62141 Study Guide

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PHYSICS 141 WINTER 2012

Mario Salvador www.windsorsos.com Page 1

CHAPTER 15S I M P L E H A R M O N I C M O T I O N

Oscillatory motion is motion that follows a pattern: repeating itself at regular intervals. It repeats itself after

one complete oscillation or cycle. There are two important properties of periodic motion: frequency and

period. Frequency, , is the number of cycles per second, measured in hertz, , or . The period, , isthe time required to complete a cycle, measured in seconds. The two are related by the formula:

Oscillatory motion that can be modeled by a sinusoidal function is referred to as simple harmonic motion.

We often use the cosine function to model the displacement as a function of time:

In the above equation, is the displacement at time . It is measured with respect to the equilibrium point

which is taken to be zero. The amplitude of motion is given by . The phase of motion is given by

with referring to the angular frequency measured in /. The phase angle, , allows for

situations in which the objects initial position is not at the positive amplitude. Consider the following graph:

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Recall that in high school, was referred to as the horizontal stretch or compression. Similar to the formulasgiven then:

By differentiating the displacement function with respect to time, we obtain formulas for velocity and

acceleration:

The objects speed is always at a maximum when it crosses the equilibrium point. Since the range of the sine

function is between -1 and 1, we can see that the maximum speed is given by:

SECTION 15.3 | BLOCK-SPRING SYSTEMS

Notice above that the acceleration of an object in simple harmonic motion is proportional to the displacementbut opposite in direction:

This also occurs for a mass (or block) connected to a spring on a horizontal frictionless surface:

A block spring system may be the most fundamental model of simple harmonic motion.

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In the -direction, there is only the spring force, for being the displacement. By Newtons SecondLaw, . Since the system is in simple harmonic motion, we get: and obtain:

The frequency in the equation above is called the natural frequency. The equilibrium position of the spring

acts as the equilibrium position of the systems periodic motion for horizontal block-spring systems. But, thesame is not true for vertical block-spring systems since a gravity force opposes the spring force:

The system is instead at equilibrium when

in which

represents the displacement from the

equilibrium position of the spring . We can then determine that the natural angular frequency is:

As the speed and the position of the block change, so does the kinetic, spring potential and mechanical

energy. We can express these quantities as functions of time:

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SAMPLE PROBLEM 1: (2010 Problem Solving 1)A mass hangs from a spring and is set into verticaloscillation with a frequency of 1.50. When the mass is at the highest point of its motion, the spring isstretched by 3.00. A stopwatch is started during the motion and it is found that the mass is5.00below the equilibrium position and is moving upwards when 0.100. The equation of motion is

cos with

positive upwards. Calculate (a) the angular frequency, (b) the extension of

the spring when the mass is at the equilibrium position, (c) the amplitude of the motion, (d) the phaseangle, , and (e) the speed of the mass when 0.100.

SOLUTION 1a | The natural frequency is given as 1.50 so: 2 21.50 9.42/

SOLUTION 1b | Manipulating the formula above: 9.89.42 11.0

SOLUTION 1c | Since the string is stretched by 3.00 when the mass reaches its highest point, we caneasily see that the amplitude of motion, 11.0 3.00 8.00.

SOLUTION 1d | The problem indicates that 0.100 5.00. Using previous calculations5 8 cos9.420.100 0.942 2.244.04 1.303.10

Substituting both values into sin 9.420.08 sin0.942, we find that 3.10 gives a positive velocity. This fits the problem since the mass was moving upwards at that time.So 3.10.

SOLUTION 1e | Using the velocity calculated in 1d:0.100 0.592/

SECTION 15.6 | PENDULUMS

Consider a pendulum consisting of a mass, called a bob, at the end of a massless string of length . We referto this as a simple pendulum. For small oscillations, their periods can be found using:

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CHAPTER 16W A V E S

Although we studied sinusoidal functions in the last chapter, it is important to understand that we did not

actually study waves in nature. The graph of a system in simple harmonic motion, specifically its position as a

function of time, simply takes the shape of a wave. In this chapter, we look at a type of waves in nature

known as mechanical waves. These waves travel through a certain medium and are governed by Newtons

Laws.

