n̂

A

E

E

General electric flux definition

E ˆE n A E A cos

• The field is not uniform

• The surface is not perpendicular to the field

AIf the surface is made up of a mosaic of N little surfaces

N

E i i ii 1

ˆE n A

E ˆE ndA E dA

Magnetic Flux

B ˆB ndA B dA

B B A

Whenever possible:

Units: [Weber] = [Wb]=[T-m2]

Gauss’s Law

Gauss asserts that the calculation for the flux through a closed surface from a point charge is true for any charge distribution!!!

enclosedE enclosed

o

qE dA 4 kq

This is true so long as Q is the charge enclosed by the surface of integration.

Gauss’s Law for magnetism

B B dA 0

This is true because we cannot isolate a magnetic pole and because magnetic field lines are continuous. The net number of field lines passing through any surface is always zero!

Faraday’s Law

When the magnet moves, a current is induced as if there was a source of emf (like a battery) in the circuit!

Active Figure 31.1

(SLIDESHOW MODE ONLY)

Faraday’s Law

Active Figure 31.2

(SLIDESHOW MODE ONLY)

Faraday’s Law of Induction

• An induced emf is produced by a changing magnetic field.

• Lenz’s Law – An induced emf is always in a direction that opposes the original change in the flux that caused the emf.

Bd

dtemf

Units: [Volts]

How can we change the flux?

• Change flux by:– Change area

– Change angle

– Change field

Bd

dtemf

B ˆB ndA B dA

30. In Figure P31.30, the bar magnet is moved toward the loop. Is Va – Vb positive, negative, or zero? Explain.

Figure P31.30

Faraday’s Law – continued

• Remember B is the magnetic flux through the circuit and is found by

• If the circuit consists of N loops, all of the same area, and if B is the flux through one loop, an emf is induced in every loop and Faraday’s law becomes

B d B A

Bdε N

dt

Decaying uniform magnetic field P31.4

• Change flux by:– Change field

– Change area

– Change angle

B

R

Assume we can change field:

t /oB B e

t /B oB Ae

A

t /oB

d B Aed

dt dtemf

t /oB Ae

t /oB A

I eR R

Applications of Faraday’s Law – Pickup Coil

• The pickup coil of an electric guitar uses Faraday’s law

• The coil is placed near the vibrating string and causes a portion of the string to become magnetized

• When the string vibrates at the same frequency, the magnetized segment produces a changing flux through the coil

• The induced emf is fed to an amplifier

3. A 25-turn circular coil of wire has diameter 1.00 m. It is placed with its axis along the direction of the Earth’s magnetic field of 50.0 μT, and then in 0.200 s it is flipped 180°. An average emf of what magnitude is generated in the coil?

6. A magnetic field of 0.200 T exists within a solenoid of 500 turns and a diameter of 10.0 cm. How rapidly (that is, within what period of time) must the field be reduced to zero, if the average induced emf within the coil during this time interval is to be 10.0 kV?

1. A 50-turn rectangular coil of dimensions 5.00 cm × 10.0 cm is allowed to fall from a position where B = 0 to a new position where B = 0.500 T and is the magnetic field directed perpendicular to the plane of the coil. Calculate the magnitude of the average emf that is induced in the coil if the displacement occurs in 0.250 s.

Linear Generator

Charges stop moving when:

E BF F

qE qvB

VvB

V vB

Linear Generator with Faraday’s Law Area x

B B dA B A B x

Bd dB x

dt dt

dxB B v

dt

By Lenz’s Law, what is the direction of current?B v

IR

Active Figure 31.10

(SLIDESHOW MODE ONLY)

Power moving the bar

bar

B vF I B B

R

P F v

2 2 2B v B vP B v

R R

2 2 2 2 2 22

2

B v B vP I R R

R R

Same result!

Breaking effect if power not added

2 2

bar

B v dvF I B m

R dt

2 2dv Bv 0

dt mR

2 2Bt

mRov v e

12. A 30-turn circular coil of radius 4.00 cm and resistance 1.00 Ω is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression B = 0.010 0t + 0.040 0t2, where t is in seconds and B is in tesla. Calculate the induced emf in the coil at t = 5.00 s.

19. An automobile has a vertical radio antenna 1.20 m long. The automobile travels at 65.0 km/h on a horizontal road where the Earth’s magnetic field is 50.0 μT directed toward the north and downward at an angle of 65.0° below the horizontal. (a) Specify the direction that the automobile should move in order to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom. (b) Calculate the magnitude of this induced emf.

22. A conducting rod of length ℓ moves on two horizontal, frictionless rails, as shown in Figure P31.20. If a constant force of 1.00 N moves the bar at 2.00 m/s through a magnetic field B that is directed into the page, (a) what is the current through the 8.00-Ω resistor R? (b) What is the rate at which energy is delivered to the resistor? (c) What is the mechanical power delivered by the force Fapp?

Rotating Generators and Faraday’s Law

B B dA B A B A cos

o t 0

B B A cos t

Bd dB A cos t

dt dt

B A sin t

N B A sin t

For N loops of wire

Induced emf in a Rotating Loop

• The induced emf in the loop is

• This is sinusoidal, with max = NAB

sin

Bdε N

dtNABω ωt

Active Figure 31.21

(SLIDESHOW MODE ONLY)

Rotating Generators

DC Generators

• The DC (direct current) generator has essentially the same components as the AC generator

• The main difference is that the contacts to the rotating loop are made using a split ring called a commutator

Active Figure 31.23

(SLIDESHOW MODE ONLY)

32. For the situation shown in Figure P31.32, the magnetic field changes with time according to the expression B = (2.00t3 – 4.00t2 + 0.800)T, and r2 = 2R = 5.00 cm. (a) Calculate the magnitude and direction of the force exerted on an electron located at point P2 when t =

2.00 s. (b) At what time is this force equal to zero?

45. A proton moves through a uniform electric field E = 50.0 j V/m and a uniform magnetic field B = (0.200i + 0.300j + 0.400k)T. Determine the acceleration of the proton when it has a velocity v = 200 i m/s.

59. A circular loop of wire of radius r is in a uniform magnetic field, with the plane of the loop perpendicular to the direction of the field (Fig. P31.59). The magnetic field varies with time according to B(t) = a + bt, where a and b are constants. (a) Calculate the magnetic flux through the loop at t = 0. (b) Calculate the emf induced in the loop. (c) If the resistance of the loop is R, what is the induced current? (d) At what rate is energy being delivered to the resistance of the loop?

Eddy Currents

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)

• 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)

Self Inductance

oBo

d nIAd dI NBA dIN N N nA

dt dt dt I dt

oB nI

BN dI dIL

I dt dt

BNL

I

Define: 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 o

NBA μ A

2

IB oN μ N A

L

Inductance Units

BNL

I

dIL

dt

VL s Henry H

A / s

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?

LR Circuits

Charging

Kirchhoff Loop Equation:

o

dIV RI L 0

dt

Solution:

t /oVI 1 e

R

L

R

LR Circuits

Discharging

Kirchhoff Loop Equation:

dIRI L 0

dt

Solution:

t /oI I e

L

R

Active Figure 32.3

(SLIDESHOW MODE ONLY)

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?

Energy in a coil

dIP VI L I

dt

21U LI

2PE in an Inductor

PE in an Capacitor21

U CV2

Energy Density in a coil

21U LI

2PE in an Inductor

oBN NI / AN NBA

LI I I

2

o 2

o o

N N A1 B 1U B A

2 N 2

o

BI

N

2

o

1u B

2

2

o

1u E

2

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.

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 dr

πr π a

ln2I

B oμ bL

π a

221

ln2 4

II oμ b

U Lπ a

Mutual Inductance

2 12 1N I

2 12 12 1N M I

12 1

212 12 2 2 12

M Id

Nd dIN N M

dt dt dt

2

1 21

dIM

dt

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 A

d N N dIN N A

dt dt dt

1 212 0

N NM A M

Induced emf and Electric Fields

• An electric field is created in the conductor as a result of the changing magnetic flux

• Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space

• This induced electric field is nonconservative– Unlike the electric field produced by stationary charges

• The emf for any closed path can be expressed as the line integral of E.ds over the path

General form of Faraday’s Lawb

b aba b a

a

U UV V V E ds

q

So the electromotive force around a closed path is:

E d

And Faraday’s Law becomes:

BdE d

dt

A changing magnetic flux produces an electric field.

This electric field is necessarily non-conservative.

E produced by changing B

BdE d

dt

2d B

E 2 r rdt

d BrE

2 dt

How about outside ro ?

Maxwell’s Equations

Gauss's law electric

0 Gauss's law in magnetism

Faraday's law

Ampere-Maxwell lawI

oS

S

B

Eo o o

qd

ε

d

dd

dtd

d μ ε μdt

E A

B A

E s

B s

•The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism•Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes

• Gauss’s law (electrical):• The total electric flux through any

closed surface equals the net charge inside that surface divided by o

• This relates an electric field to the charge distribution that creates it

• Gauss’s law (magnetism): • The total magnetic flux through any

closed surface is zero• This says the number of field lines

that enter a closed volume must equal the number that leave that volume

• This implies the magnetic field lines cannot begin or end at any point

• Isolated magnetic monopoles have not been observed in nature

oS

qd

ε E A

0S

d B A

• Faraday’s law of Induction:• This describes the creation of an electric field by a

changing magnetic flux• The law states that the emf, which is the line

integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path

• One consequence is the current induced in a conducting loop placed in a time-varying B

• The Ampere-Maxwell law is a generalization of Ampere’s law

• It describes the creation of a magnetic field by an electric field and electric currents

• The line integral of the magnetic field around any closed path is the given sum

Bdd

dt

E s

I Eo o o

dd μ ε μ

dt

B s

The Lorentz Force Law

• Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be calculated

• F = qE + qv x B• This relationship is called the Lorentz force law• Maxwell’s equations, together with this force law,

completely describe all classical electromagnetic interactions

Voltage transformers

Bs s

dV N

dt

Psolenoid o P

NI A

BP P

dV N

dt

SB P

P S

Vd V

dt N N

S PN N step up transformer

P SN N step down transformer

Current transformers

SS P

P

NV V

N

Primary SecondaryP P

P P S SV I V I

P P S SN I N I

PI

SI

Example: transformers

PI

SI

p

p

s

V 110V

N 916

N 100

What is Vs ?