Magnetic Induction and the Storage of Magnetic Energy

8/14/2019 Magnetic Induction and the Storage of Magnetic Energy

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MAGNETIC INDUCTION AND THE STORAGE OF MAGNETIC ENERGY



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Define and use Faraday’s Law and

Lenz’s Law to determine the effectof changing magnetic fluxes.

Com ute for Inductances and learnways on how to store magneticenergies

ompu e or e c rcu aparameters of RL Circuits



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1830’s – Michael Faraday(England) and Joseph Henry (USA)

changing magnetic field inducesa current in the wire.

The emfs and currents caused by

changing magnetic fields areca e in uce em s an in ucecurrents.

The process itself, is referred to asmagnetic induction.

When you pull the plug of an

electric cord from its socket, yousometimes observed a small spark.This phenomenon is explained bymagnetic induction!



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The flux of a magnetic field througha surface is defined similarly to the

.

The magnetic flux Φm is defined as

The unit of flux is that of a magnetic, -squared, which is called a weber(Wb)

1 Wb = 1 T•m2

Exercise: Show that a weber persecond is a volt.



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We are often interestedin the flux through a coil

containing several turns ofwire.

If the coil contains N turns,

the flux through the coil ismes e ux rougeach turn.



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ons er a ar magne n prox m y o a

loop attached to an ammeter.



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Moving the bar magnet towards the,

without a battery. Such induced current

arises rom t e in uce em .



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From what we haveMove the Move the loo

How to change the magnetic flux?

ear er, a c ang ngmagnetic flux results to

permanentmagnet towards

the loop

towards thepermanent

magnet

an induced emf.

Current that Area of the loo

Faraday’s Law!

be changedcan be changed

Loops/B sourcescan be rotated



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EXAMPLES:

1. A uniform magnetic field makes an angle of 30o with the axis of acircular coil of 300 turns and a radius of 4 cm. The field changes

.coil.

2. An 80-turn coil has a radius of 5.0cm and a resistance of 30Ω. Atwhat rate must a perpendicular magnetic field to produce a

.

3. A solenoid of len th 25 cm and radius 0.8cm with 400 turns is inan external magnetic field of 600 G that makes an angle of 50o

with the axis of the solenoid. (a) Find the magnitude flux through

.solenoid if the external magnetic field is reduced to zero in 1.4s.



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Developed by Heinrich Lenz

Lenz’s Law gives us the direction of

e n uce curren .

“ uinduced current are in such

a direction so as to o osethe change the producesthem.”

Note: We didn’t specify just whatkind of chan e causes the inducedemf and current. The statementwas left vague to cover a variety

of conditions we will now illustrate.



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A rectangular coil of 80turns, 20 cm wide and 30

cm ong, is ocate in amagnetic field B = 0.8T,

only a portion of the coil in

the region of the magneticie . T e resistance o t ecoil is 30Ω.

n e magn u e an

direction of the inducedcurrent if the coil is movedwith a speed of 2m/s (a) tothe right, (b) up, and (c)own.



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An airport metal detectorcontains a large coil in its frame.

The coil has a property called.

When a metal asses throu hthe frame, the inductance of theframe changes.

The change in the inductance isconverted to an alarm sound!



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The Unit of Inductance

s e enry .



. -19

When the switch is closed,current rises until it reaches

its max va ue.

During the current rise, themagnetic field it produces

,magnetic flux

Thus there should be aninduced emf caused by thechanging magnetic flux

Therefore, there is self-induction in the circuit!



. -20

The general relation of-

the changing current is: The proportionality constant is the

self-inductance (L) of the circuit!

By applying Faraday’sLaw in reverse, we derive:

-constant but depends on

circuit/loop!



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1. Calculate the self-inductance of an air-core

so eno con a n ng urns e eng o esolenoid is 25.0 cm and its cross-sectional area is

4.00 cm2.

2. Find the self-inductance of a solenoid of length 10

cm, area 5 cm , and 100 turns. At what rate mustthe current in the solenoid change to induce an emf

of 20V?



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Figure shows two circuits.Circuit 1 Circuit 2

As we change the resistance incircuit 1, the current also

.it produces also changes.

The changing magnetic fluxinduces an emf on circuit 1 and

Thus circuit 2 has an inducedemf.

This phenomenon is called,mutual induction!

Circuit 1 Circuit 2



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The net induced emf on

the changing current by:Mutual Inductance

-

depends on the

geometry of the two

Self-induced

by 2

Mutually

induced by

circuits an t e

distance between

them!

two circuits are equal and

transformable formula:



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An electric toothbrush has a basedesigned to hold the toothbrushhandle when not in use. As shown in

the Figure, the handle has acylindrical hole that fits loosely over amatching cylinder on the base. When

,changing current in a solenoid insidethe base cylinder induces a current in

a coil inside the handle. This induced xcurrent charges the battery in thehandle.

