1. Monday(04/06): Intro. to Ch.31, Faraday's Law, Motional Emf, Lenz's

Law. HW: Read and Study pages 967-81, then solve problems  1, 5, 7, 13,

and 15 on pages 992-3.

2. Tuesday(04/07): Induced Emf's and Electric Fields, Generators and

Motors. HW: Read and Study pages 981-90, then solve problems 19, 21, 26,

29, and 35 on pages 994-96.

3. Wednesday(04/08): Intro. to Ch.32, Inductance, Self-Inductance, RL

circuits. HW: Read and Study pages 1003-11, then solve problems 3, 9, 15,

17, and 21 on page 1025-6.

4. Thursday(04/09): Energy in a Magnetic Field, Mutual Inductance.

HW: Read and Study pages 1011-20, then solve problems 29, 31, 33, 39,

and 41 on page 1027. LC Circuits and RLC Circuits. HW: Read and Study

pages 1020-23, then solve problems 47, 49, and 55 on page 1028.

5. Friday(04/10): No School. HW: Finish all assigned problems.

6. Monday(04/13): LAB of Efficiency of Electric Motors. HW: Process all

Lab Data. Lab report is due Wednesday.

7. Tuesday(04/14): Post-Lab Discussion. HW: Write lab report.

8. Wednesday(04/15): Review for Ch.31&32. HW: Complete Review

Handout.

9. Thursday(04/16): AP Exam Registration in Cafeteria. HW: Complete

Review Handout.

10. Friday(04/17): TEST on Chapters 31 and 32.  HW: Visit the web-site

for notes and plans for AP Exam Review.

Very Important: If you have any questions or miss a class, see

Website Notes for Ch.31: Faraday's Law.

Background: Our studies so far have been concerned with electric fields due to

stationary charges and magnetic fields produced by moving charges. This chapter

(31) deals with electric fields that originate from changing magnetic fields.

Experiments conducted by Michael Faraday in England in 1831 and independently

by Joseph Henry in the United States that same year showed that an electric current

could be induced in a circuit by a changing magnetic field.

The results of these experiments led to a basic and important law of

electromagnetism known as Faraday's Law of Induction. This law says that the

magnitude of the Emf induced in a circuit equals the time rate of change of the

magnetic flux through the circuit.

As we shall see, an induced Emf can be produced in several ways. For instance, an

induced Emf and an induced current can be produced in a closed loop of wire when

the wire moves into a magnetic field. We shall describe such experiments along

with a number of important applications that make use of the phenomenon of

electromagnetic induction. The Emf induced in a circuit is proportional to the time

rate of change of magnetic flux through the circuit.

(MAKE SURE YOU FILL-IN THE BLANKS IN THESE NOTES.)

1. An Emf can be induced in the circuit in several ways: (I) The magnitude of the

magnetic field can change as a function of time. (II) The area of the circuit can

change with time. (III) The direction of the magnetic field relative to the circuit

can change with time. (IV) Any combination of the above can change.

2. In particular, it is important to note that the magnitude of the induced Emf

depends on the rate at which the magnetic field is changing.

3. Motional Emf: A potential difference will be maintained across a conductor

moving in a magnetic field as long as the direction of motion through the field is

not parallel to the field direction. If the motion is reversed, the polarity of the

potential difference will also be reversed.

4. Lenz's Law: The polarity of the induced Emf is such that it tends to produce a

current that will create a magnetic flux to oppose the change in flux through the

circuit.

5. Maxwell's equations applied to free space are: (I) Gauss's Law __________

 ____________ which states that the total electric flux through any closed

surface equals the net charge inside that surface divided by epsilon naught.

This law describes how charge creates the electric field.

6. Gauss's Law for Magnetism ________________________which states that

the net magnetic flux through a closed surface is zero.

7. Faraday's Law of Induction _______________________which states that the

line integral of the electric field around any closed path equals the rate of change

of magnetic flux through any surface area bounded by the path. This law describes

how a changing magnetic field creates an electric field.

8. The Ampere-Maxwell Law ____________________________which states that

the line integral of the magnetic field around any closed path is determined by the

sum of the net conduction current through that path and the rate of change of

electric flux through any surface bounded by that path. This law describes how

changing electric fields create a magnetic field.

9. These four equations, together with the Lorentz Force Law ____________

 _______________ , describe all electromagnetic phenomena.

10. The total magnetic flux through a plane area, A, placed in a uniform magnetic

field depends on the angle between the direction of the magnetic field and the

direction perpendicular to the surface area.