Mechanical waves can be taken as two types, transverse and longitudinal. The elements of a transverse

waves oscillate in a direction perpendicular to the waves direction of travel. Perhaps the easiest example of

this is the waves that are generated by shaking the end of a long string or rope. We easily can visualize this

wave is in two dimensions: any piece on the string will have an horizontal and vertical position, and . Butthe vertical displacement of the string is also dependent on time. It is modeled with the formula:

,

The variable does not represent the spring constant but instead the angular wave number. It is measuredin radians per meter.

A wave can be broken into phases, the identical pieces the put together form the wave. The width of the

phase in meters is called the wavelength denoted by the variable, :

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The angular wave number and wavelength are related as follows:

We can now determine the speed of the wave:

And we could model the vertical displacement instead as:

, The indicates the waves direction of travel. For waves travelling in the positive-direction we use andfor waves travelling in the negative -direction we use a .The transverse speed is how fast an element oscillates perpendicularly to the direction of the wave. It can befound by taking the partial derivative of our vertical displacement function with respect to time:

,

Thespeed of a wave on a stretched string can be determined using the tension in the string denoted by and its linear density using the following formula:

Recall that the linear density of the spring can be found using its density and cross sectional area:

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In general, waves can be modeled mathematically using any function of the form thatsatisfies the wave equation:

This may serve as an alternative way to determine the wave speed.

SECTION 16.7 | ENERGY AND POWER OF TRAVELLING WAVES

For a wave travelling on a string, any particular element on the string will have a kinetic energy and potential

energy. Those energies are both at a maximum when the element is at 0. We particularly concernourselves with the rate at which this energy moves across the wire: the power of the wave. We can find this

power using the following formula:

SAMPLE PROBLEM 2: (2010 Problem Solving 1) The amplitude of a transverse waves travelling on astring is 5.75. The string has a diameter of 1.10 and a density of 5000/. The wave hasa frequency of 90.0, a wavelength of 60.0 and is travelling in the -direction. Assume that sin . Calculate (a) the tension in the string, (b) the angular frequency of the wave, (c)the angular wave number, (d) the displacement of a point with

1.50at

0.0600, (e) the

transverse velocity of a point with 1.50 at 0.0600 and (f) the energy transported persecond by the string. SOLUTION 2a| 900.650000.00055 13.9 SOLUTION 2b| 2 290 565/ SOLUTION 2c| . 10.5/ SOLUTION 2d| 5.75 sin10.51.50 5650.06 3.38 SOLUTION 2e|

cos 5655.75 cos10.51.50 5650.06 2.63/

SOLUTION 2f| 50000.00055900.61800.00575 1.35

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SECTION 16.10 | INTERFERENCE OF WAVES

It is possible to send more than one wave through a string simultaneously. In this case, the net vertical

displacement of the two waves at a certain horizontal position and time is simply the sum of the verticaldisplacements which would have been generated by each wave separately. This is referred to as the

principle of superposition. Mathematically, this means that the resultant wave,

, caused by two waves:

and on the same string is given by:, , ,

It is important to note that these two overlapping waves will not alter the way in which the individual waves

travel. This phenomenon is referred to as interference.

Now suppose that and are two sinusoidal waves of thesame wavelength, frequency, ampl itude anddirection of travelbut have a phase difference of

| |. Using trigonometric identities, the

resultant wave, with amplitude 2 cos is of the form:

,

For ; 0,2,4, the waves are said to be in phase and result in fully constructive interference.For

; 1,3,5, the waves are said to be out of phase resulting in fully destructive interference.

