Compilation in Physics 212

7/26/2019 Compilation in Physics 212

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surface repeatedly. 'or sound waves the disturbance is a change in air

pressure perhaps created by the oscillating cone inside a speaker. 'or

earthquakes there are several types of disturbances including disturbance of

Earths surface and pressure disturbances under the surface.

% wave is a disturbance that propagates or moves from the place it was

created. The simplest waves repeat themselves for several cycles and are

associated with simple harmonic motion.

The wave is an up and down disturbance of the water surface. It causes a

sea gull to move up and down in simple harmonic motion as the wave crests

and troughs (peaks and valleys) pass under the bird. The time for one complete

up and down motion is the waves period T. The waves frequency is f * + T as

usual. The wave itself moves to the right in the $gure. This movement of the

wave is actually the disturbance moving to the right not the water itself (or the

bird would move to the right). #e de$ne wave velocity vw to be the speed at

which the disturbance moves. #ave velocity is sometimes also called the

propagation velocity or propagation speed because the disturbance propagates

from one location to another.

Figure 16.30%n ideali,ed ocean wave passes under a sea gull that bobs up and down in

simple harmonic motion. The wave has a wavelength which is the distance between ad-acent


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Figure 16.31 In this example of a transverse wave the wave propagates hori,ontally and

the disturbance in the cord is in the vertical direction.

Figure 16.3% In this example of a longitudinal wave the wave propagates hori,ontally and

the disturbance in the cord is also in the hori,ontal direction.

#aves may be transverse longitudinal or a combination of the two. (#ater

waves are actually a combination of transverse and longitudinal. The

simpli$ed water wave illustrated in Figure 16.30 shows no longitudinal

motion of the bird.) The waves on the strings of musical instruments are

transverse2so are electromagnetic waves such as visible light.

0ound waves in air and water are longitudinal. Their disturbances are periodic

variations in pressure that are transmitted in 3uids. 'luids do not have

appreciable shear strength and thus the sound waves in them must be

longitudinal or compressional. 0ound in solids can be both longitudinal and

transverse.

In both cases (and in all other forms of wave motion) the disturbance moves

through the medium (slinky string water air whatever....) with only a minimal

motion of the medium itself. #hat is being described in the equations of wave

motion is the motion of the disturbance. The wave speed for example is the

speed at which the disturbance moves.

4y(x 44


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t)

v

y(xt)

t4 x 4

%s seen in the derivation for a wave on a string it is -ust 5ewton6s second law

applied to a small part of the string itself. The most general solution to this

equation is any function of the argument (x7vt) and v is the speed at which the

wave propogates. That is any function of x and t that has the form

The most general form of the di"erential equation that describes a mechanical

wave is written8

y(xt)f (x 7vt)

would solve the above di"erential wave equation with v being the speed of the

disturbance (ie the wave) through the medium. The speci$c form of the wave

would depend on the source of the disturbance (a hand clap would be di"erent

than a tuning fork) ! but the wave speed itself would depend on the medium

through which the disturbance travelled. The direction of propagation is

determined by the sign in the argument. f(x9vt) corresponds to a wave

traveling to the left a :!: sign is to the right.

Wave &r!'aga"i!n S'ee

The wave speed is determined by properties of the medium (except for light

which travels at speed of ;x*


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'or a transverse wave on a taut string or a longitudinal wave on a stretched

:slinky: the wave speed is dependent on the linear mass density of the string

(ie the mass per unit length ?) and the tension in the string or the slinky ('). Ie

v= F

'or sound waves through air or water the wave speed would depend on the

compressibility of the medium and the volume mass density (since a

disturbance can propagate in all directions in a three dimensional medium).

0ince water is nearly like a solid with respect to compressions it should not be

surprising that sound travels faster through water than through air (even

though the mass density is a thousand times as great).

&any important examples of wave motion involve a wave function that is

sinusoidal. That is rather than a single pulse that propagates the disturbance is

in the shape of a sine wave. The resulting wave motion description is then

y(xt)Asin(kx 7t)

It is not di@cult to show that this equation in fact is a solution to thedi"erential wave equation by -ust taking derivatives of y(xt) and substituting

into the di"erential equation. #hen that is done one can see the connection

between the wave speed and the constants k and . The wave number as k is

called is -ust 4A divided by the wavelength. The angular frequency has the

same meaning in wave motion as it does in simple harmonic motion or circular

motion. %nd the wave speed wavelength and frequency are always related in

the same way.

k

4

and 4f

4

and v f T

These relationships are always true.


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5otice that for any speci$c time t the shape of the wave as you move along

the string (ie y vs x) is -ust that of a sine function. %nd if you consider only that

part of the wave at a speci$c location x the motion of the medium itself is

simple harmonic with an amplitude % and angular frequency.

Biven the relationships between k f T and the wave function y(xt) can

be written in a variety of ways. They are all equivalent.

y (x , t)=Asin (kx 2ft)=Asink(x vt)

The amplitude of the wave is -ust the maximum displacement of any part of

the medium from the equilibrium or undisturbed position. The frequency and

period of the oscillation the wavelength the wave number etc. and the wave

speed are then related by the equations above. 'or a wave on a string the

wave speed is still dependent on the mass density and the tension by v=F . Su'er'!si"i!n an In"er(eren)e

In general the idea of superposition of waves is common to all types of wavemotion. The standing wave problem discussed in the waves!on!a!string

discussion is simply the result of two identical waves traveling in opposite

directions as shown previously. The superposition of two waves traveling in the

same direction will lead to the ideas of constructive and destructive interference

and will be important in both sound and light. >ut the mathematics is most

easily developed using the equations of harmonic waves to describe waves on a

string.

0uppose two identical waves are traveling in the same direction and they

di"er only in that there is a phase di"erence between them. That is the waves

y*(xt) and y4(xt) are the two waves described by

y*(kx t) and y4(kx t 9)


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The wave descriptions can be rewritten by adding and subtracting +4 in each

argument. 'or the sake of the derivation we can then let kx!wt9+4. The

total combined wave is then written8

y*(kx t 9+ 4+ 4)9y4(kx t 99+ 4+ 4)y*(+ 4)9

y4(9+ 4)

Csing trigonometric identities to break sin(7+4) into products of sines and

cosines yields the following result8

ytotal(xt)4Acos(+4)sin(kx t 9+ 4)

This equation represents the combined wave equation. 5otice that it simply

represents a traveling wave with the same frequency and wavelength as the

constituent waves but with an amplitude 4%cos(+4) that depends on the phase

di"erence . #hen .


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Cri"i)a# Da+'ing the condition in which the damping of an oscillator

causes it to return as quickly as possible to its equilibrium position

withoutoscillating back and forth about this position

De(!r+a"i!n displacement from equilibrium

Des"ru)"ive In"er(eren)e when two identical waves arrive at the same

point exactly out of phase1 that is precisely aligned crest to trough

E#as"i) &!"en"ia# Energ* potential energy stored as a result of

deformation of an elastic ob-ect such as the stretching of a spring

F!r)e C!ns"an" a constant related to the rigidity of a system8 the

larger the force constant the more rigid the system1 the force

constant isrepresented by k

Freuen)* number of events per unit of time

Funa+en"a# Freuen)* the lowest frequency of a periodic waveform

In"ensi"* power per unit area

L!ngi"uina# Wave a wave in which the disturbance is parallel to the

direction of propagation


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Na"ura# Freuen)* the frequency at which a system would oscillate if

there were no driving and no damping forces

N!es the points where the string does not move1 more generally nodes

are where the wave disturbance is ,ero in a standing wave

Os)i##a"e moving back and forth regularly between two points

Over Da+'ing the condition in which damping of an oscillator causes

it to return to equilibrium without oscillating1 oscillator moves more

slowlytoward equilibrium than in the critically damped system

Over"!nes multiples of the fundamental frequency of a sound &eri!i) M!"i!n motion that repeats itself at regular time intervals

&eri! time it takes to complete one oscillation

Res!nan)e the phenomenon of driving a system with a frequency equal

to the system6s natural frequency

Res!na"e a system being driven at its natural frequency

Res"!ring F!r)e force acting in opposition to the force caused by a

deformation Si+'#e ar+!ni) M!"i!n the oscillatory motion in a system where the

net force can be described by Gookes law


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Si+'#e ar+!ni) Os)i##a"!r a device that implements Gookes law

such as a mass that is attached to a spring with the other end of the

springbeing connected to a rigid support such as a wall

Si+'#e &enu#u+ an ob-ect with a small mass suspended from a light

wire or string

Su'er'!si"i!n the phenomenon that occurs when two or more waves

arrive at the same point

Transverse Wave a wave in which the disturbance is perpendicular to

the direction of propagation Uner Da+'ing the condition in which damping of an oscillator causes

it to return to equilibrium with the amplitude gradually decreasing to ,ero1

system returns to equilibrium faster but overshoots and crosses the

equilibrium position one or more times

Wave Ve#!)i"* the speed at which the disturbance moves. %lso called

the propagation velocity or propagation speed

Wave#eng"2 the distance between ad-acent identical parts of a wave

Wave a disturbance that moves from its source and carries energy


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cm. If the cork is 4 m from the edge of the pool how long does it take a

ripple passing the cork to reach the edgeN

Solution:

Biven8 wavelength of 4< cmcork is 4 m from the edge of the pool

Mequired8 time required to take a ripple passing the cork to reach the

edge

'ormula8The time taken for the ripple to reach the edge of the pool is obtained

from8

t=D

v v=D

t

#e know thatv=f

Therefore

t= D

f

t= 2m

1Hz 0.2m

t= 2m1 s

1 0.2m

t=10 s

. alculate the fundamental frequency for a string


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f 0+4JKT+mL *+

4DO by using the approximate correction of


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'ormula8

=. % sound wave traveling at ;< m+s has a frequency of


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4< vibrations

Mequired8 wavelength of sound

'ormula8

#avelength DistanceNo . of osciations

25

20

*.4 ms!*.

*


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Biven8 frequency of *< G,wavelength


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Chapter 2+ )o$n!

DISCUSSION AND DERIVATION OF

FORMULAS IN SOUND

SOUND

This study of sound will concentrate on only a few main ideas ! sound as

an example of a longitudinal wave exhibits all the properties that all waves

exhibit including a speed that depends on the medium that carries the wave

and both interference and di"raction. 0ound level measurements (the

decibel scale) are related to the energy density in the wave and the

apparent frequency of the sound one hears depends on both the speed of

the source and the speed of the listener relative to the speed of sound in air

(the Foppler e"ect). The most general study of sound would include

discussions of how sound propagates through air as well as in liquids and


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solids our perceptions of sound ! which would require understanding the

physiology of hearing and would ultimately lead to the study of musical

instruments and the complex study of acoustics.

S!un as a $aveThe general principles studied in the discussion of wave motion apply

equally well to sound. That includes of course the most general

relationships between wave speed wavelength and frequency8 That is

v f

where represents the wavelength and f is the frequency. The wave speed v

is determined by properties of the medium ! which we will consider is air. The

wave itself is longitudinal rather than transverse as are waves on strings and

the surface waves on water. >ut the waves propagate in all directions as a

speed determined by the compressibility and mass density of the air. The

propagation of a disturbance in air is described by a di"erential equation of

the same form as for transverse waves on a taut string. 0o the solutions to

the equation are necessarily of the same form as well. That is a disturbance

in air is governed by a wave equation of the form

!2y (x , t)

! t2 =v2

!2y (x ,t)

! x2

which has solutions representing waves traveling a speed v that depends on

the bulk modulus > (whichin turn depends on pressure and temperature) and

on the mass density of the air. That is the disturbance that propagates

through the air that we call :sound: can be described with the same type ofwave equation that was used to describe waves on strings. %nd the speed of

those waves depends on properties of the medium through which they

travel.


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v="#where plays the same role as the tension in the string and is the mass

per unit volume of the air rather than the linear mass density of the string

which supported a transverse wave.

In air the wave speed can be related to the air temperature by

v

-

T (;;*

m+s)

T

4/

;M

where is a constant for air M is the Cniversal Bas onstant & is the

average molecular mass of air and the temperature is the %bsolute

temperature on the Pelvin scale. The equation can be simpli$ed in terms of

the speed of sound in air at 4/; P (ie ice point).

S!un In"ensi"*

The intensity of a wave is simply the energy per unit time that is

transferred per unit area of a surface that the wave impinges on. >ut energy

per time is -ust the power that is delivered by the source. %nd that energy is

distributed over an ever increasing area as the wave propagates away from

the source. %ssuming a point source of sound with waves spreading outward

in spherical wave fronts ! and assuming no energy dissipation as the wave

propagates through the air ! the intensity decreases as the inverse square of

the distance from the source as the energy is spread over an ever increasing

spherical surface. 0o the intensity or power per unit area is simply given by


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Intensity in watts+m4 I Qavg+D r4

where Qavgis the power emitted by the source and r is the distance from the

source. The intensity I would be measured in watts per square meter In

practice the intensity of a sound is much more complicated since the above

expression assumes a point source of sound that spreads out 0ound source

uniformly in all directions. Ignored in this expression is the absorption of

sound by the air itself and re3ections from surfaces that the sound

encounters.

Braphing the intensity as a function of distance from the source shows

how intensity diminishes as the energy of the sound waves is spread over an

ever increasing area. %s the distance from the sound source increases the

intensity decreases until it would be undetectable. The weakest sound

intensity that most humans can hear is about *

called the threshold of hearing and is assigned the symbol Io. %ll other sound

intensities can be related to Io. That is a sound intensity one hundred times

the threshold of hearing would be written I*


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ideas. #e will consider interference between two identical sound waves that

are constrained to travel in a straight line but in opposite directions. This is

identical in principle to the standing waves on a string problem. 0econdly we

will consider two identical sound waves that arrive at the same point from

two di"erent sources which are separated from each other (as the sound

from two loudspeakers driven by the same signal). In general those sources

can either be exactly in phase ! or there could be a phase di"erence between

the signals produced at the speakers. This problem is identical to the

superposition and interference of two waves on a string traveling in the same

direction. The resulting wave depended on the phase di"erence between the

two signals. 'inally when two sources have slightly di"erent frequencies

there is no $xed phase di"erence between them and the super!position of

the two signals results in a varying intensity wave with a :beat frequency:

that depends on the di"erence in the individual frequencies that are being

added.

The general principle of course is the superposition principle ! that is

waves can be added to form a new wave which depends on the properties

and characteristics of the original waves. This idea treated earlier applies

equally well to longitudinal waves. %dding two waves y*(k*x7 *t) and

y4(k4x74t9) results in a wave equation that represents one of three

possibilities8 % standing wave if y* andy4 have the same frequency (and

hence wavelengths) and are traveling in opposite directions1 either

constructive or destructive interference at a particular location depending

on the phase angle at that location1 and a wave whose amplitude varies if

the two frequencies are di"erent. #e will consider each of those cases.

S"aning Wave Res!nan)es in Air C!#u+ns

The standing sound wave problem requires a one!dimensional sound

wave ! as in a sound wave created inside a hollow tube or pipe. onsider a

sound wave traveling in a tube which encounters an identical wave in the


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opposite direction. This is identical to the situation when a wave on a string

:sees: its re3ection from the end of the string ! which leads to standing

waves. The di"erence here is that when a string is $xed at both ends there

must be a node at each end of the string. >ut a tube can have either a node

or an antinode at the ends depending on whether it is closed or open at that

end.

0ince sound is a longitudinal wave air must be able to move along the

axis of the tube. This results in the special condition that an open end of a

tube must be at an antinode of any standing wave whereas a closed end

must be a node (since air cannot move in!and!out of a closed ended tube).0o

the relationship between the length of a tube and the wavelength of the

sound when a standing wave occurs depends on whether the tube has open

or closed ends. 0tanding waves can be supported in a tube (or air column)

only if the tube lengths are the wavelengths of the sound in the tube are

related in certain speci$c ways.

O'en9O'en :!r C#!se9C#!se; Tu,e

If a tube is open at both ends antinodes must appear at the ends of the

tube for a standing wave to occur ! and that means the tube length must be

a multiple of half!wavelengths (ie the distance between antinodes will

always be +4 or or ;+4 etc). Sf course if the tube is closed at both ends

nodes appear at the ends and the condition is the same (and identical to the

standing wave resonances on a taut string). >ut with a tube closed at both

ends it is not obvious how the wave would be generated nor whether the

resonance could be heard outside the tubeR The condition for resonance and

the frequencies at which standing waves can be supported in a tube of

length J are given by


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L=n

2 and fn=

v

=( v2L )n

where v is the speed of sound in air (ie in the air column contained within

the tube). %gain this is the same condition as for standing waves on a string

! and the frequencies are -ust multiples of the fundamental frequency given

by (v+4J) for the tube open at both ends.

If the tube has one open and one closed end there must be a node at

the closed end and an antinode at the open end for resonance to occur.

0ince the distance between a node and an antinode is +D ;+D +D etc. the

resonance condition is di"erent than for the open!open tube. In this case the

length of the tube is necessarily an odd multiple of quarter wavelengths or

L=m

4=(2n+1)

4

where m is an odd integer represented by (4n9*) for any integer n. 0o the

frequencies at which resonance can occur are

fn= v=( v4L )

m=(2n+1) f1

where f* is the fundamental frequency given by (v+DJ) for the tube closed at

one end.

DO&&LER EFFECT

#hen a car or train passes you the frequency of the sound that you hear

varies from a higher pitch as it approaches to a lower pitch as it recedes fromyou. (Fo not confuse this e"ect with the change in intensity or sound level as

it approaches and then recedes.) This Foppler e"ect is a common occurance

in everyday life when a source of sound is moving with respect to an

observer. There are really two di"erent causes for the e"ect. #hen the


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source of the sound is moving the sound waves are compressed in front of

the source and expanded behind. >ut the sound still travels with respect to

the air at the speed of sound. 0o an observer in front of the source hears a

higher frequency than the source is emitting since the wave fronts arrive

closer together than if the source were not moving whereas an observer

behind the moving source hears a lower frequency.

If the source is stationary but the observer is moving a similar e"ect

occurs for a di"erent reason. The wavefronts spread out from the source

uniformly. >ut if the observer is moving toward the source he or she

encounters wavefronts more often ! hence hears a higher frequency. If the

observer is moving away from the source the wavefronts arrive less often !hence a lower frequency.

%ll of this can be summari,ed mathematically. The frequency one hears (f)

is always the speed at which the wave is traveling with respect to the

observer (vrel) divided by the wavelength () that is encountered by the

observer. That is

f.vrel

#.

The relative pee! vrelis -ust the di"erence betweeen the speed of

sound relative to the air and the speed of the observer ($ob) relative to

the air or

vrel.vo/$ob

where vois the speed of sound and $obis the speed of the observer. #hetherthe 9 or ! sign is used depends on the direction of the observer6s motion

relative to the source ! to be discussed shortly.


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The wavelength that the observer encounters is the wavelength in front or

behind the source depending on whether the source is moving toward or

away from the observer. Gence

(vo/$o$rce)#fo

where fois the frequency the source produces and $o$rceis the speed of the

source. ombining these results yields. 5otice that in this form all cases are

included. That is if the source is moving the denominator is modi$ed by the

speed of the source. If the observer is moving the numerator is modi$ed.

%nd of course if both are moving then both are a"ected. The choice of

whether to use the 9 or ! sign in each case can be made by deciding

whether the speed of the source or of the observer has the e"ect of

increasing or decreasing the frequency that would be heard depending on

whether the motion of either tends to decrease or increase the separation

between the source and observer. If the gap between them is closing the

sound is Foppler shifted to a higher frequency whereas if the gap is

increasing the sound is Foppler shifted to a lower frequency ! regardless of

which is moving. The choice of sign is then made to yield that result.


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DEFINITION OF TERMS

(Sound)

A)!us"i) I+'ean)e property of medium that makes the

propagation of sound waves more di@cult an"in!e point of

maximum displacement

/!$ Wa


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In"ensi"* Re>e)"i!n C!e?)ien" a measure of the ratio of the

intensity of the wave re3ected o" a boundary between two media

relative to theintensity of the incident wave

In"ensi"* the power per unit area carried by a wave#!uness the

perception of sound intensity

N!e point of ,ero displacement

N!"e basic unit of music with speci$c names combined to generatetunes

Over"!nes all resonant frequencies higher than the fundamental

&2!n the numerical unit of loudness

&i")2 the perception of the frequency of a sound

S!ni) /!!+ a constructive interference of sound created by an

ob-ect moving faster than sound

S!un In"ensi"* Leve# a unit less quantity telling you the level of

the sound relative to a $xed standard


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S!un &ressure Leve# the ratio of the pressure amplitude to a

reference pressure

S!un a disturbance of matter that is transmitted from its source

outward

Ti+,re number and relative intensity of multiple sound frequencies

T!ne number and relative intensity of multiple sound frequencies

U#"ras!un sounds above 4


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SOLVED &RO/LEMS

(Sound)

*. %n echo was heard ;.D seconds after the sound was produced.

Temperature of the air is 4U . Gow far was the re3ectorN

Solution:

Biven8 Time8 ;.D seconds

Temperature8 %5@ C

Mequired8 Fistance

'ormula8

elocity8 ;;* 9


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4. The $rst overtime of an open pipe produces D beats per second with a

tuning fork of D


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Biven8 Jength8 ; cm 3.6 +

#eight8

Mequired8 Tension

'ormula8

;4 x /.4

4;D< m+s

v=Tm

2340= T

4.2x105

T=9.44 k' (Answer)

D. %n open pipe is < cm long and a closed pipe is 4= cm long. They are

sounded together to produced their fundamentals. Gow many beats

are heard per second if of sound in air is ;DD meter per secN

Solution:

Biven8

Spen Qipe8

Jength8 < cm 6 +


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losed Qipe8

Jength8 4= cm %.B +

elocity8 3 +8s

Mequired8 >eats

'ormula8

Spen Qipe8

J8 m

8 *4 m

f*

;DD f (*4)

f* 4=. G,

losed Qipe8

J8 4.= m

8 . m

f4

;DD f (.)f4 H.D G,

>eats f4 ! f* H.D V 4=. ;


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0(elocity of the air)8 33 +8s

Mequired8 v s (elocity of the source)

'ormula8

n2=n

1

v

vvs

350=(320Hz) 343m/ s343m /sv

v=29.4m /s (Answer)

. %n iron wire is stretched between two supports *4< cm apart. It has a

density of /. gm+cmW and a modulus of elasticity of * x *


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v=461880.2154

4J (4)(*.4) %. +

v=f

461880.2154=f2.4

f=1.92x 103 sec(Answer)

/. %n open organ pipe whose length is H< cm is blown at a temperature of

4U . #hat is the frequency of the second overtoneN

Solution:

Biven8 Jength8 40 )+ 0.4 +

Temperature8 %5 C

Mequired8 0econd overtone (;f)

'ormula8

elocity8 ;;* 9


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f=192.22m

3 f=576.66/sec (Answer )

=. % wood cutter makes * strokes per minute. If the sound of each stroke

reaches the observer as the axe makes the next stroke and air

temperature is 4


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Biven8 0ound power Q %rea % m

Mequired8 Fistance

'ormula8 0ound intensity is given by

$=*

A

$=0.5+104

5m

1x 105/m2

*


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t=1

2(3.4 )

t=1.7 sec

**. % train moving at 4 m+s is traveling towards &s. &iller who is

standing in the middle of the tracks. #hat frequency does &s. &iller

hear if the train has a horn frequency of *


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*4. % burglar alarm is wailing with a frequency of *4


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vs .< m+s

Mequired8 n4

'ormula8

n2=n1(v vo

v ) n2=700Hz( 340m /s+5.0m /s340m/ s ) 710.%4 Hz (Answer)

*D. Zou are in a car traveling at mph (4D. m+s). % second car is

moving toward you at the same speed. Its horn is sounding at D/ G,.

#hat frequency do you hearN

Solution:

Biven8

n* =


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*. Zou are in a car traveling at 4< m+s. %n ambulance is behind you

traveling ; m+s in the same direction. #hat frequency do you hear if

the siren has a frequency of < G,N elocity of sound in air is ;D; m+s.

Biven8

n* < G,

v ;D; m+s

vs ; m+svo 4< m+s

Mequired8 n;

'ormula8

n3=n

1(v vo

vvs ) n3=550Hz( 343m / s+20m/s343m /s35m / s )

n3 6B.%1 Hz(Answer)

*. #hat will be the frequency if the ambulance takes overN

Solution:

Biven8

n* < G,

v ;D; m+s

vs ; m+svo 4< m+s

Mequired8 n;'ormula8


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n3=n

1(v vovvs )n3=550Hz( 343m /s20m/ s343m / s+35m/s)

n3 469.97 Hz(Answer)

*/. 'or a pipe of length J


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Chapter 12 y!rotatic


FORMULAS IN FLUIDS

F#ui

&atter most commonly exists as a solid liquid or gas1 these states are

known as the three common phae of matter. 0olids have a de$nite

shape and a speci$c volume liquids have a de$nite volume but their

shape changes depending on the container in which they are held and

gases have neither a de$nite shape nor a speci$c volume as their

molecules move to $ll the container in which they are held. (0ee Figure

11.%.) Jiquids and gases are considered to be 3uids because they yield to

shearing forces whereas solids resist them. 5ote that the extent to which

3uids yield to shearing forces (and hence 3ow easily and quickly) depends

on a quantity called the viscosity which is discussed in detail in Vis)!si"*

an La+inarF#!$ &!iseui##es La$.#e can understand the phases of

matter and what constitutes a 3uid by considering the forces between

atoms that make upmatter in the three phases.


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Figure 11.% (a) %toms in a solid always have the same neighbors held near

home by forces represented here by springs. These atoms are essentially in

contact with oneanother. % rock is an example of a solid. This rock retains its

shape because of the forces holding its atoms together. (b) %toms in a liquid are

also in close contact but can slide over one another. 'orces between them

strongly resist attempts to push them closer together and also hold them in

close contact. #ater is an example of a liquid. #ater can 3ow but it also

remains in an open container because of the forces between its atoms. (c)

%toms in a gas are separated by distances that are considerably larger than the

si,e of the atoms themselves and they move about freely. % gas must be held in

a closed container to prevent it from moving out freely.

%toms in oli!are in close contact with forces between them that

allow the atoms to vibrate but not to change positions with neighboring

atoms. (These forces can be thought of as springs that can be stretched

or compressed but not easily broken.) Thus a solid reit all types of

stress. % solid cannot be easily deformed because the atoms that make

up the solid are not able to move about freely. 0olids also resist

compression because their atoms form part of a lattice structure in which

the atoms are a relatively $xed distance apart. Cnder compression the

atoms would be forced into one another. &ost of the examples we have

studied so far have involved solid ob-ects which deform very little when

stressed.

GDROSTATIC &RESSURE

Gydrostatic Qressure ! due to a column of 3uid of height hand mass

density # is

% . # &h

DENSITG


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Fensity is mass per unit volume.

m

.

where8

is the symbol for density1

m is the mass1 and

0 is the volume occupied by the substance.

The 0I Cnit of density is the ecause the relative density of water is * the density of any substance

in kg m!;is its relative density multiplied by *


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MANOMETER

Figure 11.16 %n open!tube manometer has one side open to the atmosphere. (a) 'luid depth must be the same on both

sides or the pressure each side exerts at the bottom will be unequal and there will be 3ow from the deeper side. (b) %

positive gauge pressure %g h& transmitted to one side of the manometer can support a column of 3uid of height h . (c)

0imilarly atmospheric pressure is greater than a negative gauge pressure %g by an amount h& . The -ars rigidity

prevents atmospheric pressure from being transmitted to the peanuts.

onsider the C!shaped tube shown in Figure 11.16 for example. This

simple tube is called a manometer. &ercury manometers are often used

to measure arterial blood pressure. %n in3atable cu" is placed on the

upper arm as shown in 'igure **.*/. >y squee,ing the bulb the person

making the measurement exerts pressure which is transmitted

undiminished to both the main artery in the arm and the manometer.

#hen this applied pressure exceeds blood pressure blood 3ow below the

cu" is cut o". The person making the measurement then slowly lowers

the applied pressure and listens for blood 3ow to resume. >lood pressure

pulsates because of the pumping action of the heart reaching a

maximum called systolic pressure and a minimum called diastolic

pressure with each heartbeat. 0ystolic pressure is measured by noting

the value of h when blood 3ow $rst begins as cu" pressure is lowered.Fiastolic pressure is measured by noting h when blood 3ows without

interruption. The typical blood pressure of a young adult raises the

mercury to a height of *4< mm at systolic and =< mm at diastolic. This is

commonly quoted as *4< over =


52/113

the elasticity of the arteries in maintaining the pressure between beats.

The density of the mercury 3uid in the manometer is *;. times greater

than water so the height of the 3uid will be *+*;. of that in a water

manometer. This reduced height can make measurements di@cult so

mercury manometers are used to measure larger pressures such as blood

pressure. The density of mercury is such that *.< mm Gg *;; Qa.

/OGLES LAW

>oyles Jaw states that when the temperature is kept constant the

volume of a given mass of an ideal gas varies inversely as the pressure to

which it is sub-ected1 therefore the product Qressure x olume of a given

mass of gas remains constant. Thus for a given mass of an ideal gas

ARCIMEDES &RINCI&LE

*.=constant(at constant temperat2re )

% body wholly or partly immersed in a 3uid is buoyed up by a force equal

to the weight of the 3uid it displaces. The buoyant force can be

considered to act vertically upward through the center of gravity of the

displaced 3uid.

F"=/2oyant force=wei'3t of )ispace) f2i)

The buoyant force on an ob-ect of volume 0that is totally immersed in

a 3uid of density#f is

#f 0& and the weight of the ob-ect is#o

g where #o is the density of the ob-ect. Therefore the net upward

force on the submerged ob-ect is


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3net '$pwar!( . 0& ' #f#o

&ASCALS &RINCI&LE

#hat happens to a pressure in an enclosed 3uidN 0ince atoms in a 3uid

are free to move about they transmit the pressure to all parts of the 3uid

and to the walls of the container. Memarkably the pressure is transmitted

undiminished. This phenomenon is called &as)a#s 'rin)i'#e because it

was $rst clearly stated by the 'rench philosopher and scientist >laise

Qascal (*4;V*4)8A chan&e in pre$re applie! to an encloe! 4$i! i

tranmitte! $n!iminihe! to all portion of the 4$i! an! to the wall of it

container5

Qascals principle an experimentally veri$ed fact is what makes pressure

so important in 3uids. 0ince a change in pressure is transmitted

undiminished in an enclosed 3uid we often know more about pressure

than other physical quantities in 3uids. &oreover Qascals principle

implies that the total pressure in a 3uid is the sum of the pressures from

di"erent sources. #e shall $nd this fact2that pressures add2very useful.


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DEFINITION OF TERMS

(Fluids)

Ar)2i+ees 'rin)i'#e the buoyant force on an ob-ect equals theweight of the 3uid it displaces

a,s!#u"e 'ressure the sum of gauge pressure and atmospheric

pressure

a2esive (!r)es the attractive forces between molecules of di"erent

types

,u!*an" (!r)e the net upward force on any ob-ect in any 3uid

)a'i##ar* a)"i!n the tendency of a 3uid to be raised or lowered in a

narrow tube

)!2esive (!r)es the attractive forces between molecules of thesame type

)!n"a)" ang#e the anglebetween the tangent to the liquid surface

and the surface

ensi"* the mass per unit volume of a substance or ob-ect

ias"!#i) 'ressure the minimum blood pressure in the artery


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ias"!#i) 'ressure minimum arterial blood pressure1 indicator for

the 3uid balance

>uis liquids and gases1 a 3uid is a state of matter that yields to

shearing forces

gauge 'ressure the pressure relative to atmospheric pressure

g#au)!+a condition caused by the buildup of 3uid pressure in the

eye

in"ra!)u#ar 'ressure 3uid pressure in the eye

+i)"uri"i!n re>e stimulates the feeling of needing to urinate

triggered by bladder pressure

&as)a#s &rin)i'#e a change in pressure applied to an enclosed 3uid

is transmitted undiminished to all portions of the 3uid and to the walls

of itscontainer

'ressure the force per unit area perpendicular to the force over

which the force acts

'ressure the weight of the 3uid divided by the area supporting it


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s'e)i-) gravi"* the ratio of the density of an ob-ect to a 3uid

(usually water)

sur(a)e "ensi!n the cohesive forces between molecules which cause

the surface of a liquid to contract to the smallest possible surface area

s*s"!#i) 'ressure the maximum blood pressure in the artery

s*s"!#i) 'ressure maximum arterial blood pressure1 indicator for the

blood 3ow


57/113

SOLVED &RO/LEMS

(Fluids)

*. The density of steel is /=


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Solution:

Biven8 4


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Mequired8 force that is exerted on the piston

'ormula8

F=*A

F=(300*a )(0.5m2 )

F=*A

F=150 (*a )m2

F=150N/m2 m2

F=150N (answer )

D. % manometer connected to a pipe indicates a negative gauge pressure

of


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*atmosp3ere=1 1x105N/m2

*a/so2te=*'a2'e+*atmosp3eric

#'3+ *atmosp3eric

*;. x *


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. % block of wood of mass ;. kg 3oats in water. alculate the buoyant

force on the block.

Solution:

Biven8 mass of ;. kg

Mequired8 buoyant force on the block

'ormula8The wooden block is 3oating so the buoyant force is equal to the

weight of the block or

F=m '

' (;. kg) (H.= m + s4)

' ;D.; 5 (answer )

/. % retaining wall m high and 4.m wide retains water up to its top. 'ind

the total pressure per meter length of the wall and the point at which

the resultant cuts the base. %lso $nd the resultant thrust on the base

of the wall per meter length. %ssume weight of masonry as 4; P5+m;.