12. Since the magnetic flux, Φ, is the product of the magnetic field, B, the area, A,

and the cosine of the angle between the magnetic field and the normal to the

surface, there are three possible ways the flux can change with time; the field, B,

or the area, A, or the angle θ.

13. When the magnetic field, B, changes with time, the Emf is expressed as:

Emf = dB/dt(Acos(θ)), where dB/dt is the rate of change of the magnetic field.

This kind of Emf is produced in transformers.

14. When the area, A, changes with time, the Emf is expressed as:

Emf = B(dA/dt)cos(θ), where dA/dt is the rate of change of the area.

15. If the area is a rectangle of length, L, and of width, W, and the side, L, is

moving with a velocity, v, in the plane of the area, then the dA/dt = L x v , the

cross-product, and the Emf is then given by: Emf = BLv, when v is perpendicular

to B. This kind of Emf is produced when wires move in magnetic fields.

18. When the angle, theta, changes with time, the Emf is expressed as:

Emf = BAd(cos(θ))/dt, where d(cos(θ))/dt is the rate of change of the cos function

as the angle theta changes.

19. If dθ/dt is the rate of change of the angle, in radians/sec, called omega, ω,

then the induced Emf is written as: Emf = ωBAsin(θ). This kind of Emf is produced

in electrical generators.

20. The negative sign in Faraday's law indicates the direction of the Emf, but it is

difficult to properly interpret this sign. A better way to find the direction of the

induced Emf is to use Lenz's law.

21. When we use Lenz's law, we endow the conducting loop with feelings; in fact,

we make the loop a violent reactionary.

22. Whenever the number of field lines through the loop is changed, the loop

screams "NO", and tries to keep the number of field lines (its flux) the same.

It does this by trying to drive current in the direction that produces a magnetic

field to oppose the change.

23. If the flux through the loop is decreasing, current flows in the loop in the

direction that replaces the lost field lines. If the flux is increasing, the current flow

makes magnetic field in the opposite direction to the increasing field.

WEBSITE NOTES: Ch.32: Inductance.

The study of inductance presents a very challenging but rewarding segment of

electricity. It is challenging in the sense that, at first, it will seem that new concepts

are being introduced. You will realize as this chapter progresses that these "new

concepts" are merely extensions and enlargements of fundamental principles that

you learned previously in the study of magnetism and electron physics. The study of

inductance is rewarding in the sense that a thorough understanding of it will enable

you to acquire a working knowledge of electrical circuits more rapidly.

1. The property of inductance might be described as "when any piece of wire is wound

into a coil form it forms an inductance which is the property of opposing any change in

current".

2. Alternatively it could be said "inductance is the property of a circuit by which energy

is stored in the form of an electromagnetic field".

3. We said a piece of wire wound into a coil form has the ability to produce a counter

emf (opposing current flow) and therefore has a value of inductance.

4. When the current in a coil changes with time, an Emf is induced in the coil according

to Faraday's Law. This self-induced Emf is given by the equation ε_L = -L dI/dt , with L

being the self inductance of the coil.

5. The standard value of inductance is the Henry (V·s/A), a large value which like the

Farad for capacitance is rarely encountered in electronics today.

6. Typical values of units encountered are milli-henries mH, one thousandth of a henry

or the micro-henry uH, one millionth of a henry.

7. A small straight piece of wire exhibits inductance (probably a fraction of a uH)

although not of any major significance until we reach UHF frequencies.

8. The inductance of any coil is L = NΦ_B/I , with   Φ_B the magnetic flux and N

the number of turns.

9. The inductance of an air-core solenoid is L = μ_oN²A/L , with A being the cross-

sectional area and L the length.

10. The value of an inductance varies in proportion to the number of turns squared.

If a coil was of one turn its value would be one unit.

11. Having two turns the value would be four units while three turns would produce

nine units although the length of the coil also enters into the equation.

12. Remember that inductance measures the electromagnetic induction of an electric

circuit component; it is a property of the component itself rather than of the circuit

as a whole.

13. The self-inductance, L, of a circuit component determines the magnitude of the

electromotive force (emf) induced in it as a result of a given voltage.

14. Faraday's law applied to an inductor states that a changing current induces a

back EMF that opposes the change. Or V = V_a-V_b = L(di/dt) . Where V is the voltage

across the inductor and L is the inductance measured in henry (H).

15. The inductance will tend to smooth sudden changes in current just as the

capacitance smoothes sudden changes in voltage. Of course, if the current is constant

there will be no induced EMF. So unlike the capacitor which behaves like an open-circuit

in DC circuits, an inductor behaves like a short-circuit in DC circuits.

16. Applications using inductors are less common than those using capacitors, but

inductors are very common in high frequency circuits. We will again skip over the

unpleasantness - that non-ideal inductors have some resistance and some capacitance.

17. Inductors are never pure inductances because there is always some resistance in and

some capacitance between the coil windings.

18. When choosing an inductor (occasionally called a choke) for a specific application, it

is necessary to consider the value of the inductance, the DC resistance of the coil, the

current-carrying capacity of the coil windings, the breakdown voltage between the coil

and the frame, and the frequency range in which the coil is designed to operate.

19. To obtain a very high inductance it is necessary to have a coil of many turns. The

inductance can be further increased by winding the coil on a closed-loop iron or ferrite

core. To obtain as pure an inductance as possible, the DC resistance of the windings

should be reduced to a minimum.

20. This can be done by increasing the wire size, which of course, increases the size

of the choke. The size of the wire also determines the current-handling capacity of the

choke since the work done in forcing a current through a resistance is converted to heat

in the resistance.

21. Magnetic losses in an iron core also account for some heating, and this heating

restricts any choke to a certain safe operating current. The windings of the coil must

be insulated from the frame as well as from each other.

22. Heavier insulation, which necessarily makes the choke more bulky, is used in

applications where there will be a high voltage between the frame and the winding.

The losses sustained in the iron core increases as the frequency increases.

23. Large inductors, rated in henries, are used principally in power applications.

The frequency in these circuits is relatively low, generally 60 Hz or low multiples

thereof. In high-frequency circuits, such as those found in FM radios and television

sets, very small inductors (of the order of micro-henries) are frequently used.

24. If a resistor and an inductor are connected in series to a battery of Emf, ε , and

a switch is closed at t = 0, the current in the circuit varies with time according to

the equation, I(t) = ε/R(1 - e^-t/τ ) .  Here τ , tau, is the time constant of the RL circuit.

25. If the battery is removed from the circuit, the current decays exponentially with time

according to the equation  I(t) = (ε/R)·e^-t/τ .

26. Don't forget, the initial current in the circuit is I_o, given by I_o = ε/R .

27. The energy stored in the magnetic field of an inductor carrying a current I is given

by U_B = ½LI² .

28. The energy per unit volume where the magnetic field is B is given by the equation 

u_B = B²/2μ_o .

29. In an LC circuit with zero resistance, the charge on the capacitor vary in time

according to Q = Q_maxcos(ωt + φ) . The current also varies with time given by

I = dQ/dt = -ωQ_maxsin(ωt + φ) . Here Q_max is the maximum charge on the capacitor,

φ (Phi) is the phase angle, and ω is the angular frequency of the oscillation.

30. It can be shown that  ω = 1/√(LC) .

31. The energy in an LC circuit continuously transfers energy stored in the capacitor

and energy stored in the inductor. The total energy of the LC circuit at any time t is,

U = U_c + U_L = (Q²_max/2C)cos²( ωt) + (LI²_max/2)sin²( ωt) .

32. At t = 0, all of the energy is stored in the electric field of the capacitor

U_c = Q²_max/2C . Eventually, all of the energy is transferred to the inductor

U_L = LI²_max/2 .  However, the total energy remains constant because the losses are

neglected in the ideal LC circuit.

33. The charge and current in an RLC circuit exhibit a damped harmonic behavior for

small values of R.  This is analogous to the damped harmonic motion of a mass-spring

system in which friction is present.

And still, we need these steps to solve any problem in Physics:

(i) read the problem and identify the given variables

(ii) determine what you are asked to solve for

(iii) find the correct equation to use

(iv) use Algebra, Trigonometry, and/or Calculus to isolate the unknown

(v) substitute-in the given information and simplify.

Check your homework.

THE AP PHYSICS C ARCHIVES Ch.1: Physics Intro. Ch.2: Linear Motion. Ch.3: Vectors. Ch.4: 2-Dim Motion. Ch.5&6: Newton's Laws. Ch.7&8: Work&Energy. Ch.9: Momentum. Ch.10&11: Rotary Motion. Ch.12: Elasticity. Ch.13: Gravitation. Ch.15: SHM. Mechanics Review. Ch.23: Electric Fields. Ch.24: Gauss's Law. Ch.25: Electric Potential. Ch.26: Capacitance. Ch.27: Current/Resistance. Ch.28: DC Circuits. Ch.29: Magnetic Effects. Ch.30: Magnetic Field Sources.