SECTION 16.13 | STANDING WAVES AND RESONANCE

For two sinusoidal waves of thesame wavelength, frequency and amplitude BUT travel ling in opposite

directions the resultant wave is referred to as a standing wave. The wave is of the form:

, The wave appears to be 'standing still no longer travelling in either direction. At certain positions callednodes, the string does not move transversally at all. Directly between these positions areantinodes, where

the string oscillates between a maxima and minima. Their positions are at:

, , , ,, , , , ,

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A standing wave can be generated by fixing one end of the string. When a certain pattern or oscillation

mode is created, the standing wave is said to be produced at resonance and the string is said to resonate at

certain resonant frequencies. The possible resonant frequencies are assigned aharmonic number andare given by:

,, ,

For example, 1 is called the first harmonic frequency or the fundamental frequency. For 2, the termsecond harmonic frequency is used or in trickier terms, the first harmonic above the fundamental. Thefundamental frequency always refers to the lowest possible frequency.

SAMPLE PROBLEM 3 : (Fundamentals of Physics 8th Edition Walker: 16-50) A rope, under a tension of200 and fixed at both ends, oscillates in a second harmonic standing wave pattern. The displacementof the rope is given by:

0.10sin/2sin12where 0 at one end of the rope, is in metres, and is in seconds. What are (a) the length of therope, (b) the speed of the waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a

third-harmonic standing wave pattern, what will be the period of oscillation?

SOLUTION 3a| Since it is oscillating at a second harmonic, 2. We can also see from the function that /2 and 12. So:1

2

1

2 4.0

SOLUTION 3b| 24/ SOLUTION 3c| . 1.4 SOLUTION 3d| 9.0 0.11

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CHAPTER 17S O U N D W A V E S

A longitudinal mechanical wave refers to waves in which elements oscillate in a direction parallel to thedirection of the wave. The most common example of this wave in nature are sound waves.

SECTION 17.3 | THE SPEED OF SOUND

The speed of sound changes through different mediums. In air, we generally assume it is343/. Still, if is the bulk modulus of the medium and the density, the speed of sound is given by:

/

SECTION 17.5 | INTERFERENCE

As explained in the last chapter, two waves of the same wavelength, frequency and amplitude will interfere

with each other and their resultant wave is dependent on the phase difference,. This phase difference isdirectly related to the path length difference, | |:

In particular with sound, we can find positions about two points sources at which sound is maximum or minimum:

, , , / .,., . ,/

SAMPLE PROBLEM 4: (2010 Problem Solving 1) Two loudspeakers atand are driven by a common oscillator. The distance is 6.00. Adetector moves along , parallel to, with being equidistant from and , and being perpendicular to both and . Thedetector indicates a minimum between and , the next minimum at ,and the next minimum at . The wavelength of the sound is0.800.Calculate (a) the distance and (b) the distance .

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SOLUTION 4a| Since is equidistant from and , there is a maximum at corresponding to 0. Since there is a minimum betweenand , the minimum at corresponds to:

3

2 3

2 | | 1.20 1.2

By Pythagorean Theorem, we also know that: || || |||| 1.2 6 ||Solving for , we find that 14.4

SOLUTION 4b| The minimum at corresponds to: 52 | | 2.00 || || 2.00

Let the to be . We then obtain that sin |||| and by cosine law on triangle:

| || || 2|||cos90 || 2 36 || 12||sinBy expanding and simplifying and , we can reduce this equation to: || 3||=8By Pythagorean Theorem on triangle :

|| 14.4 || 8 3|| 207.36 || || 6|| 17.92 0By quadratic formula: || 2.19

SECTION 17.6 | INTENSITY AND SOUND LEVEL

To measure the intensity of a sound wave is how powerful the sound is in an area. In more complicated terms,it is the rate of energy transfer through the area of a detecting surface. For point like sources of power,we can find the intensity of sound at a distance away to be:

For a sound wave of displacement amplitude :

Intensity is measured in /.

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It is often more convenient to measure a sound in terms of its sound level, which is a comparison of a sounds

intensity to a standard reference intensity 10/. Sound level, , is measured in decibels, ,and is found using the formula:

SAMPLE PROBLEM 5: (2010 Problem Solving 1) A point source ofsound waves is placed at point . At point, sound level is76.0. At point , the sound level is 71.0. 4.00and 12.0. Calculate (a) the distance and (b) the poweremitted by the source.