Solution:

Biven8 m high4.m wideweight of masonry as 4; P5+m;

Mequired8 total pressure per meter length of the wallresultant thrust on the base of the wall per meter length

'ormula8

#e know that total pressure per meter length of the wall


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Qoints at which the resultant cuts the base

#e also know that weight of masonry per meter length of the wall

and distance between the mid!point 'M(of the wall and the point

where resultant cuts the base '-(.

Mesultant thrust on the base of the wall per meter length

#e know that resultant thrust on the base of the wall per meter length

=. % pipe of cross sectional area =< cm4has a constriction where the area

is reduced to 4< cm4. If the velocity of the 3uid in the larger area is


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Biven8 area =< cm41reduced to 4< cm4

velocity of the 3uid in the larger area is


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.=m

#

.= 2k'

8700 k' /m3

.=2.299x104 m3

alculate the buoyant force8

F=#'.

F= (**


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wt=m '

wt=(600k' )(9.8m/s2)

wt=5880N


66/113

Chapter 26 Ma&netim

DISCUSSION AND DERIVATION OFFORMULAS IN MAHNETISM Magne"s

Figure %%.3 &agnets come in various shapes si,es and strengths. %ll have

both a north pole and a south pole. There is never an isolated pole (a monopole).

%ll magnets attract iron such as that in a refrigerator door. Gowever

magnets may attract or repel other magnets. Experimentation shows that

all magnets have two poles. If freely suspended one pole will point toward

the north. The two poles are thus named the n!r"2 +agne"i) '!#e and

the s!u"2 +agne"i) '!#e (or more properly north!seeking and south!

seeking poles for the attractions in those directions).


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Figure %%. Sne end of a bar magnet is suspended from a thread that

points toward north. The magnets two poles are labeled 5 and 0 for north!

seeking and south!seekingpoles respectively.

E#e)"r!+agne"s

7lectroma&net

Early in the *Hth century it was discovered that electrical currents cause

magnetic e"ects. The $rst signi$cant observation was by the Fanish

scientist Gans hristian Sersted (*///V*=*) who found that a compass

needle was de3ected by a current!carrying wire. This was the $rst

signi$cant evidence that the movement of charges had any connection with

magnets. E#e)"r!+agne"is+ is the use of electric current to make

magnets. These temporarily induced magnets are called e#e)"r!+agne"s.

Electromagnets are employed for everything from a wrecking yard crane

that lifts scrapped cars to controlling the beam of a H

particle accelerator to the magnets in medical imaging machines (0ee

Figure %%.4).


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Figure %%.4 Instrument for magnetic resonance imaging (&MI). The

device uses a superconducting cylindrical coil for the main magnetic

$eld. The patient goes into this [tunnel\ on the gurney. (credit8 >ill

&chesney 'lickr)

Figure %%.10 showsthat the response of iron $lings to a current!

carrying coil and to a permanent bar magnet. The patterns are similar. In

fact electromagnets and ferromagnets have the same basic

characteristics2for example they have north and south poles that

cannot be separated and for which like poles repel and unlike poles

attract.

Figure %%.10 Iron $lings near (a) a current!carrying coil and (b) a

magnet act like tiny compass needles showing the shape of their $elds.


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Their response to a current!carryingcoil and a permanent magnet is seen

to be very similar especially near the ends of the coil and the magnet.

Magne"i) Fie#s an Magne"i) Fie# Lines

Einstein is said to have been fascinated by a compass as a child

perhaps musing on how the needle felt a force without direct physical

contact. Gis ability to think deeply and clearly about action at a distance

particularly for gravitational electric and magnetic forces later enabled

him to create his revolutionary theory of relativity. 0ince magnetic forces

act at a distance we de$ne a +agne"i) -e# to represent magnetic

forces. The pictorial representation of +agne"i) -e# #ines is very

useful in visuali,ing the strength and direction of the magnetic $eld. %s

shown in Figure %%.15 the ire)"i!n !( +agne"i) -e# #ines is

de$ned to be the direction in which the north end of a compass needle

points. The magnetic $eld is traditionallycalled the B9-e#.

Figure %%.15 &agnetic $eld lines are de$ned to have the direction that

a small compass points when placed at a location. (a) If small compasses

are used to map themagnetic $eld around a bar magnet they will point

in the directions shown8 away from the north pole of the magnet toward

the south pole of the magnet. (Mecall that the Earths north magnetic


70/113

pole is really a south pole in terms of de$nitions of poles on a bar

magnet.) (b) onnecting the arrows gives continuous magnetic $eld

lines. The strength of the $eld is proportional to the closeness (or

density) of the lines. (c) If the interior of the magnet could be probed

the $eld lines would be found to form continuous closed loops.

0mall compasses used to test a magnetic $eld will not disturb it.

(This is analogous to the way we tested electric $elds with a small test

charge. In both cases the $elds represent only the ob-ect creating them

and not the probe testing them.) Figure %%.16 shows how the magnetic

$eld appears for a current loop and a long straight wire as could beexplored with small compasses. % small compass placed in these $elds

will align itself parallel to the $eld line at its location with its north pole

pointing in the direction of . 5ote the symbols used for $eld into and out

of the paper.

Figure %%.16 0mall compasses could be used to map the $elds shown here.

(a) The magnetic $eld of a circular current loop is similar to that of a bar

magnet. (b) % long andstraight wire creates a $eld with magnetic $eld lines

forming circular loops. (c) #hen the wire is in the plane of the paper the $eld


71/113

is perpendicular to the paper. 5ote that the symbols used for the $eld pointing

inward (like the tail of an arrow) and the $eld pointing outward (like the tip of

an arrow).

The strength of the $eld is proportional to the closeness of the lines. It is

exactly proportional to the number of lines per unit area perpendicular to

the lines (called the areal density).&agnetic $eld lines can never cross

meaning that the $eld is unique at any point in space. &agnetic $eld lines

are continuous forming closed loops without beginning or end. They go

from the north pole to the south pole. The last property is related to the fact

that the north and south poles cannot be separated. It is a distinct

di"erence from electric $eld lines which begin and end on the positive and

negative charges. If magnetic monopoles existed then magnetic $eld lines

would begin and end on them.

Magne"i) Fie# S"reng"2 F!r)e !n a M!ving C2arge in a

Magne"i) Fie#

#hat is the mechanism by which one magnet exerts a force onanotherN The answer is related to the fact that all magnetism is caused by

current the 3ow of charge. Ma&netic 8el! exert force on movin&

char&e and so they exert forces on other magnets all of which have

moving charges.

Right Hand Rule 1

The magnetic force on a moving charge is one of the most

fundamental known. &agnetic force is as important as the electrostatic or

oulomb force. Zet the magnetic force is more complex in both the number


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of factors that a"ects it and in its direction than the relatively simple

oulomb force. The magnitude of the +agne"i) (!r)e 3 on a charge 9

moving at a speed vin a magnetic $eld of strength

3 9v sin (44.*)

where is the angle between the directions of vand /. This force is often

called the L!ren" (!r)e. In fact this is how we de$ne the magnetic $eld

strength 2in terms of the force on a charged particle moving in a

magnetic $eld. The 0I unit for magnetic $eld strength is called the "es#a

(T) after the eccentric but brilliant inventor 5ikola Tesla (*=V*HD;). To

determine how the tesla relates to other 0I units we solve 3 9vsin for

.

3 (44.4)

9v sin

>ecause sin is unitless the tesla is

* 5* T * 5

(44.;)

E ]

m+s

% ]

m

(note that +s %).