We can model the base as a solenoidof length x with Nbase turns (Fig.32.15b), carrying a current I, and having a cross sectional area A. Thehandle coil contains Nhandle turns and completely surrounds the base coil.

system.

5 The Storage of



5. The Storage of

Magnetic Energy25

An inductor stores magneticenergy through the current

ui ing up in it, just as acapacitor stores electrical.

Consider, the circuit at the right.

carrying a current I is given by:

,is given by: This is the energy that is stored in a

magnetic field, regardless of theconfiguration!



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RL Circuits contain a

res s or an aninductor.

I flows in a single

direction But changes its value, it

6 The Growth of



6. The Growth of

I in RL Circuits27

We assume that the inductor.

After the switch is closed, theem o e a ery equa es

to the back emf of thein uctor, an current ui saccording to:

As current builds up, theImax is the maximum current in the circuit

equivalent to 0/R.

n uctor s ac em sreduced to zero! is the time constant equilvalent to L/R

6 The Decay of



6. The Decay of

I in RL Circuits28

As the switches are

recon gure , e maxcurrent is drained by the

resistor R according to:

inductor acts like a battery,

w a essen ng curren

pump abilities!

Io is the initial current in the circuit equivalent to 0/R.

is the time constant equilvalent to L/R



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1. A basic RL circuits consists of the following: a= =, ,

resistor (R = 6 Ω). Find the time constant, and if the,

half its maximum value.

2. If the batter in the exam le above is carefull

removed after the current reaches its maximumvalue when will the current deca to 10% of theoriginal maximum value?

Chapter Six is

pretty much a



pretty much a

chapter.

1. Sources of AC

2. R in AC

3. n

4. C in AC

.

6. The Series RLC

in AC7. Resonance in

AC

8. rans ormers

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IN THIS FINAL CHAPTER, YOU SHOULD BE ABLE



TO…

Understand the sourcesof alternating currents

na yze e e av orsof R, L, and C if

alternating currents

Understand theoperating principles of

transformers

Define phasors and Analyze the behavior of

2 when analyzing for thebehaviors of R, L, and C

specific R, L, Ccombinations

TIME VARYING VALUES



TIME-V ARYING V ALUES

| To identify time

varying values, we use

lower case letters!

| To identify fixed

,

upper case letters

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WHY STUDY ALTERNATING CURRENTS?



WHY STUDY A LTERNATING CURRENTS?

| More than 99% of theelectrical energy used todayis produced by electricalgenerators in the form of alternating current (ac).

| AC’s advantage over DC

can be transported overlong distances at very high

volta e and low currents toreduce energy losses due toJoule heat!

| AC can then betransformed, with almost,

safer voltages andcorrespondingly higher

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ALTERNATING CURRENTS



A LTERNATING CURRENTS

| Alternating Currents –

vary periodically over

time!

general are sinusoidal in

nature and generallysupply alternating

voltages of the form:

| Because voltage changes, In our country most

it is positive ½ the period,

and negative ½ the5

AC’s have frequencies

of 60 Hz or angular

fre uenc of 377 rad/s

per o

1 AC SOURCES



1. AC SOURCES

| There are many kindso sources

| The most commonprobably are the AC

outlets in our homes!

| But how do we

actuall roduce

alternating currents?

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2 RESISTORS IN AC



2. RESISTORS IN AC

| Consider the circuit to ther g .

| The instantaneous voltage

and current through theresistor are given by:

| Notice that vR and iR are“ ”

another!7

2 RESISTORS IN AC



2. RESISTORS IN AC

| There is still power lossn res s ors w en

current passes through

em ecause o e

voltage drop!

|

This power has 3forms:

y Instantaneous

y Average

y Maximum8

ROOT MEAN SQUARED RMS VALUES



ROOT MEAN SQUARED RMS V ALUES

|

Most AC ammeters and

measure rms values of currents and voltages, instead

o t e maximum va ues.

| o t ere is a necessity tointerconvert between rms and

the maximum value over thesquare root of 2!

| Example:9

EXERCISES



EXERCISES

| 1. Find Pav in terms of Irms and R

| 2. Find Pav in terms of ξmax and Imax

. av rms rms

| . n rms n erms o rms an

| 5. A 12-Ω resistor is connected across a sinusoidal

emf that has a peak value of 48V. Find (a) the

rms current, (b) the average power, (c) the

maximum power.10

3 INDUCTORS IN AC



3. INDUCTORS IN AC

| Consider the circuit to ther g .


and current are given by:

| Notice that vL and iL are

, L L

by π/2 rads

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χL is ca e in uctive reactance, it

has the unit of ohms!

s means a n uc ors reacdifferently to current by offering

resistance!

