1

Chapter 23

Magnetic Flux and Faraday’s Law of Induction

2

Overview of Chapter 23

• Induced Electromotive Force • Magnetic Flux • Faraday’s Law of Induction • Lenz’s Law • Mechanical Work and Electrical Energy • Generators and Motors • Inductance • RL Circuits • Energy Stored in a Magnetic Field • Transformers

3

23-1 Induced Electromotive Force

Faraday’s experiment: closing the switch in the primary circuit induces a current in the secondary circuit, but only while the current in the primary circuit is changing.

4

23-1 Induced Electromotive Force• Current in the secondary circuit is

0 if current in the primary circuit is not changing. !

• Current flows in the secondary circuit when current in the primary is changing. !

• Flows in opposite direction depending on whether the magnetic field is increasing or decreasing. !

• Magnitude of induced current proportional to rate at which the magnetic field is changing….

5

23-1 Induced Electromotive Force

Note the motion of the magnet in each image

6

23-2 Magnetic Flux

Magnetic flux is used in the calculation of the induced emf.

7

23-3 Faraday’s Law of InductionFaraday’s law: An emf is induced only when the magnetic flux through a loop changes with time.

8

23-3 Faraday’s Law of InductionMany devices that operate on the basis of Faraday’s law.

An electric guitar pickup:

9

23-4 Lenz’s Law

Lenz’s Law

• Induced current always flows in a direction that opposes the change that caused it.

• If magnetic field is increasing (or decreasing), the magnetic field created by the induced current will be the opposite direction…

10

23-4 Lenz’s Law

• This conducting rod completes the circuit. !

• As it falls, the magnetic flux decreases, and a current is induced.

11

23-4 Lenz’s Law

The force due to the induced current is upward, slowing the fall.

12

23-4 Lenz’s Law

Currents can also flow in bulk conductors. These induced currents, called eddy currents, can be powerful brakes.

13

23-5 Mechanical Work and Electrical Energy

This diagram shows the variables we need to calculate the induced emf.

14

23-5 Mechanical Work and Electrical Energy

Change in flux: Induced emf:

Electric field caused by the motion of the rod:

15

23-5 Mechanical Work and Electrical Energy

• If the rod is to move at a constant speed, an external force must be exerted on it.

!• This force should have equal magnitude and opposite direction to the magnetic

force:

16

23-5 Mechanical Work and Electrical Energy

Mechanical power delivered by the external force is:

Compare this to the electrical power in the light bulb:

Therefore, mechanical power has been converted directly into electrical power.

17

23-6 Generators and Motors

An electric generator converts mechanical energy into electric energy:

An outside source of energy is used to turn the coil, thereby generating electricity.

18

23-6 Generators and Motors

The induced emf in a rotating coil varies sinusoidally:

19

23-6 Generators and MotorsAn electric motor is exactly the opposite of a generator – it uses the torque on a current loop to create mechanical energy.

20

23-7 Inductance

When the switch is closed in this circuit, a current is established that increases with time.

21

23-7 Inductance

Inductance is the proportionality constant that tells us how much emf will be induced for a given rate of change in current:

Solving for L,

22

23-7 Inductance

Given the definition of inductance, the inductance of a solenoid can be calculated:

When used in a circuit, such a solenoid (or other coil) is called an inductor.

23

23-8 RL Circuits

• Switch is closed, the current immediately starts to increase. • Back emf in the inductor is large, as the current is changing rapidly. • As time goes on, the current increases more slowly, and the potential

difference across the inductor decreases.

24

23-8 RL CircuitsThis shows the current in an RL circuit as a function of time. The time constant is:

25

23-9 Energy Stored in a Magnetic Field

• It takes energy to establish a current in an inductor; this energy is stored in the inductor’s magnetic field. !

• Considering the emf needed to establish a particular current, and the power involved, we find:

26

23-9 Energy Stored in a Magnetic Field

We know the inductance of a solenoid; therefore, the magnetic energy stored in a solenoid is:

Dividing by the volume to find the energy density gives:

Result is valid for any magnetic field, regardless of source.

27

23-10 TransformersA transformer is used to change voltage in an alternating current from one value to another.

28

23-10 Transformers

By applying Faraday’s law of induction to both coils, we find:

Here, p stands for the primary coil and s the secondary.

29

23-10 Transformers

The power in both circuits must be the same; therefore, if the voltage is lower, the current must be higher.

30

Answer: 1.6 x 10-4 Wb

31

Answer: 1.9 Wb

32

Answer: 14 V

33

Answer: 3.8 x 103

34

Answer: 32.58 mT

35

Answer: 15.5 mV

L = 353.2 µH

36

Answer: a) 9.95 x 108 J/m3

Answer: b) 1.5 x 1010 V/m