lecture powerpoints chapter 21 physics: principles with...

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Lecture PowerPoints

Chapter 21 Physics: Principles with Applications, 7th edition

Giancoli

Chapter 21 Electromagnetic Induction and

Faraday’s Law


Contents of Chapter 21

•  Induced EMF

•  Faraday’s Law of Induction; Lenz’s Law

•  EMF Induced in a Moving Conductor

•  Changing Magnetic Flux Produces an Electric Field

•  Electric Generators



•  Back EMF and Counter Torque; Eddy Currents

•  Transformers and Transmission of Power

•  Information Storage: Magnetic and Semiconductor; Tape, Hard Drive, RAM

•  Applications of Induction: Microphone, Seismograph, GFCI

•  Inductance



•  Energy Stored in a Magnetic Field

•  LR Circuit

•  AC Circuits and Reactance

•  LRC Series AC Circuit

•  Resonance in AC Circuits


21-1 Induced EMF

Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:


21-1 Induced EMF

He found no evidence when the current through the left-hand loop was steady, but did see a current induced in the right-hand loop when the switch was turned on or off.


21-1 Induced EMF In addition, a current will be induced in a wire loop if a magnet is moved through the loop, but not when the magnet is held steady. Changing magnetic field can produce an electric current.


21-1 Induced EMF

Therefore, a changing magnetic field induces an emf. This phenomenon is called an electromagnetic induction.

Faraday’s experiment used a magnetic field that was changing because the current producing it was changing; the previous graphic shows a magnetic field that is changing because the magnet is moving.

Which factors affect the magnitude of the induced emf?

1)  The more rapidly the magnetic field is changing, the greater is induced current;

2)  The induced emf depends on the area of the circuit loop


21-2 Faraday’s Law of Induction; Lenz’s Law

The induced emf in a wire loop is proportional to the rate of change of magnetic flux through the loop.

Magnetic flux:

Unit of magnetic flux: weber, Wb: 1 Wb = 1 T·m2


(21-1)


The magnetic flux is analogous to the electric flux—it is proportional to the total number of lines passing through the loop.



Faraday’s law of induction:

If the circuit contains N loops that are closely wrapped so that the same flux passes through each, then


(21-2a)

(21-2b)


The minus sign gives the direction of the induced emf:

Lenz law: A current produced by an induced emf moves in a direction so that the magnetic field it produces tends to restore the changed field.

An induced emf is always in a direction that opposes the original change in flux that caused it.


Question 1 •  A coil lies flat on a level tabletop in a region where

the magnetic field vector points straight up. The magnetic field suddenly grows stronger. When viewed from above, what is the direction of the induced current in this coil as the field increases?

A) counterclockwise B) clockwise C) clockwise initially, then counterclockwise before

stopping D) There is no induced current in this coil.



Magnetic flux will change if the area of the loop changes:



Magnetic flux will change if the angle between the loop and the field changes:



Problem Solving: Lenz’s Law

1.  Determine whether the magnetic flux is increasing, decreasing, or unchanged.

2.  The magnetic field due to the induced current points in the opposite direction to the original field if the flux is increasing; in the same direction if it is decreasing; and is zero if the flux is not changing.

3.  Use the right-hand rule to determine the direction of the current.

4.  Remember that the external field and the field due to the induced current are different.


Example 1

•  A 2.00-m long metal wire is formed into a square and placed in the horizontal xy-plane. A uniform magnetic field is oriented at 30° above the horizontal with a strength of 0.344 T. What is the magnetic flux through the square due to this field?

•  Answer: 0.0430 T · m2

•  Solution:


€

Φ = BAcosθ; θ = 60 ; A = a2;a = l / 4 = 2m / 4 = 0.5m;

Φ = 0.344T⋅ 0.5m( )2 ⋅ cos60 = 0.043 T⋅ m2

30o

θ=90o-30o

Example 2 •  As shown in the figure, a wire and a 10-Ω resistor are used to

form a circuit in the shape of a square with dimensions 20 cm by 20 cm. A uniform but non-steady magnetic field is directed into the plane of the circuit. The magnitude of the magnetic field is steadily decreased from 2.70 T to 0.90 T in a time interval of 96 ms. What is the induced current in the circuit, and what is its direction through the resistor?

