Chapter 23

Magnetic Flux and Faraday’s

Law of Induction

Units of Chapter 23

• Induced Electromotive Force

• Magnetic Flux

• Faraday’s Law of Induction

• Lenz’s Law

• Mechanical Work and Electrical Energy

• Generators and Motors

Lenz’s Law Lenz’s Law

An induced current always flows in a direction

that opposes the change that caused it.

Therefore, if the magnetic field is increasing, the

magnetic field created by the induced current will

be in the opposite direction; if decreasing, it will

be in the same direction.

Motional emf: Qualitative

This conducting rod

completes the circuit.

As it falls, the magnetic

flux decreases, and a

current is induced.

Motional emf: Qualitative

The force due to the

induced current is

upward, slowing the fall.

Since an emf is

produced in the system,

it is referred to as

motional emf.

Mechanical Work and Electrical Energy

This diagram shows the variables we need to

calculate to calculate the induced emf.

Motional emf: Quantitative

A rod is

pushed by

an external

force, so it

moves to the

right with

constant

speed v.

Motional emf: Quantitative

1) Calculate the change in flux:

2) Using Faraday’s law calculate the

magnitude of the induced emf.

3) Electric field caused by the motion of

the rod

Motional emf: Quantitative

Change in flux:

Induced emf:

Electric field caused by the motion of the

rod: EUREKA

A CHANGING MAGNETIC FLUX DOES

INDEED CREATE AN ELECTRIC FIELD.

Only one resistance, the current is: R

vBI in

10/30/2013 9

A. Evil physics

professor

B. Applied force on

bar pulling to right

C. Applied force on

bar pulling to left

D. Magnetic forces

Consider the setup shown in the figure.

What can cause the bar to move with

velocity v:

10/30/2013 10

A. It is too small to have

an effect.

B. It has an effect in the

same direction as the

applied force.

C. It has an effect in the

opposite direction as

the applied force.

I

Consider the setup shown in the figure where

current I is flowing through the mobile bar. What

is the effect of the magnetic force acting on the

bar?

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.

1) Calculate the magnetic force:

Mechanical Work and Electrical Energy

2) Calculate mechanical power delivered by the

external force:

3) Compare this to the electrical power in

the light bulb:

EUREKA, mechanical power has been

converted directly into electrical power.

13 10/30/2013

PROBLEM: Lighting a bulb The graph below shows a circuit consisting of a flashlight

bulb, rated 3.0V/1.5 W, and ideal wires with no resistance.

The right wire of the circuit, which is 10 cm long, is pulled

at constant speed v through a perpendicular magnetic

field of strength 0.10 T.

a. What speed must

the wire have to

light the bulb to full

brightness?

b. What force is

needed to keep the

wire moving?

14 10/30/2013

Work it at home: wire in the

magnetic field Over a region where the vertical component of the Earth's magnetic

field is 40.0µT directed downward, a 5.00 m length of wire is held in

an east-west direction and moved horizontally to the north with a

speed of 10.0 m/s. Calculate the potential difference between the

ends of the wire, and determine which end is positive.

Also, work out all the examples

and active examples from the

book

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.

Generators and Motors

The induced emf in a rotating coil varies

sinusoidally:

The induced emf in the coil alternates in sign, which

means that the current in the coil alternates in direction

–alternating-current generator, ac generator.

10/30/2013 17

Torque on a current-carrying wire in a magnetic field.

Recall that a generator has a current-carrying coil in a

magnetic field.

A. No problem – that was in

Chap. 29 and no longer

relevant.

B. Generator coil will

experience a torque

which makes the coil

spin faster.

C. Generator coil will

experience a torque

which makes the coil

spin slower.

10/30/2013 18

Electric generators and motors

Hybrid vehicles are designed so that their

electric motors are equipped with circuits to

take advantage of “regenerative” braking,

effectively converting the motor to a generator

to recharge the batteries when the breaks are

activated.

Other uses for inductors

• Rechargeable electric toothbrush

• Induction heating cooking

10/30/2013 PHY 114 A Spring 2012 -- Lecture

13

19

Uses of Faraday’s law continued:

http://theinductionsite.com/how-induction-works.shtml

Motors

An electric motor is exactly the opposite of a

generator – it uses the torque on a current loop

to create mechanical energy.

Inductance

When the switch is closed in this circuit, a

current is established that increases with

time.

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,

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.

24 10/30/2013

Home Example: solenoid

A solenoid of radius 2.5cm has 400 turns and a length of 20 cm.

Find (a) its inductance and (b) the rate at which current must

change through it to produce an emf of 75mV.

RL Circuits

When the switch is closed, the current

immediately starts to increase. The 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.

R is resistor and L inductor.

RL Circuits

This shows the current in an RL circuit as a

function of time.

The characteristic time is:

10/30/2013 27

A. Why physics class is so beautiful.

B. Why physics class is so terrible.

C. RC circuit.

D. Money in the bank.

The LR circuit reminds us of:

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:

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:

This result is valid for any magnetic field,

regardless of source.

Transformers

A transformer is used to change voltage in an

alternating current from one value to another.

Transformers

By applying Faraday’s law of induction to both

coils, we find:

Here, p stands for the primary coil and s the

secondary.

Transformers

The power in both circuits must be the same;

therefore, if the voltage is lower, the current

must be higher.

Summary of Chapter 23

• A changing magnetic field can induce a current

in a circuit. The magnitude of the induced

current depends on the rate of change of the

magnetic field.

• Magnetic flux:

• Faraday’s law gives the induced emf:

Summary of Chapter 23

• Lenz’s law: an induced current flows in the

direction that opposes the change that created

the current.

• Motional emf:

• emf produced by a generator:

• An electric motor is basically a generator

operated in reverse.

• Inductance occurs when a coil with a changing

current induces an emf in itself.

Summary of Chapter 23

• Definition of inductance:

• Inductance of a solenoid:

• An RL circuit has a characteristic time

constant:

Summary of Chapter 23

• Current in an RL circuit after closing the switch:

• Magnetic energy density:

• Transformer equation: