Chapter Thirty Four Notes:
Electric Current
Electric current is related to the voltage that produced it and the resistance
that opposed it.
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In household circuits, the energy is supplied by a local
utility company which is responsible for making sure that the
hot and the neutral plates within the circuit panel box of your
home always have an electric potential difference of about 110
Volts to 120 Volts (in the United States). In typical lab activities,
an electrochemical cell or group of cells (i.e., a battery) is used
to establish an electric potential difference across the
two ends of the external circuit of about 1.5 Volts (a
single cell) or 4.5 Volts (three cells in a pack).
Analogies are often made between an electric circuit
and the water circuit at a water park or a roller
coaster ride at an amusement park. In all three cases, there is
something which is moving through a complete loop - that is,
through a circuit. And in all three cases, it is essential that the
circuit include a section where energy is put into the water, the
coaster car or the charge in order to move it uphill against its
natural direction of motion from a low potential energy to a
high potential energy.
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A water park ride has a water pump which pumps the water from
ground level to the top of the slide. A roller coaster ride has a
motor-driven chain that carries the train of coaster cars from
ground level to the top of the first drop. And an electric circuit has
an electrochemical cell, battery (group of cells) or some other energy
supply that moves the charge from ground level (the negative
terminal) to the positive terminal. By constantly supplying the energy
to move the charge from the low energy, low potential terminal to
the high energy, high potential terminal, a continuous flow of charge
can be maintained.
By establishing this difference in electric potential, charge is
able to flow downhill through the external circuit. This motion of the
charge is natural and does not require energy. Like the movement of
water at a water park or a roller coaster car at an amusement park,
the downhill motion is natural and occurs without the need for
energy from an external source. It is the difference in potential whether gravitational potential or electric potential - which causes
the water, the coaster car and the charge to move. This potential
difference requires the input of energy from an external source. In
the case of an electric circuit, one of the two requirements to
establish an electric circuit is an energy source.
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In conclusion, there are two requirements which must be met in
order to establish an electric circuit. The requirements are
There must be an energy supply capable doing work on charge to
move it from a low energy location to a high energy location and
thus establish an electric potential difference across the two ends of
the external circuit.
There must be a closed conducting loop in the external circuit which
stretches from the high potential, positive terminal to the low
potential, negative terminal.
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If the two requirements of an electric circuit are met, then charge
will flow through the external circuit. It is said that there is a current
- a flow of charge. Using the word current in this context is to
simply use it to say that something is happening in the wires charge is moving. Yet current is a physical quantity which can be
measured and expressed numerically. As a physical quantity, current
is the rate at which charge flows past a point on a circuit. As
depicted in the diagram below, the current in a circuit can be
determined if the quantity of charge Q passing through a cross
section of a wire in a time t can be measured. The current is simply
the ratio of the quantity of charge and time.
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Current is a rate quantity. There are several rate quantities in
physics. For instance, velocity is a rate quantity - the rate at which
an object changes its position. Mathematically, velocity is the
position change per time ratio. Acceleration is a rate quantity - the
rate at which an object changes its velocity. Mathematically,
acceleration is the velocity change per time ratio. And power is a
rate quantity - the rate at which work is done on an object.
Mathematically, power is the work per time ratio. In every case of a
rate quantity, the mathematical equation involves some quantity
over time. Thus, current as a rate quantity would be expressed
mathematically as
Note that the equation above uses the symbol I to represent the
quantity current.
As is the usual case, when a quantity is introduced in Physics, the
standard metric unit used to express that quantity are introduced as
well. The standard metric unit for current is the ampere. Ampere is
often shortened to Amp and is abbreviated by the unit symbol A. A
current of 1 ampere means that there is 1 coulomb of charge
passing through a cross section of a wire every 1 second.
 1 ampere = 1 coulomb / 1 second
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To test your understanding, determine the current for the following
two situations. Note that some extraneous information is given in
each situation. Click the Check Answer button to see if you are
correct.
A 2 mm long cross section of wire is isolated A 1 mm long cross section of wire is isolated
and 20 C of charge are determined to pass
through it in 40 s.
and 2 C of charge are determined to pass
through it in 0.5 s.
I = _____ Ampere
I = _____ Ampere
Check Answer
Check Answer
A 2 mm long cross section
of wire is isolated and 20 C
of charge are determined to
pass through it in 40 s.
A 1 mm long cross section
of wire is isolated and 2 C of
charge are determined to
pass through it in 0.5 s.
Answer: I = Q / t
= (20 C) / (40 s) = 0.50 Ampere
Answer: I = Q / t
= (2 C) / (0.5 s) = 4.0 Ampere
• Any
elastic material can transmit
sound.
• Steel is a very good conductor of
sound.
• Water is not as good a conductor
as steel, but is better than air.
• Air is a poor conductor of sound
The Speed of Sound
A sound wave is a pressure disturbance which travels through a
medium by means of particle-to-particle interaction. As one particle
becomes disturbed, it exerts a force on the next adjacent particle,
thus disturbing that particle from rest and transporting the energy
through the medium. Like any wave, the speed of a sound wave
refers to how fast the disturbance is passed from particle to particle.
While frequency refers to the number of vibrations which an
individual particle makes per unit of time, speed refers to the
distance which the disturbance travels per unit of time. Always be
cautious to distinguish between the two often confused quantities of
speed (how fast...) and frequency (how often...).
Since the speed of a wave is defined as the distance which a point on
a wave (such as a compression or a rarefaction) travels per unit of
time, it is often expressed in units of meters/second (abbreviated
m/s). In equation form, this is
 speed = distance/time
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The faster a sound wave travels, the more distance it will cover in
the same period of time. If a sound wave is observed to travel a
distance of 700 meters in 2 seconds, then the speed of the wave
would be 350 m/s. A slower wave would cover less distance perhaps 660 meters - in the same time period of 2 seconds and
thus have a speed of 330 m/s. Faster waves cover more distance in
the same period of time.
 Factors Affecting Wave Speed
The speed of any wave depends upon the properties of the medium
through which the wave is traveling. Typically there are two essential
types of properties which affect wave speed - inertial properties and
elastic properties. Elastic properties are those properties related to
the tendency of a material to maintain its shape and not deform
whenever a force or stress is applied to it. A material such as steel
will experience a very small deformation of shape (and dimension)
when a stress is applied to it. Steel is a rigid material with a high
elasticity. On the other hand, a material such as a rubber band is
highly flexible; when a force is applied to stretch the rubber band, it
deforms or changes its shape readily. A small stress on the rubber
band causes a large deformation.
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Steel is considered to be a stiff or rigid material, whereas a rubber
band is considered a flexible material. At the particle level, a stiff or
rigid material is characterized by atoms and/or molecules with
strong attractions for each other. When a force is applied in an
attempt to stretch or deform the material, its strong particle
interactions prevent this deformation and help the material maintain
its shape. Rigid materials such as steel are considered to have a high
elasticity. (Elastic modulus is the technical term). The phase of
matter has a tremendous impact upon the elastic properties of the
medium. In general, solids have the strongest interactions between
particles, followed by liquids and then gases. For this reason,
longitudinal sound waves travel faster in solids than they do in
liquids than they do in gases. Even though the inertial factor may
favor gases, the elastic factor has a greater influence on the speed
(v) of a wave, thus yielding this general pattern:
 vsolids > vliquids > vgases
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Inertial properties are those properties related to the material's
tendency to be sluggish to changes in it's state of motion. The
density of a medium is an example of an inertial property.
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The greater the inertia (i.e., mass density) of individual particles of
the medium, the less responsive they will be to the interactions
between neighboring particles and the slower that the wave will be.
As stated above, sound waves travel faster in solids than they do in
liquids than they do in gases. However, within a single phase of
matter, the inertial property of density tends to be the property
which has a greatest impact upon the speed of sound. A sound wave
will travel faster in a less dense material than a more dense material.
Thus, a sound wave will travel nearly three times faster in Helium as
it will in air. This is mostly due to the lower mass of Helium particles
as compared to air particles.
The speed of a sound wave in air depends upon the properties of the
air, namely the temperature and the pressure. The pressure of air
(like any gas) will affect the mass density of the air (an inertial
property) and the temperature will affect the strength of the particle
interactions (an elastic property). At normal atmospheric pressure,
the temperature dependence of the speed of a sound wave through
air is approximated by the following equation:
 v = 331 m/s + (0.6 m/s/C)•T
where T is the temperature of the air in degrees Celsius. Using this
equation to determine the speed of a sound wave in air at a
temperature of 20 degrees Celsius yields the following solution.
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• v = 331 m/s + (0.6 m/s/C)•T
v = 331 m/s + (0.6 m/s/C)•(20 C)
 v = 331 m/s + 12 m/s
 v = 343 m/s
While the intensity of a sound is a very objective quantity which can
be measured with sensitive instrumentation, the loudness of a sound
is more of a subjective response which will vary with a number of
factors. The same sound will not be perceived to have the same
loudness to all individuals. Age is one factor which effects the human
ear's response to a sound. Quite obviously, your grandparents do not
hear like they used to. The same intensity sound would not be
perceived to have the same loudness to them as it would to you.
Furthermore, two sounds with the same intensity but different
frequencies will not be perceived to have the same loudness. Because
of the human ear's tendency to amplify sounds having frequencies in
the range from 1000 Hz to 5000 Hz, sounds with these intensities
seem louder to the human ear. Despite the distinction between
intensity and loudness, it is safe to state that the more intense
sounds will be perceived to be the loudest sounds.
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Nearly all objects, when hit or struck or plucked or strummed or
somehow disturbed, will vibrate. If you drop a meter stick or pencil
on the floor, it will begin to vibrate. If you pluck a guitar string, it
will begin to vibrate. If you blow over the top of a pop bottle, the air
inside will vibrate. When each of these objects vibrate, they tend to
vibrate at a particular frequency or a set of frequencies. The
frequency or frequencies at which an object tends to vibrate with
when hit, struck, plucked, strummed or somehow disturbed is
known as the natural frequency of the object. If the amplitude of the
vibrations are large enough and if natural frequency is within the
human frequency range, then the vibrating object will produce
sound waves which are audible.
All objects have a natural frequency or set of frequencies at which
they vibrate. The quality or timbre of the sound produced by a
vibrating object is dependent upon the natural frequencies of the
sound waves produced by the objects.
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Some objects tend to vibrate at a single frequency and they
are often said to produce a pure tone. A flute tends to vibrate
at a single frequency, producing a very pure tone. Other
objects vibrate and produce more complex waves with a set
of frequencies which have a whole number mathematical
relationship between them; these are said to produce a rich
sound. A tuba tends to vibrate at a set of frequencies which
are mathematically related by whole number ratios; it
produces a rich tone. Still other objects will vibrate at a set of
multiple frequencies which have no simple mathematical
relationship between them. These objects are not musical at
all and the sounds which they create could be described as
noise. When a meter stick or pencil is dropped on the floor, it
vibrates with a number of frequencies, producing a complex
sound wave which is clanky and noisy.
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If you were to take a guitar string and stretch it to a given length
and a given tightness and have a friend pluck it, you would hear a
noise; but the noise would not even be close in comparison to the
loudness produced by an acoustic guitar. On the other hand, if the
string is attached to the sound box of the guitar, the vibrating string
is capable of forcing the sound box into vibrating at that same
natural frequency. The sound box in turn forces air particles inside
the box into vibrational motion at the same natural frequency as the
string. The entire system (string, guitar, and enclosed air) begins
vibrating and forces surrounding air particles into vibrational
motion. The tendency of one object to force another adjoining or
interconnected object into vibrational motion is referred to as a
forced vibration. In the case of the guitar string mounted to the
sound box, the fact that the surface area of the sound box is greater
than the surface area of the string, means that more surrounding air
particles will be forced into vibration. This causes an increase in the
amplitude and thus loudness of the sound.
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This same principle of a forced vibration is often demonstrated in a
Physics classroom using a tuning fork. If the tuning fork is held in
your hand and hit with a rubber mallet, a sound is produced as the
tines of the tuning fork set surrounding air particles into vibrational
motion. The sound produced by the tuning fork is barely audible to
students in the back rows of the room. However, if the tuning fork is
set upon the whiteboard panel or the glass panel of the overhead
projector, the panel begins vibrating at the same natural frequency
of the tuning fork. The tuning fork forces surrounding glass (or
vinyl) particles into vibrational motion. The vibrating whiteboard or
overhead projector panel in turn forces surrounding air particles into
vibrational motion and the result is an increase in the amplitude and
thus loudness of the sound. This principle of forced vibration
explains why demonstration tuning forks are mounted on a sound
box, why a commercial music box mechanism is mounted on a
sounding board, why a guitar utilizes a sound box,
and why a piano string is attached to a sounding
board. A louder sound is always produced when
an accompanying object of greater surface area
is forced into vibration at the same natural frequency.
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Now consider a related situation which resembles another common
Physics demonstration. Suppose that a tuning fork is mounted on a
sound box and set upon the table; and suppose a second tuning
fork/sound box system having the same natural frequency (say 256
Hz) is placed on the table near the first system. Neither of the tuning
forks is vibrating. Suppose the first tuning fork is struck with a
rubber mallet and the tines begin vibrating at its natural frequency 256 Hz. These vibrations set its sound box and the air inside the
sound box vibrating at the same natural frequency of 256 Hz.
Surrounding air particles are set into vibrational motion at the same
natural frequency of 256 Hz and every student in the classroom
hears the sound. Then the tines of the tuning fork are grabbed to
prevent their vibration and remarkably the sound of 256 Hz is still
being heard. Only now the sound is being
produced by the second tuning fork - the
one which wasn't hit with the mallet. Amazing!!
The demonstration is often repeated to
assure that the same surprising results are
observed. They are! What is happening?
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In this demonstration, one tuning fork forces another tuning fork
into vibrational motion at the same natural frequency. The two forks
are connected by the surrounding air particles. As the air particles
surrounding the first fork (and its connected sound box) begin
vibrating, the pressure waves which it creates begin to impinge at a
periodic and regular rate of 256 Hz upon the second tuning fork
(and its connected sound box). The energy carried by this sound
wave through the air is tuned to the frequency of the second tuning
fork. Since the incoming sound waves share the same natural
frequency as the second tuning fork, the tuning fork easily begins
vibrating at its natural frequency. This is an example of resonance when one object vibrating at the same natural frequency of a second
object forces that second object into vibrational motion.
The result of resonance is always a large vibration. Regardless of the
vibrating system, if resonance occurs, a large vibration results.
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Wave interference is the phenomenon which occurs when two waves
meet while traveling along the same medium. The interference of
waves causes the medium to take on a shape which results from the
net effect of the two individual waves upon the particles of the
medium. As mentioned in the last chapter, if two upward displaced
pulses having the same shape meet up with one another while
traveling in opposite directions along a medium, the medium will
take on the shape of an upward displaced pulse with twice the
amplitude of the two interfering pulses. This type of interference is
known as constructive interference. If an upward displaced pulse
and a downward displaced pulse having the same shape meet up
with one another while traveling in opposite directions along a
medium, the two pulses will cancel each other's effect upon the
displacement of the medium and the medium will assume the
equilibrium position. This type of interference is known as
destructive interference.
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The diagrams below show two waves - one is blue and the other is
red - interfering in such a way to produce a resultant shape in a
medium; the resultant is shown in green. In two cases (on the left
and in the middle), constructive interference occurs and in the third
case (on the far right, destructive interference occurs.
But how can sound waves which do not possess upward and
downward displacements interfere constructively and destructively?
Sound is a pressure wave which consists of compressions and
rarefactions. As a compression passes through a section of a
medium, it tends to pull particles together into a small region of
space, thus creating a high pressure region. And as a rarefaction
passes through a section of a medium, it tends to push particles
apart, thus creating a low pressure region. The interference of sound
waves causes the particles of the medium to behave in a manner
that reflects the net effect of the two individual waves upon the
particles.
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The animation below shows two sound waves interfering
constructively in order to produce very large oscillations in pressure
at a variety of anti-nodal locations. Note that compressions are
labeled with a C and rarefactions are labeled with an R.
Now if two sound waves interfere at a given location in such a way
that the compression of one wave meets up with the rarefaction of a
second wave, destructive interference results. The net effect of a
compression (which pushes particles together) and a rarefaction
(which pulls particles apart) upon the particles in a given region of
the medium is to not even cause a displacement of the particles. The
tendency of the compression to push particles together is canceled
by the tendency of the rarefactions to pull particles apart; the
particles would remain at their rest position as though there wasn't
even a disturbance passing through them. This is a form of
destructive interference.