4. CAPACITORS IN AC



4. C APACITORS IN AC

| Consider the figure to ther g .


and current are given by:

| Notice that vC and iC are

“ ”, C

vC with π/2 radians.

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C is ca e capacitive reactance, it

has the unit of ohms!

s means a capac ors reacdifferently to current by offering

resistance!

L, C IN AC: EXAMPLES



L, C IN AC: EXAMPLES

| 1. A 40mH inductor is placed across an ac generator thathas a maximum emf of 120V. Find the inductive reactanceand the maximum current when the frequency is

(a) 60 Hz

(b) z

current?

| 2. A 20-μF capacitor is placed across a generator that has amaximum emf of 100V. Find the capacitive reactance and

(a) 60 Hz(b) 5000 Hz

What can you conclude about the relation of capacitive reactance and13

THE BEHAVIORS OF L AND C IN AC CIRCUITS



THE BEHAVIORS OF L AND C IN AC CIRCUITS

| Alternating current behaves differently thandirect current in inductors and ca acitors.

| When a capacitor becomes fully charged in a dcc rcu t, t stops t e current, t at s, t acts e anopen circuit.

,flows onto or off the plates of the capacitor and athigher frequencies, the capacitor, will hardly impede

, ,circuit!

| Conversely, an inductor coil usually has a very smallresistance and is essentially a short circuit for dc.

u w en e curren s a erna n , a ac em sgenerated in an inductor, and at higher frequencies,the back emf is so large, the inductor acts like an 14

open c rcu

5. LC IN AC



| Consider the circuit to theright.

| When the switch is closed, the

discharges producing a backemf on the inductor, which in

current, recharging thecapacitor.

Oscillation of i

| Thus once the capacitorcompletely discharges, it isonce aga n c arge y e

inductor.

urren n an c rcu

| Conversely, once the inductorreaches zero current, currentwill a ain flow throu h it from 15

the capacitor!

5. LC IN AC EXAMPLE



| A 2-μF capacitor is charged to 20V and is thenconnec e across a -μ n uc or. a a s e

frequency of oscillation? (b) What is the

max mum va ue o e curren

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6. RLC IN AC SERIES



|

Consider the figure toe r g .

| The circuit has acurrent given by:

| Where Z is impedance

| n s t e p ase

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6. RLC IN AC SERIES



|

The Average Power for RLC in AC, seriesconnec on can e represen e y:

| Where cos δ is called the ower factor.

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7. RESONANCE IN SERIES RLC IN AC



| Resonance is thecondition in which wehave the smallestpossible impedance that

maximum current.y Zmin can only happen

zero.

y Reactances can only be

,frequency equates to thenatural frequency of thecircuit!

| At resonance, we havemax mum current anpower and the powerfactor is one! 19

7. SERIES RLC IN AC: EXAMPLES



1. A series RLC Circuit with L = 2H, C = 2μF,an = s r ven y a genera or w a

maximum emf of 100 V and a variable

requency. n a e resonance requency

(f 0), (b) the maximum current at resonance, (c)

, ,

the average power delivered.

2. A series RLC Circuit with L = 2H, C = 2μF,

an = 20 is riven y a generator wit a

maximum emf of 100 V and a variablerequency. n e max mum vo age across

the resistor, the inductor and the capacitor.20

8. TRANSFORMERS



| A transformer is a device used

to raise or lower the voltage in acircuit without an appreciableloss of power

y A simple transformer consistingof two wire coils around acommon iron core.

y The coil carrying the inputpower is called the primary.

y

The coil carrying the outputpower s ca e e secon ary.

| The transformer o erates on theprinciple of mutual induction

| e ron core ncreases emagnetic field for a givencurrent and guides it so that

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through one coil goes throughthe other coil.

6. TRANSFORMERS



|

For a transformer with1

and N2 turns in the

,

across the secondary coilis related to the

generator emf across the

primary coil by:

| If there are no losses,

due to Joule Heating

(which is due toneg g e res s ance n

the coils), RMS Power22

6. TRANSFORMERS: EXAMPLES



|

1. A doorbell requires 0.4A at 6V. It is connectedo a rans ormer w ose pr mary con a n ng

2000turns, is connected to a 120-V ac line. (a)

ow many urns s ou ere e n e

secondary? (b) What is the current in the

| 2. A transmission line has a resistance of

0.02Ω/km. Calculate the I2R power loss if 200kW

o power is transmitte rom a power generator

to a city 10km away at (a) 240 V and (b) 4.4 kV

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Magnetic Induction and the Storage of Magnetic Energy