current flowing from the battery through the LR circuit shown. Then at t = 0 we flip the switch… This just takes the battery out of the circuit.. R L I Switch V0V0

LR Circuits The decaying current generates an emf and this drives the current through the resistance: This is our old friend which has solution. R L I Switch V0V0

LR Circuits The equation has solution so the decay time:. 3L/R3L/R 2L/R2L/R L/R 0 I(t)I(t) t I0I0 0.37I 0 R A L I BC

LR Circuits continued… Suppose with no initial current we now reconnect to the battery. How fast does the current build up? Remember that now the inductance is opposing the battery:. R A L I(t)I(t) S V0V0 BC

LR Circuits continued… Suppose with no initial current we now reconnect to the battery. How fast does the current build up? Remember that now the inductance is opposing the battery:. R A L I(t)I(t) S V0V0 BC

LR Circuits continued… We must solve the equation or This differs from the earlier equation by having a constant term added on the right. It’s like which you can easily check has solution.. R A L I(t)I(t) S V0V0 BC

LR Circuits continued… We’re solving We know the solution to is, where A is a constant to be fixed by the initial conditions. Equating gives and A is fixed by the requirement that the current is zero initially, so.

LR Circuits continued… We’ve solved and found Initially the current is zero but changing rapidly—the inductance emf is equal and opposite to the battery.. 3L/R 2L/R L/R 0 I(t)I(t) V0/RV0/R R A L I(t)I(t) V0V0 BC

Clicker Question The switch S is closed…. R L S V0V0 R

Clicker Question The switch S is closed and current flows. The initial current, immediately after the switch is closed, is: A B C. R L I(t)I(t) S V0V0 R

Clicker Answer The switch S is closed and current flows. The initial current, immediately after the switch is closed, is: A B C. R L I(t)I(t) S V0V0 R The current through the inductance takes time to build up—it begins at zero. But the current through the other R starts immediately, so at t = 0 there is current around the lower loop only.

Clicker Question The switch S is closed and current flows. What is the current a long time later? A B C. R L I(t)I(t) S V0V0 R

Clicker Answer The switch S is closed and current flows. What is the current a long time later? A B C. R L I(t)I(t) S V0V0 R After the current has built up to a steady value, the inductance plays no further role as long as the current remains steady.

Clicker Question After this long time, the switch is suddenly opened! What are the currents immediately after the switch is opened? A round the upper loop B round the upper loop C all currents zero. R L S V0V0 R

Clicker Question After this long time, the switch is suddenly opened! What are the currents immediately after the switch is opened? A round the upper loop B round the upper loop C all currents zero. R L S V0V0 R

Clicker Answer After this long time, the switch is suddenly opened! What are the currents immediately after the switch is opened? A round the upper loop B round the upper loop C all currents zero. R L V0V0 R The inductance will not allow sudden discontinuous change in current, so the current through it will be the same just after opening the switch as it was before. This current must now go back via the other resistance.

Clicker Question The two circuits shown have the same inductance and the same t = 0 current, no battery, and resistances R and 2 R. In which circuit does the current decay more quickly? A. R B.2 R C.Both the same.

Clicker Answer The two circuits shown have the same inductance and the same t = 0 current, no battery, and resistances R and 2 R. In which circuit does the current decay more quickly? A. R B.2 R The decay is by heat production I 2 R..

LC Circuits Question Suppose at t = 0 the switch S is closed, and the resistance in this circuit is extremely small. What will happen? A.Current will flow until the capacitor discharges, after which nothing further will happen. B.Current will flow until the capacitor is fully charged the opposite way, then a reverse current will take it back to the original state, etc.. L Q0Q0 -Q 0 initial charge C S

LC Circuits Answer: B This is an oscillator! The emf V = Q / C from the capacitor builds up a current through the inductor, so when Q drops to zero there is substantial current. As this current decays, the inductor generates emf to keep it going—and with no resistance in the circuit, this is enough to fully charge the oscillator. We’ll check this out with equations.. L Q-Q C S I

LC Circuit Analysis The current. With no resistance, the voltage across the capacitor is exactly balanced by the emf from the inductance: From the two equations above,. L Q-Q C S I S in the diagram is the closed switch

Force of a Stretched Spring If a spring is pulled to extend beyond its natural length by a distance x, it will pull back with a force where k is called the “spring constant”. The same linear force is also generated when the spring is compressed. A Natural length Extension x Spring’s force Quick review of simple harmonic motion from Physics 1425…

Mass on a Spring Suppose we attach a mass m to the spring, free to slide backwards and forwards on the frictionless surface, then pull it out to x and let go. F = ma is: A Natural length m Extension x Spring’s force m frictionless Quick review of simple harmonic motion from Physics 1425…

Solving the Equation of Motion For a mass oscillating on the end of a spring, The most general solution is Here A is the amplitude,  is the phase, and by putting this x in the equation, mω 2 = k, or Just as for circular motion, the time for a complete cycle Quick review of simple harmonic motion from Physics 1425…

Back to the LC Circuit… The variation of charge with time is We’ve just seen that has solution from which. L Q-Q C S I

Where’s the Energy in the LC Circuit? The variation of charge with time is so the energy stored in the capacitor is The current is the charge flowing out so the energy stored in the inductor is. Compare this with the energy stored in the capacitor! L Q-Q C S I

Clicker Question Suppose an LC circuit has a very large capacitor but a small inductor (and no resistance). During the period of one oscillation, is the maximum energy stored in the inductor A.greater than B.less than C.equal to the maximum energy stored in the capacitor?

Clicker Answer Suppose an LC circuit has a very large capacitor but a small inductor (and no resistance). During the period of one oscillation, is the maximum energy stored in the inductor A.greater than B.less than C.equal to the maximum energy stored in the capacitor?

Energy in the LC Circuit We’ve found t he energy in the capacitor is The energy stored in the inductor is So the total energy is Total energy is of course constant: it is cyclically sloshed back and forth between the electric field and the magnetic field.. L Q-Q C S I

Energy in the LC Circuit Energy in the capacitor: electric field energy Energy in the inductor: magnetic field energy.