Unit 9: Electromagnetic Induction Magnetic flux Electromagnetic - - PowerPoint PPT Presentation

• Magnetic flux
• Electromagnetic induction phenomena.
• Faraday’s law and Lenz’s law.
• Examples of Faraday’s and Lenz’s laws.
• Mutual inductance and self-inductance.
• The transformer.
• Stored energy on an inductor.

Michael Faraday 1791-1867

Unit 9: Electromagnetic Induction

Hall of Royal Society

• London. Today

S B   d d   

S d

B 

• Unit: Weber
• Wb = T m2
• Given a surface element dS, magnetic flux through

such surface element is defined as (inner product):

• Tipler, chapter 28.1

Magnetic flux

• If surface is finite (surface S):



 

S

S d B   

• Induced currents on a loop are due to electromotive

forces depending on speed of change of magnetic flux passing throug the surface enclosed by the loop:

dt d  

Tipler, chapter 28.2

Faraday’s law

Derivative of flux against time give us the modulus of induced electromotive force, but not the polarity. The polarity of emf comes from Lenz’s law.

Faraday’s law Experimental law

• If R is the resistance of loop, a

current will flow through the loop:

R i  

• Electromagnetic induction phenomena are known

from 1830 when Michael Faraday and Joseph Henry discovered that a varying magnetic field produces electric currents on a coil (or a loop).

Current  0 Moving a magnet into a coil produces an electric current on coil

Tipler, chapter 28, Introduction

We’ll see that a varying magnetic field isn’t the only way to induce a current. Most important are changes on magnetic flux

Electromagnetic induction phenomena

• As magnetic flux through a loop surface comes from:

 

  

S S

dS B d   cos S B  

Tipler, chapter 28.2

• Changes on flux can be due to:
• a) Changes on magnetic field, B
• b) Changes on surface of loop, S
• c) Changes on angle between B and S, 

Why could magnetic flux change?

N S

Iinduced

• a) Changes on magnetic field, B

Why could magnetic flux change?

 v

I i

• b) Changes on surface of loop, S

Why could magnetic flux change?

I produces a magnetic field through loop. When side of loop moves (speed v), magnetic flux changes and i appears.

Why could magnetic flux change?

• c) Changes on angle between B and S, 

A.C. Generator

B 

S  

Lenz’s law

• Polarity of induced emf () and intensity of current

are always opposites to changes produced on magnetic flux through loop.

• It’s very important to note that induced current is

not opposite to flux, but to changes on flux.

• Induced current always tries to keep existing flux.

Tipler, chapter 28.3

• Examples. Applications of induced currents

Generating an alternating current Tipler, chapter 28.5

B 

S  

Eddy currents. Induction heating Data reading on a magnetic disk head https://www.youtube.com/watch?v=RlvnTk3BJKU

wt NBS cos   S B N    NBSwsenwt dt d     

• If we have two different and close loops (or circuits) electric

current along one of circuits (I1) creates a magnetic field and a magnetic flux through the second circuit (21). The rate of 21 to I1 is the mutual inductance coefficient between circuits 1 and 2 (M21):

I1 B1

1 2

B1

Tipler, chapter 28.6

due to I1

Mutual Inductance

1 21 21 1 21 21

I M I M     

I2

1 2

Tipler, chapter 28.6

due to I2

Mutual Inductance M

• In the same way could be defined

2 12 12

I M  

• It can be proved that M12=M21=M
• M is mutual inductance between circuits 1 and 2.

B2 B2

Inductance unit: Henry (H)

1 Henry=1Wb/1A

Symbol of mutual inductance

• If we have only one loop or circuit (instead two) the rate of

magnetic flux to the intensity is the self-inductance of such

• circuit. The devices having a high self-inductance are the

coils, solenoids or inductors.

• On a coil with a current I, such current produces a magnetic

field and a magnetic flux through coil. Rate of flux to I is called self-inductance of coil, L: On circuits it’s drawn with symbol: L

Tipler, chapter 28.6

Self-Inductance

Physical device is called “Inductor”

LI I L      I B 

• A solenoid with a current I, produces a magnetic field inside

solenoid:

L

Tipler, chapter 28.6

Example: Self-Inductance of a solenoid

I l N B  

l N: turns

• Assuming B is uniform inside solenoid, magnetic flux

through solenoid will be:

l IA N A l NI N NBA

2

     

A: cross section

• Thus:

l A N I L

2

   

Self-inductance of a solenoid

According faraday’s law a emf appears: Polarity of induced e.m.f.:

Electromotive force on an self-inductance.

dt d   

    L dI dt

L

 dt dI

I

L

I

• +

L

 dt dI

I

L

I

+

• dt

dI L   dt dI L  

dt dI L  

General rule: •

dI/dt must have its own sign (>0 or <0 according I)

• The + terminal is that where current is entering by.
• For steady currents (D.C.) ε=0. In this case, the

inductor behaves as a short-circuit.

N1 N2 V1 ~ V2 ~

The transformer

• If current on primary isn’t constant, it produces a flux varying on time.
• This flux is completely driven by the ferromagnetic material, producing an induced emf on

terminals of secondary. The magnetic field created on primary is driven by the ferromagnetic material and it creates a magnetic flux on secondary.

• It works due to electromagnetic induction and then it doesn’t work on D.C.

Primary winding Secondary winding

1 1 1 1

d dB V N S dt dt     

1 1 2 2

V N V N 

Voltage ratio or ratio

• f transformation

equals the turns ratio On an ideal transformer the power on primary equals the power on secondary On an real transformer there are three type of losses: On windings: Joule heating On core: Magnetic Histeresys and Eddy currents In order to minimize Eddy currents the core is built with sheets isolated between them

B



i

1 1

N BS  

S cross section of solenoids

2 2

N BS  

2 2 2 2

d dB V N S dt dt     

When switch is closed (t>0):

RL circuit. Stored energy on an inductor

Tipler, chapter 28.7

t I   

L R

εg

I

L

dI L dt  

+

• t=0

The intensity increases from 0 (on t=0) up to the steady current (dI/dt=0 and therefore εL=0 on t=). At this time the inductor behaves as a shortcircuit and I(t=)=g/R.

g

dI L IR dt    

g

t I R     

If we multiply the equation of this circuit I times, the power consumed by the inductor plus the power lost on resistor by Joule heating equals the generated power on the battery:

2 g

dI LI I R I dt   

Pg PR PL

x I

2 I L L

LI 2 1 LIdI dt dt dI LI dt P W

  

   

Then, when intensity reachs its value I, the stored energy on inductor is: 2

2 1 LI WL 

Stored energy on an inductor