•  Answer: 75 mA, from b to a •  Solution:


€

I =ER; E = −N ΔΦ

Δt; Φ = B⋅ Acosθ

ΔΦ = Bf Acosθ − BiAcosθ = ΔB⋅ Acosθ

I = −N ΔB⋅ AcosθR⋅ Δt

= −1 0.9 − 2.7( )⋅ 0.22

10⋅ 96⋅ 10−3= 0.075 A

Where do we use the electromagnetic induction?

•  Everywhere! –  For example, at home – induction stove. The ac current

sets up a changing magnetic field, that passed through the pan bottom. This changing magnetic field induces a current in the pan, and since the pan has a reasonable resistance, electric energy is transformed to thermal energy, which heats the pant and its contents.

– Electrical generators, induction motors, induction welding and many others…


21-3 EMF Induced in a Moving Conductor

This image shows another way the magnetic flux can change:



The induced current is in a direction that tends to slow the moving bar—it will take an external force to keep it moving.



The induced emf has magnitude

Measurement of blood velocity from induced emf:


(21-3)

Example 3


As shown in the figure, a region of space contains a uniform magnetic field. The magnitude of this field is 2.8 T, and it is directed straight into the plane of the page in the region shown. Outside this region the magnetic field is zero. A rectangular loop measuring 0.20 m by 0.60 m and having a resistance of 2 Ω is being pulled into the magnetic field by an external force, as shown.

–  (a) What is the direction (clockwise or counterclockwise) of the current induced in the loop?

–  (b) Calculate the magnitude of the external force Fext that is required to move the loop at a constant speed of 3.9 m/s.

•  ANSWER: Counterclockwise; 0.6 N

€

E =ΔΦB

Δt= Blv; F = IlB; I =

ER

21-4 Changing Magnetic Flux Produces an Electric Field

A changing magnetic flux induces an electric field; this is a generalization of Faraday’s law. The electric field will exist regardless of whether there are any conductors around.


€

E =Fq

=qvBq

= vB

21-5 Electric Generators

A generator is the opposite of a motor—it transforms mechanical energy into electrical energy. This is an ac generator:


The axle is rotated by an external force such as falling water or steam. An emf is induced in the rotating coil. The brushes are in constant electrical contact with the slip rings.

Electric generators

•  According to the Lenz’s law, the current in wire b is directed towards us. The current through brush b has the same direction.


•  After one-half revolution, wire b will be where wire a is now in this Figure, and the current at brush b will be inward. Thus the current produced is alternating. In USA and Canada, frequency is 60 Hz


A dc generator is similar, except that it has a split-ring commutator instead of slip rings.



A sinusoidal emf is induced in the rotating loop (N is the number of turns, and A the area of the loop):


(21-5)

€

Vrms =NBωA2

Example

•  A simple generator has a square armature 6 cm on a side. The armature has 85 turns of 0.59-mm-diameter copper wire and rotates in a 0.65 T magnetic field. The generator is used to power the lightbulb rated at 12 V and 25 W. At what rate should the generator rotate to provide 12 V to the bulb?

•  Answer: 17 Hz


21-6 Back EMF and Counter Torque; Eddy Currents

An electric motor turns because there is a torque on it due to the current. We would expect the motor to accelerate unless there is some sort of drag torque.

That drag torque exists, and is due to the induced emf, called a back emf.


€

τ = NIABsinθ

€

Ii =Vi

R=120V5Ω

= 24 A

Vf =120V −108V =12V

I f =12V5Ω

= 2.4 A


A similar effect occurs in a generator—if it is connected to a circuit, current will flow in it, and will produce a counter torque. This means the external applied torque must increase to keep the generator turning.



Induced currents can flow in bulk material as well as through wires. These are called eddy currents, and can dramatically slow a conductor moving into or out of a magnetic field.