%nother smaller unit called the gauss (B) where * B *


73/113

superconducting electromagnets may attain *< T or more. The Earths

magnetic $eld on its surface is only about Y*


74/113

DEFINITION OF TERMS

(Magnetism)

De-ni"i!n !( Ter+s

B9-e#another term for magnetic $eld

A+'eres #a$ the physical law that states that the magnetic

$eld around an electric current is proportional to the current1each segment of currentproduces a magnetic $eld like that of a

long straight wire and the total $eld of any shape current is the

vector sum of the $elds due to each segment

/i!"9Savar" #a$ a physical law that describes the magnetic $eld

generated by an electric current in terms of a speci$c equation

Curie "e+'era"ure the temperature above which a ferromagnetic

material cannot be magneti,edire)"i!n !( +agne"i) -e# #ines

the direction that the north end of a compass needle points

!+ains regions within a material that behave like small bar

magnets

e#e)"r!+agne" an ob-ect that is temporarily magnetic when an

electrical current is passed through it e#e)"r!+agne"is+ the use

of electrical currents to induce magnetism

(err!+agne"i) materials such as iron

cobalt nickel and gadolinium thatexhibit strong magnetic e"ects gauss B

/D o m p I l a t I o n I n Q h y s I c s 4 * 4


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the unit of the magnetic $eld strength1

* B *


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+agne"ie to be turned into a magnet1 to be induced to be

magnetic

+agne"!)ari!gra+ :MCH; a recording of the hearts magnetic

$eld as it beats

+agne"!en)e'2a#!gra+ :MEH; a measurement of the brains

magnetic $eld

+e"er common application of magnetic torque on a current!

carrying loop that is very similar in construction to a motor1 by

design the torque isproportional to :and not so the needle

de3ection is proportional to the current

+!"!r loop of wire in a magnetic $eld1 when current is passed

through the loops the magnetic $eld exerts torque on the loopswhich rotates a shaft1 electrical energy is converted to

mechanical work in the process

n!r"2 +agne"i) '!#e the end or the side of a magnet that is

attracted toward Earths geographic north pole

nu)#ear +agne"i) res!nan)e :NMR; a phenomenon in which an

externally applied magnetic $eld interacts with the nuclei of certain

atoms

'er+ea,i#i"* !( (ree s'a)e the measure of the ability of a

material in this case free space to support a magnetic $eld1 the

constant

/ o m p I l a t I o n I n Q h y s I c s 4 * 4


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o


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SOLVED &RO/LEMS

(Magnetism)

1. If k.m+d4is equal to 5+%mp.m $nd the magnetic $eld produced by

them*and m4at point %.

%. 'ind the forces exerted by 0 poles of magnets given below.

/= o m p I l a t I o n I n Q h y s I c s 4 * 4


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F


80/113

/3% and wires are given below. 'ind the magnetic $eld of % > and

at points ^ and Z.

Dire)"i!ns !( +agne"i) -e#s a" '!in" are (!un using rig2"

2an ru#e.

/A !u"$ar

//in$ar

=< o m p I l a t I o n I n Q h y s I c s 4 * 4


81/113

/Cin$ar

///J/C9/A

/%


82/113

Inu)e e+(9:;8:";.N

C2ange in F#u

%91

10 sin)e )r!ss se)"i!n area !( s!#en!i an +agne"i) -e#

#ines are 'ara##e# "! ea)2 !"2er.

%/.A

/.A90/.A

9/.A.N8"

6. 'ind the magnetic 3ux through a square with side of ; cm whichis located near a long straight conductor with electric current of

* %. Sne side of the square is parallel to the conductor with

distance of D cm between the side and the conductor. The opposite

side of the square is located cm away from the conductor.

a;cm


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:*%

!*Dcm


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`;i`


85/113

hange in 'lux1

4!*

*.%

>.%!.%

!>.%.5+t

4. % coil of wire of one turn has a cross!sectional area of = o m p I l a t I o n I n Q h y s I c s 4 * 4


86/113

1%. % solenoid has =< cm diameter number of loop is D and

magnetic $eld inside it is *4 .*


87/113

Chapter 2= 7lectrotatic


FORMULAS IN ELECTROSTATICSE#e)"ri) C2arge an E#e)"ri) Fie#

The image of %merican politician and scientist >en-amin 'ranklin

(*/


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Figure 1B.% #hen >en-amin 'ranklin demonstrated that lightning was

related to static electricity he made a connection that is now part of the

evidence that all directlyexperienced forces except the gravitational force are

manifestations of the electromagnetic force.

&uch has been written about 'ranklin. Gis experiments were only

part of the life of a man who was a scientist inventor revolutionary

statesman and writer. 'ranklins experiments were not performed in

isolation nor were they the only ones to reveal connections.

'or example the Italian scientist Juigi Balvani (*/;/V*/H=) performed

a series of experiments in which static electricity was used to stimulate

contractions of leg muscles of dead frogs an e"ect already known in

humans sub-ected to static discharges. >ut Balvani also found that if he

-oined two metal wires (say copper and ,inc) end to end and touched the

other ends to muscles he produced the same e"ect in frogs as static

discharge. %lessandro olta (*/DV*=4/) partly inspired by Balvanis

work experimented with various combinations of metals and developed

the battery.

Furing the same era other scientists made progress in discovering

fundamental connections. The periodic table was developed as the

systematic properties of the elements were discovered. This in3uenced

the development and re$nement of the concept of atoms as the basis of

matter. 0uch submicroscopic descriptions of matter also help explain a

great deal more.

%tomic and molecular interactions such as the forces of friction

cohesion and adhesion are now known to be manifestations of thee#e)"r!+agne"i) (!r)e. 0tatic electricity is -ust one aspect of the

== o m p I l a t I o n I n Q h y s I c s 4 * 4


89/113

electromagnetic force which also includes moving electricity and

magnetism.

%ll the macroscopic forces that we experience directly such as the

sensations of touch and the tension in a rope are due to the

electromagnetic force one of the four fundamental forces in nature. The

gravitational force another fundamental force is actually sensed through

the electromagnetic interaction of molecules such as between those in

our feet and those on the top of a bathroom scale. (The other two

fundamental forces the strong nuclear force and the weak nuclear force

cannot be sensed on the human scale.)

This chapter begins the study of electromagnetic phenomena at a

fundamental level. The next several chapters will cover static electricity

moving electricity and magnetism2collectively known as

electromagnetism. In this chapter we begin with the study of electric

phenomena due to charges that are at least temporarily stationary

called electrostatics or static electricity.

S"a"i) E#e)"ri)i"* an C2arge C!nserva"i!n !( C2arge

#hat makes plastic wrap clingN 0tatic electricity. 5ot only are

applications of static electricity common these days its existence has

been known since ancient times. The $rst record of its e"ects dates to

ancient Breeks who noted more than .. that polishing amber

temporarily enabled it to attract bits of straw (see Figure 1B.3). The very

wordelectric

derives from the Breek word for amber (electron

).

=H o m p I l a t I o n I n Q h y s I c s 4 * 4


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&any of the characteristics of static electricity can be explored by rubbing

things together. Mubbing creates the spark you get from walking across a

wool carpet for example. 0tatic cling generated in a clothes dryer and the

attraction of straw to recently polished amber also result from rubbing.

0imilarly lightning results from air movements under certain weather

conditions. Zou can also rub a balloon on your hair and the static

electricity created can then make the balloon cling to a wall. #e also have

to be cautious of static electricity especially in dry climates. #hen we

pump gasoline we are warned to discharge ourselves (after sliding across

the seat) on a metal surface before grabbing the gas no,,le. %ttendants in

hospital operating rooms must wear booties with aluminum foil on the

bottoms to avoid creating sparks which may ignite the oxygen being used.

0ome of the most basic characteristics of static electricity include8

The e"ects of static electricity are explained by a physical quantity not

previously introduced called electric charge.

There are only two types of charge one called positive and the other

called negative.

Jike charges repel whereas unlike charges attract.

The force between charges decreases with distance.

Gow do we know there are two types of e#e)"ri) )2argeN #hen

various materials are rubbed together in controlled ways certaincombinations of materials always produce one type of charge on one

H< o m p I l a t I o n I n Q h y s I c s 4 * 4


91/113

material and the opposite type on the other. >y convention we call

one type of charge [positive\ and the other type [negative.\ 'or

example when glass is rubbed with silk the glass becomes positively

charged and the silk negatively charged. 0ince the glass and silk haveopposite charges they attract one another like clothes that have

rubbed together in a dryer. Two glass rods rubbed with silk in this

manner will repel one another since each rod has positive charge on

it. 0imilarly two silk cloths so rubbed will repel since both cloths have

negative charge. Figure 1B. shows how these simple materials can

be used to explore the nature of the force between charges.

Figure 1B. % glass rod becomes positively charged when rubbed with silk

while the silk becomes negatively charged. (a) The glass rod is attracted to

the silk because theircharges are opposite. (b) Two similarly charged glass

rods repel. (c) Two similarly charged silk cloths repel.

E#e)"ri) Fie# C!n)e'" !( a Fie# Revisi"e

ontact forces such as between a baseball and a bat are explained

on the small scale by the interaction of the charges in atoms and

molecules in close proximity. They interact through forces that include

H* o m p I l a t I o n I n Q h y s I c s 4 * 4


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the C!u#!+, (!r)e. %ction at a distance is a force between ob-ects that

are not close enough for their atoms to [touch.\ That is they are

separated by more than a few atomic diameters.

'or example a charged rubber comb attracts neutral bits of paper

from a distance via the oulomb force. It is very useful to think of an

ob-ect being surrounded in space by a (!r)e -e#. The force $eld carries

the force to another ob-ect (called a test ob-ect) some distance away.