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Now if a particular location along the medium repeatedly
experiences the interference of a compression and rarefaction
followed up by the interference of a rarefaction and a compression,
then the two sound waves will continually cancel each other and no
sound is heard. The absence of sound is the result of the particles
remaining at rest and behaving as though there were no disturbance
passing through it. Amazingly, in a situation such as this, two sound
waves would combine to produce no sound. As mentioned in in the
last chapter when talking about standing waves, locations along the
medium where destructive interference continually occurs are known
as nodes.
 Two Source Sound Interference
A popular Physics demonstration involves the interference of two
sound waves from two speakers. The speakers are set approximately
1 meter apart and produced identical tones. The two sound waves
traveled through the air in front of the speakers, spreading our
through the room in spherical fashion. A snapshot in time of the
appearance of these waves is shown in the diagram on the next
page.
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In the diagram, the compressions of a wavefront are represented by
a thick line and the rarefactions are represented by thin lines. These
two waves interfere in such a manner as to produce locations of
some loud sounds and other locations of no sound. Of course the
loud sounds are heard at locations where compressions meet
compressions or rarefactions meet rarefactions and the "no sound"
locations appear wherever the compressions of one of the waves
meet the rarefactions of the other wave. If you were to plug one ear
and turn the other ear towards the place of the speakers and then
slowly walk across the room parallel to the plane of the speakers,
then you would encounter an amazing phenomenon. You would
alternatively hear loud sounds as you approached anti-nodal
locations and virtually no sound as you approached nodal locations.
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Destructive interference of sound waves becomes an important issue
in the design of concert halls and auditoriums. The rooms must be
designed in such as way as to reduce the amount of destructive
interference. Interference can occur as the result of sound from two
speakers meeting at the same location as well as the result of sound
from a speaker meeting with sound reflected off the walls and
ceilings. If the sound arrives at a given location such that
compressions meet rarefactions, then destructive interference will
occur resulting in a reduction in the loudness of the sound at that
location. One means of reducing the severity of destructive
interference is by the design of walls, ceilings, and baffles that serve
to absorb sound rather than reflect it.
The destructive interference of sound waves can also be used
advantageously in noise reduction systems. Ear phones have been
produced which can be used by factory and construction workers to
reduce the noise levels on their jobs. Such ear phones capture sound
from the environment and use computer technology to produce a
second sound wave which one-half cycle out of phase. The
combination of these two sound waves within the headset will result
in destructive interference and thus reduce a worker's exposure to
loud noise.
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A final application of physics to the
world of music pertains to the topic
of beats. Beats are the periodic and
repeating fluctuations heard in the
intensity of a sound when two sound
waves of very similar frequencies
interfere with one another. The
diagram illustrates the wave
interference pattern resulting from two waves (drawn in red and blue)
with very similar frequencies.
A beat pattern is characterized by a
wave whose amplitude is changing at a regular rate. Observe that the
beat pattern (drawn in green) repeatedly oscillates from zero amplitude
to a large amplitude, back to zero amplitude throughout the pattern.
Points of constructive interference (C.I.) and destructive interference
(D.I.) are labeled on the diagram. When constructive interference
occurs between two crests or two troughs, a loud sound is heard. This
corresponds to a peak on the beat pattern (drawn in green).
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When destructive interference between a crest and a trough occurs,
no sound is heard; this corresponds to a point of no displacement
on the beat pattern. Since there is a clear relationship between the
amplitude and the loudness, this beat pattern would be consistent
with a wave which varies in volume at a regular rate.
A piano tuner frequently utilizes the phenomenon of beats to tune a
piano string. She will pluck the string and tap a tuning fork at the
same time. If the two sound sources - the piano string and the
tuning fork - produce detectable beats then their frequencies are
not identical. She will then adjust the tension of the piano string and
repeat the process until the beats can no longer be heard. As the
piano string becomes more in tune with the tuning fork, the beat
frequency will be reduced and approach 0 Hz. When beats are no
longer heard, the piano string is tuned to the tuning fork; that is,
they play the same frequency. The process allows a piano tuner to
match the strings' frequency to the frequency of a standardized set
of tuning forks.
 Important Note: Many of the previous diagrams represent a sound wave by a sine wave. Such
a wave more closely resembles a transverse wave and may mislead people into thinking that
sound is a transverse wave. Sound is not a transverse wave, but rather a longitudinal
wave. Nonetheless, the variations in pressure with time take on the pattern of a sine wave and
thus a sine wave is often used to represent the pressure-time features of a sound wave.