Force F is opposite to the force that rotates the wheel, slowing it down.

21-7 Transformers and Transmission of Power

A transformer is a device for increasing or decreasing an ac voltage. It consists of two coils, either interwoven or linked by an iron core. A changing emf in one induces an emf in the other. When an ac voltage is applied to the primary coil, the changing magnetic field it produces will induce an ac voltage of the same frequency in the secondary coil.

The ratio of the emfs is equal to the ratio of the number of turns in each coil:


(21-6) €

VS = NSΔΦB

Δt; VP = NP

ΔΦB

Δt⇒

VS

VP

=NS

NP


This is a step-up transformer—the emf in the secondary coil is larger than the emf in the primary:



Energy must be conserved; therefore, in the absence of losses, the ratio of the currents must be the inverse of the ratio of turns:


(21-7)

Example 4

•  An ideal transformer has 60 turns on its primary coil and 300 turns on its secondary coil. If 120 V at 2.0 A is applied to the primary, what voltage and current are present in the secondary?

•  Answer: 600 V; 0.4 A


Example 5 •  For the transmission of electric power from power plant to

home, where the electric power sent by plant is 100 kW, about how far away could the house be from the power plant before power loss is 50%? Assume the wires have a resistance per unit length of 5x10-5 Ω/m.

•  Answer: 6x109 m •  Formulas to use:


€

P =VI;VS

VP

=NS

NP

; ISIP

=NP

NS

;

P = I2R; R = ρl


Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac.


21-9 Applications of Induction: Microphone, Seismograph, GFCI

This microphone works by induction; the vibrating membrane induces an emf in the coil


21-9 Applications of Induction: Microphone, Seismograph, GFCI

A seismograph has a fixed coil and a magnet hung on a spring (or vice versa), and records the current induced when the earth shakes.


21-10 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 as the mutual inductance, is the same:


(21-8a)

(21-8b)

21-10 Inductance

Unit of inductance: the henry, H.

1 H = 1 V·s/A = 1 Ω·s

A transformer is an example of mutual inductance.


21-10 Inductance

A changing current in a coil will also induce an emf in the same coil:

Here, L is called the self-inductance.


(21-9)

Example

•  The mutual inductance between two coils is 10 mH. The current in the first coil changes uniformly from 2.7 A to 5.0 A in 160 ms. If the second coil has a resistance of 0.60 Ω, what is the magnitude of the induced current in the second coil?

•  Answer: 0.24 A •  Formulas to use:


€

E2 = −M ΔI1Δt

; E2 = I2 R

21-11 Energy Stored in a Magnetic Field

Just as we saw that energy can be stored in an electric field, energy can be stored in a magnetic field as well, in an inductor, for example.

Analysis shows that the energy density of the field is given by:


(21-10)

21-12 LR Circuit

A circuit consisting of an inductor and a resistor will begin with most of the voltage drop across the inductor, as the current is changing rapidly. With time, the current will increase less and less, until all the voltage is across the resistor.


21-12 LR Circuit

This plot shows the current as a function of time in an LR circuit that has just been connected across an emf.


21-12 LR Circuit

If the circuit is then shorted across the battery, the current will gradually decay away.


Example: •  The series circuit shown in the figure contains an ideal battery

with a constant terminal voltage VB=60 V, an ideal inductor L=50 H, a resistor R=11 Ω, and a switch S. Initially, the switch is open, and there is no current in the inductor. At time t=0 s the switch is suddenly closed. What is the current in the circuit 4.09 s after closing the switch?

•  Answer: 3.2 A


Summary of Chapter 21

•  Magnetic flux:

•  Changing magnetic flux induces emf:

•  Induced emf produces current that opposes original flux change

•  Changing magnetic field produces an electric field

•  Electric generator changes mechanical energy to electrical energy; electric motor does the opposite


Summary of Chapter 21

•  Transformer uses induction to change voltage:

•  Mutual inductance:

•  Energy density stored in magnetic field:


lecture powerpoints chapter 21 physics: principles with...

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