C!n)e'" !( a Fie#

% $eld is a way of conceptuali,ing and mapping the force that

surrounds any ob-ect and acts on another ob-ect at a distance without

apparent physical connection. 'or example the gravitational $eldsurrounding the earth (and all other masses) represents the gravitational

force that would be experienced if another mass were placed at a given

point within the $eld.

Ear"2s E#e)"ri) Fie#

% near uniform electric $eld of approximately *< 5+ directed

downward surrounds Earth with the magnitude increasing slightly as we

get closer to the surface. #hat causes the electric $eldN %t around *


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including the electric $eld surrounding Earth. In fair weather the

ionosphere is positive and the Earth largely negative maintaining the

electric $eld (Figure 1B.3(a)).

In storm conditions clouds form and locali,ed electric $elds can be

larger and reversed in direction (Figure 1B.3(b)). The exact charge

distributions depend on the local conditions and variations of Figure

1B.3(b) are possible.

If the electric $eld is su@ciently large the insulating properties of the

surrounding material break down and it becomes conducting. 'or air this

occurs at around ;Y*


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0o far we have considered excess charges on a smooth symmetrical

conductor surface. #hat happens if a conductor has sharp corners or is

pointedN Excess charges on a non uniform conductor become

concentrated at the sharpest points. %dditionally excess charge may

move on or o" the conductor at the sharpest points.

To see how and why this happens consider the charged conductor in

Figure 1B.35. The electrostatic repulsion of like charges is most e"ective

in moving them apart on the 3attest surface and so they become least

concentrated there. This is because the forces between identical pairs of

charges at either end of the conductor are identical but the components

of the forces parallel to the surfaces are di"erent. The component parallel

to the surface is greatest on the 3attest surface and hence more

e"ective in moving the charge.

The same e"ect is produced on a conductor by an externally applied

electric $eld as seen in Figure 1B.35 (c). 0ince the $eld lines must be

perpendicular to the surface more of them are concentrated on the most

curved parts.

A''#i)a"i!ns !( C!nu)"!rs

Sn a very sharply curved surface such as shown in Figure 1B.36 the

charges are so concentrated at the point that the resulting electric $eld

can be great enough to remove them from the surface. This can be useful.

HD o m p I l a t I o n I n Q h y s I c s 4 * 4


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Jightning rods work best when they are most pointed. The large charges

created in storm clouds induce an opposite charge on a building that can

result in a lightning bolt hitting the building. The induced charge is bled

away continually by a lightning rod preventing the more dramatic

lightning strike.

Sf course we sometimes wish to prevent the transfer of charge rather

than to facilitate it. In that case the conductor should be very smooth and

have as large a radius of curvature as possible. (0ee Figure 1B.37.)

0mooth surfaces are used on high!voltage transmission lines for example

to avoid leakage of charge into the air.

%nother device that makes use of some of these principles is a Faraa*

)age. This is a metal shield that encloses a volume. %ll electrical charges

will reside on the outside surface of this shield and there will be no

electrical $eld inside.

% 'araday cage is used to prohibit stray electrical $elds in the

environment from interfering with sensitive measurements such as

the electrical signals inside a nerve cell.

Furing electrical storms if you are driving a car it is best to stay

inside the car as its metal body acts as a 'araday cage with ,ero

electrical $eld inside. If in the vicinity of a lightning strike its e"ect

is felt on the outside of the car and the inside is una"ected provided

you remain totally inside. This is also true if an active ([hot\)

H o m p I l a t I o n I n Q h y s I c s 4 * 4


96/113

electrical wire was broken (in a storm or an accident) and fell on

your car.

Figure 1B.36 % very pointed conductor has a large charge concentration at the

point. The electric $eld is very strong at the point and can exert a force large

enough to transfer charge on or o" the conductor. Jightning rods are used to

prevent the buildup of large excess charges on structures and thus are pointed.

C!nu)"!rs an Insu#a"!rs

Qolari,ation is the separation of positive and negative charges in a neutral

ob-ect.

H o m p I l a t I o n I n Q h y s I c s 4 * 4
http://var/www/apps/conversion/tmp/scratch_4/HYPERLINK%23page635http://var/www/apps/conversion/tmp/scratch_4/HYPERLINK%23page635


97/113

% conductor is a substance that allows charge to 3ow freely through its

atomic structure.

%n insulator holds charge within its atomic structure.

Sb-ects with like charges repel each other while those with unlike charges

attract each other.

% conducting ob-ect is said to be grounded if it is connected to the

Earth through a conductor. Brounding allows transfer of charge to and

from the earths large reservoir.

Sb-ects can be charged by contact with another charged ob-ect and obtain

the same sign charge.

If an ob-ect is temporarily grounded it can be charged by induction and

obtains the opposite sign charge.

Qolari,ed ob-ects have their positive and negative charges concentrated in

di"erent areas giving them a non!symmetrical charge.

Qolar molecules have an inherent separation of charge.

C!u#!+,s La$

H/ o m p I l a t I o n I n Q h y s I c s 4 * 4


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'renchman harles oulomb was the $rst to publish the mathematical

equation that describes the electrostatic force between two ob-ects.

oulombs law gives the magnitude of the force between point charges. It

is

where 9*and 94are two point charges separated by a distance r and k

H.


99/113

The electrostatic force $eld surrounding a charged ob-ect extends out into

space in all directions.

The electrostatic force exerted by a point charge on a test charge at a

distance r depends on the charge of both charges as well as the

distance between the two.

The electric $eld Eis de$ned to be

E 9,F

where Fis the oulomb or electrostatic force exerted on a small positivetest charge 9. Ehas units of 5+.

The magnitude of the electric $eld Ecreated by a point charge >is

E k`r>4`.

where ris the distance from >. The electric $eld Eis a vector and $elds

due to multiple charges add like vectors.

E#e)"ri) Fie# Lines Mu#"i'#e C2arges

Frawings of electric $eld lines are useful visual tools. The properties of

electric $eld lines for any charge distribution are that8

'ield lines must begin on positive charges and terminate on negative

charges or at in$nity in the hypothetical case of isolated charges.

The number of $eld lines leaving a positive charge or entering a negative

charge is proportional to the magnitude of the charge.

HH o m p I l a t I o n I n Q h y s I c s 4 * 4


100/113

The strength of the $eld is proportional to the closeness of the $eld lines

2more precisely it is proportional to the number of lines per unit area

perpendicular to the lines.

The direction of the electric $eld is tangent to the $eld line at any point in

space.

'ield lines can never cross.

E#e)"ri) F!r)es in /i!#!g*

&any molecules in living organisms such as F5% carry a charge.

%n uneven distribution of the positive and negative charges within a polar

molecule produces a dipole.

The e"ect of a oulomb $eld generated by a charged ob-ect may be

reduced or blocked by other nearby charged ob-ects.

>iological systems contain water and because water molecules are polar

they have a strong e"ect on other molecules in living systems.

*


101/113

C!nu)"!rs an E#e)"ri) Fie#s in S"a"i) Eui#i,riu+

% conductor allows free charges to move about within it.

The electrical forces around a conductor will cause free charges to move

around inside the conductor until static equilibrium is reached.

%ny excess charge will collect along the surface of a conductor.

onductors with sharp corners or points will collect more charge at those

points.

% lightning rod is a conductor with sharply pointed ends that collect

excess charge on the building caused by an electrical storm and allow it

to dissipate back into the air.

Electrical storms result when the electrical $eld of Earths surface in

certain locations becomes more strongly charged due to changes in the

insulating e"ect of the air.

% 'araday cage acts like a shield around an ob-ect preventing electric

charge from penetrating inside.

*


102/113

A''#i)a"i!ns !( E#e)"r!s"a"i)s

Electrostatics is the study of electric $elds in static equilibrium.

In addition to research using equipment such as a an de Braa"

generator many practical applications of electrostatics exist

including photocopiers laser printers ink!-et printers and electrostatic

air $lters.

DEFINITION OF TERMS

(Magnetism)

De-ni"i!n !( Ter+s

a+'#i"ue +!u#a"i!n :AM; a method for placing information on

electromagnetic waves by modulating the amplitude of a carrier

wave with an audio signal resulting in a wave with constant

frequency but varying amplitude

*


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a+'#i"ue the height or magnitude of an electromagnetic wave

)arrier $ave an electromagnetic wave that carries a signal by

modulation of its amplitude or frequency

e#e)"ri) -e# #ines a pattern of imaginary lines that extend

between an electric source and charged ob-ects in the surrounding

area with arrowspointed away from positively charged ob-ects and

toward negatively charged ob-ects. The more lines in the pattern the

stronger the electric $eld in that region

e#e)"ri) -e# s"reng"2 the magnitude of the electric $eld denoted

7!$eld

e#e)"ri) -e# a vector quantity (E)1 the lines of electric force per

unit charge moving radially outward from a positive charge and intoward anegative charge

e#e)"r!+agne"i) s'e)"ru+ the full range of wavelengths or

frequencies of electromagnetic radiation

e#e)"r!+agne"i) $aves radiation in the form of waves of electric

and magnetic energy

e#e)"r!+!"ive (!r)e :e+(; energy produced per unit charge

drawn from a source that produces an electrical current

e"re+e#* #!$ (reuen)* :ELF; electromagnetic radiation with

wavelengths usually in the range of < to ;


104/113

(reuen)* +!u#a"i!n :FM; a method of placing information on

electromagnetic waves by modulating the frequency of a carrier

wave with anaudio signal producing a wave of constant amplitude

but varying frequency

(reuen)* the number of complete wave cycles (up!down!up)

passing a given point within one second (cycles+second)

ga++a ra* ( ray)1 extremely high frequency electromagnetic

radiation emitted by the nucleus of an atom either from naturalnuclear decay or induced nuclear processes in nuclear reactors and

weapons. The lower end of the !ray frequency range overlaps the

upper end of the ^!ray range but rays can have the highest

frequency of any electromagnetic radiation

2er" an 0I unit denoting the frequency of an electromagnetic

wave in cycles per second

in(rare raia"i!n :IR; a region of the electromagnetic spectrum

with a frequency range that extends from -ust below the red region

of the visible light spectrum up to the microwave region or from


105/113

+agne"i) -e# s"reng"2 the magnitude of the magnetic $eld

denoted!$eld

+agne"i) -e# a vector quantity (/)1 can be used to determine the

magnetic force on a moving charged particle

+ai+u+ -e# s"reng"2 the maximum amplitude an

electromagnetic wave can reach representing the maximum amount

of electric force and+ormagnetic 3ux that the wave can exert

+i)r!$aves electromagnetic waves with wavelengths in the range

from * mm to * m1 they can be produced by currents in macroscopic

circuitsand devices

!s)i##a"e to 3uctuate back and forth in a steady beat

RLC )ir)ui" an electric circuit that includes a resistor capacitor and

inductor

raar a common application of microwaves. Madar can determine

the distance to ob-ects as diverse as clouds and aircraft as well as

determinethe speed of a car or the intensity of a rainstorm rai! $aves electromagnetic waves with wavelengths in the range

from * mm to *


106/113

in a vacuum such as space the speed of light is a

constant ; x *


107/113

9ra* invisible penetrating form of very high frequency

electromagnetic radiation overlapping both the ultraviolet range and

the

C!u#!+, (!r)eanother term for the electrostatic force

C!u#!+, in"era)"i!n the interaction between two charged

particles generated by the oulomb forces they exert on one another

C!u#!+,s #a$ the mathematical equation calculating the

electrostatic force vector between two charged particles conductor8 a

material that allows electrons to move separately from their atomic

orbits

)!nu)"!ran ob-ect with properties that allow charges to move

about freely within it

i'!#ea molecules lack of symmetrical charge distribution causing

one side to be more positive and another to be more negative

e#e)"ri) )2argea physical property of an ob-ect that causes it to

be attracted toward or repelled from another charged ob-ect1 each

charged ob-ect generates and is in3uenced by a force called an

electromagnetic force

e#e)"ri) -e# #inesa series of lines drawn from a point charge

representing the magnitude and direction of force exerted by that

charge electric $eld8 a three!dimensional map of the electric force

extended out into space from a point charge

e#e)"r!+agne"i) (!r)e one of the four fundamental forces of

nature1 the electromagnetic force consists of static electricity

moving electricity and magnetism

e#e)"r!n a particle orbiting the nucleus of an atom and carrying the

smallest unit of negative charge

e#e)"r!s"a"i) eui#i,riu+ an electrostatically balanced state inwhich all free electrical charges have stopped moving about

*


108/113

electrostatic force8 the amount and direction of attraction or

repulsion between two charged bodies

e#e)"r!s"a"i) 're)i'i"a"!rs$lters that apply charges to particles

in the air then attract those charges to a $lter removing them from

the airstream electrostatic repulsion8 the phenomenon of two ob-ects

with like charges repelling each other

e#e)"r!s"a"i)sthe study of electric forces that are static or slow!

moving

Faraa* )agea metal shield which prevents electric charge from

penetrating its surface

-e#a map of the amount and direction of a force acting on other

ob-ects extending out into space

(ree )2argean electrical charge (either positive or negative) which

can move about separately from its base molecule

(ree e#e)"r!nan electron that is free to move away from its atomic

orbit

gr!unewhen a conductor is connected to the Earth allowing

charge to freely 3ow to and from Earths unlimited reservoir

grounded8 connected to the ground with a conductor so that charge

3ows freely to and from the Earth to the grounded ob-ect induction8

the process by which an electrically charged ob-ect brought near a

neutral ob-ect creates a charge in that ob-ect

in


109/113

#aser 'rin"eruses a laser to create a photoconductive image on a

drum which attracts dry ink particles that are then rolled onto a

sheet of paper to print a high!quality copy of the image

#a$ !( )!nserva"i!n !( )2argestates that whenever a charge is

created an equal amount of charge with the opposite sign is created

simultaneously

'2!"!)!nu)"!ra substance that is an insulator until it is exposed

to light when it becomes a conductor point charge8 % charged

particle designated generating an electric $eld

'!#ar +!#e)u#e8 a molecule with an asymmetrical distribution of

positive and negative charge polari,ation8 slight shifting of positive

and negative charges to opposite sides of an atom or molecule

'!#arie a state in which the positive and negative charges within

an ob-ect have collected in separate locations

'r!"!na particle in the nucleus of an atom and carrying a positive

charge equal in magnitude and opposite in sign to the amount of

negative charge carried by an electron

s)reening the dilution or blocking of an electrostatic force on a

charged ob-ect by the presence of other charges nearby static

electricity8 a buildup of electric charge on the surface of an ob-ect

"es" )2arge8 % particle (designated q ) with either a positive or

negative charge set down within an electric $eld generated by a

point charge

Van e Hraa= genera"!r8 a machine that produces a large amount

of excess charge used for experiments with high voltage vector

addition8 mathematical combination of two or more vectors

including their magnitudes directions and positions vector8 aquantity with both magnitude and direction

*


110/113

er!gra'2*8 a dry copying process based on electrostatics

)a'a)i"an)e amount of charge stored per unit volt capacitor8 a

device that stores electric charge

e-,ri##a"!ra machine used to provide an electrical shock to a

heart attack victim6s heart in order to restore the heart6s normal

rhythmic pattern dielectric strength8 the maximum electric $eld

above which an insulating material begins to break down and

conduct

ie#e)"ri)an insulating material

e#e)"ri) '!"en"ia#potential energy per unit charge

e#e)"r!n v!#" the energy given to a fundamental charge

accelerated through a potential di"erence of one volt equipotential

line8 a line along which the electric potential is constant

gr!uning$xing a conductor at ,ero volts by connecting it to the

earth or ground

+e)2ani)a# energ*8 sum of the kinetic energy and potential energy

of a system1 this sum is a constant

'ara##e# '#a"e )a'a)i"!r8 two identical conducting plates separated

by a distance

'!#ar +!#e)u#ea molecule with inherent separation of charge

'!"en"ia# i=eren)e :!r v!#"age;change in potential energy of a

charge moved from one point to another divided by the charge1

units of potential di"erence are -oules per coulomb known as volt

s)a#arphysical quantity with magnitude but no direction

ve)"!rphysical quantity with both magnitude and direction

**< o m p I l a t I o n I n Q h y s I c s 4 * 4


111/113

AC )urren"8 current that 3uctuates sinusoidally with time expressed

as I I< sin 4Aft where I is the current at time t I< is the peak

current and f is the frequency in hert,

AC v!#"age8 voltage that 3uctuates sinusoidally with time

expressed as < sin 4Aft where is the voltage at time t < is

the peak voltage and f is the frequency in hert,

a#"erna"ing )urren"8 (%) the 3ow of electric charge that

periodically reverses direction

a+'ere(amp) the 0I unit for current1 * % * +s

,i!e#e)"ri)i"* electrical e"ects in and created by biological

systems direct current8 (F) the 3ow of electric charge in only one

direction

ri(" ve#!)i"*8 the average velocity at which free charges 3ow in

response to an electric $eld electric current8 the rate at which charge

3ows I +t

e#e)"ri) '!$er8 the rate at which electrical energy is su

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Compilation in Physics 212

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