SOLUTION 5a| For

representing the sound level at

and

the

sound level at: 10 log ||||By Pythagorean Theorem: || || || and || || || and so:

76 71 5 10 log || |||| || 10log|| 144|| 16

10|| 1610 || 144 || 1441610101 6.57 SOLUTION 5b| We will consider the intensity at:

10. 4 4|| 10. 29.6SECTION 17.7 | SOUNDS AND MUSIC

Many instruments produce music by generating a standing wave pattern through a pipe, some closed at one

end and others open at both ends. With representing the speed of sound, the resonant frequencies in pipesof length

are as follows:

,, , , ,,

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Remember that the harmonic refers to the possible harmonic. For example, the second harmonicfrequency for a pipe open at both ends corresponds to 2 but for a pipe closed at one end, 3.When we hear two sounds of different frequencies we hear an average of the two frequencies along with avariation in the intensity of the combined sound at wavering beats. This phenomenon is used in tuninginstruments. The beat frequency heard is found by:

| | SAMPLE PROBLEM 6: (Fundamentals of Physics 8th Edition Walker: 17-48) A tube 1.20 m long is closed

at one end. The wire is 0.330 m long and has a mass of 9.60 g. It is fixed at both ends and oscillates in

its fundamental mode. By resonance, it sets the air column in the tube into oscillation at that columnsfundamental frequency. Find (a) that frequency and (b) the tension in the wire.

SOLUTION 6a| For a pipe closed at one end: 4 134341.2 71.5

SOLUTION 6b| For a wire resonating at its fundamental frequency: 2 1/2 4

Since the wire will oscillate at the same frequency as the pipe:

4

471.5

0.009600.330 64.7

SECTION 17.9 | THE DOPPLER EFFECT

The speed of sound changes in different situations. For example, since the speed of sound is dependent on

the air it passes through, a wind can help carry or slow down the speed of sound. As well, if the source of thesound or the listener/detector is moving, the sound frequency heard differs from its actual frequency. This

phenomenon is termed the Doppler effect. The detected frequency and the emitted frequency arerelated as follows:

The signs on the speed of the detector, , and the speed of the source, , change based on the directionthey are travelling relative to each other. If they move closer to the other, the detected frequency should

increase. If they move further away from the other, the detected frequency increases.

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SAMPLE PROBLEM 7: (2002 Test 1) A listener hears the siren from a moving police car, , which ismoving away from the listener with a speed of 20.0/. The listener is moving at 10.0/towards the car. There is a wind blowing with a speed of 10.0/ from to . The siren isemitting sound at a frequency of 800. The velocity of sound in air is 343/. Calculate thefrequency heard by the listener.

SOLUTION 7| Sincethe wind helps carry the sound from the source to the listener, 343 10 353.The source is moving away from the listener, so 20 must be +ve to decrease the detectedfrequency.

The listener is travelling towards to the source, so 10 must be 've to increase the detectedfrequency.

This produces:

800

3531035320 779

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CHAPTER 21E L E C T R I C C H A R G E

Objects in nature have a characteristic known as electric charge based on an imbalance between the

electrons on protons of the fundamental particles making up those objects. If there is a balance, the object is

said to be neutral. If there are more electrons, it is said to be negative and if there are more protons it is

said to be positive. Electric charge is a measure of that an imbalance and is quantized. This means that any

charge, , can be written using the formula:

The constant is called the elementary charge and 1.602 10. You can see that charge ismeasured in coulombs, . The imbalance, or number of electrons, is represented by.Charges exert a force on each other called electrostatic force. Charges with the same electrical sign repel

or push each other away. Charges with opposite electrical signs attract or pull each other closer. If the

charges are separated by a distance of , then this force can be calculated using Coulombs Law:

||||

The constant comes from the permittivity constant, 8.85 10:

.

It is worth repeating that forces are vector quantities and so the magnitude and direction of the net force on a

charge must be found by adding forces using components. Free body diagrams of the central object (theobject on which we are focusing on) are not required but are helpful. To calculate the direction of

electrostatic forces remember that: like charges repel and opposites attract.

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CHAPTER 22E L E C T R I C F I E L D

Take a point charge at some point. In certain situations, it becomes useful to calculate the electrostaticforce on a charged object independent of the charge of . We refer to this calculation as the electric fieldat. Since we are independent of the charge of :

The electric field due to a point charge can then be given by:

||

It is important to note that this is a vector quantity so direction is important. The direction is determined by

imagining q to be a positive point charge (attraction force from negative charges, repulsion for positivecharges). Furthermore, if we wish to calculate the electric field due to a system of point charges, thenvector

addition is required.

SAMPLE PROBLEM 8: Question about electrostatic force and electric field. SOLUTION 8|

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SECTION 22.9 | ELECTRIC DIPOLES

An electric dipole consists of two charges, both of charge magnitude , but opposite in direction. They are

separated by a distance . Every dipole is an associated quantity called the dipole moment denoted by .

The direction of is from the negative charge to the positive charge and its magnitude can be calculatedusing:

The electric field can be created by this dipole can be calculated at any point a distance from the centerof the dipole (refer to the figure below). The magnitude of the electric field at this point is given by:

We can create a uniform electric field by setting up two differently charged large plates, and . In the

figure below, the electric field points to the right. This means that a positively charged particle (which we

assume for the measurement of electric field) would travel towards the right and a negatively charged

particle towards the left.

To generate this field, the charge of R must be less than the charge ofL. Here are three examples:

i. 20 20 ii. 20 40iii. 20 10

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If we place a dipole in the electric field, then a torque (turning force) acts on the dipole to more align the

dipole moment with the field :

And the dipole will have a potential energy associated with it for different configurations:

The potential energy is a maximum when the dipole moment and field are anti-parallel and a minimum when

they are parallel (or aligned).

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CHAPTER 24E L E C T R I C P O T E N T I A L

Recall from Chapter 8, that with every force there comes an associated energy. In many cases, there is an

associated potential energy that changes with different configurations of a system. For example, the

gravitational potential energy is associated with the gravitational force and changes with respect to the

height of the object measured from some reference point. Another example would be the spring potential or

elastic potential energy associated with the spring force that changes with respect to the stretched or

compressed position of the object with reference to the springs natural length or equilibrium position.

Instead of calculating the potential energy at some point, it is often more useful to calculate thechange in

potential, , between two points or positions. We relate to the work done by the force, , and thework done applied,

, when moving the object from one point to another as follows:

The electrostatic force, , has an associated electric potential energy, sometimes denoted by . Themagnitude of the electric potential energy of the object is influenced by the magnitude of its charge, say.Direction is not applicable since energy is a scalar quantity. If we wish to calculate the electric potential

energy independent of the charge of , much like we do for the electric field, we are determining what isreferred to as the electric potential at point

. We denote the electric potential difference with

:

SECTION 24.5, 24.10 | ELECTRIC FIELDS AND ELECTRIC POTENTIAL

Recall: since the electrostatic force is a variable force, we calculate the work done by the force using:

So, to calculate the electric potential difference, we can divide by on both sides:

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If we are given the graph of the electric field function with respect to position (an vs curve), the formulaabove indicates that the potential difference is calculated using the negative area underneath the curve.

Consider a situation in which the electric field is uniform: the curve would be a horizontal line. This would

give:

We can thus calculate the uniform electric field according to:

This would also be the negative slope of a V vs x curve.SECTION 24.7| ELECTRIC POTENTIAL FROM POINT CHARGES

If we integrate the electric field from a point charge at some point , we would obtain the electric potentialat that point :

Since this is a scalar quantity, the electric potential from a finite system of charges is simply given by:

SECTION 24.8 | ELECTRIC POTENTIAL FROM AN ELECTRIC DIPOLE

The electric potential from a dipole at some point , a distance from the center of the dipole is given by:

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The angle is the angle made between the dipole moment and the radial line from the center to the point,.SECTION 24.11 | ELECTRIC POTENTIAL FROM A SYSTEM OF CHARGES

If we wish to calculate the electric potential energy of a system, we can do this by calculating the

work required to assemble the system. A manipulation of the formula above gives:

In this case, always refers to the charge that is being moved. will be zero as we are moving the chargefrom an initial position of infinity. is the potential from the charges already in place at the point where weare moving the new charge.

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CHAPTER 25C A P A C I T A N C E

In everyday devices, we often store charge in capacitors. One such device is the flash on a camera. When

flash is required, the battery cannot supply all the necessary energy in such a short amount of time. So, a

capacitor is charged, and once enough energy is available, the camera emits the burst of bright light.

Capacitors take on many different configurations but we will concern ourselves only with parallel plate

capacitors in this course. As the name indicates, it is made up by two parallel plates each brought to a

charge of magnitude . One plate is positively charged and the other is negatively charged. We separatethe two plates by a distance, . If the area of each of the plates is, then the capacitance, measured infarads, , is given by:

Sometimes, we insert a dielectric material in between the plates of a capacitor in order to increase its

capacitance. If we completely fill the region in between the plates, then capacitance becomes:

The charged plates are each equipotential surfaces and there is a potential difference between the two

plates. Instead of using the notation like in the previous chapter, we simply use. there is a potentialdifference between the two plates. We also call this quantity, voltage. The charge of the plates of a

capacitor can be given by:

To charge a capacitor, work from some external agent is required in order for electrons to flow. It is equal to

the electric potential energy of a charged capacitor which can be determined by:

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SECTION 25.4 CAPACITORS IN CIRCUITS

A capacitor may be hooked up by wires to a battery thus forming an electriccircuit, a device through which

charge can flow. Charge flows because a battery maintains a certain potential difference between across its

terminals. Several capacitors may be connected in the same circuit. There are two special types of

connections: parallel and series.

A parallel connection is formed when a wire essentially splits into different wires and later joins back

together. Capacitors on each of those wires are said to be in parallel. The charge along the original wire

will split across the subsequent wires, not necessarily equally. However, the voltage across each of

those wires remains the same as the original wire.

Capacitors in parallel can be reduced to an equivalent capacitor using:

A series connection is accomplished by connecting capacitors along the same wire. In this case, the charge is

the same for each capacitor but, the potential difference across the ends of the wires is split between

the connected capacitors. Again, it is not necessarily divided equally.

Capacitors in series can be reduced to an equivalent capacitor using:

Capacitors may also be connected without a battery. Prior to connecting, each of the capacitors often have a

certain charge already on their plates. To analyze the circuit, we can use two concepts:

(1) Charge is conserved. The net charge of an isolated system will always remain the same. For twocapacitors then:

(2) The connected capacitors will reach an equilibrium if they are not connected by a battery and the

voltage on each capacitor will be the same:

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CHAPTER 26C U R R E N T A N D R E S I S T A N C E

As studied in the previous chapter, when an electric potential is maintained across a conducting wire, an

electric field is set up through the wire and charge begins to flow. However, the flow through a certain point

may happen at different rates, a measured quantity we call current, denoted by .

As opposed to coulombs per second, the units for current is there ampere, denoted by

. By convention, we

say that the current is flowing in the same direction that positively charged particles (or positive charge carriers)

would be forced to move. Thus, conduction electrons would flow in the opposite direction. We may choose to

analyze the current density,, which is the amount of current flowing per area. This has the same direction asthe current. If the current density is uniform, and is perpendicular to the area, then:

But in general:

The flow of positive carriers has a slow drift speed, , in the direction of the electric field. If representsthe carrier charge density, the amount of charge per unit volume, then we can also calculate the current

charge density using:

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SECTION 26.4 RESISTANCE AND RESISTIVITY

If we maintain the same potential difference between the ends of geometrically similar wires made of two

different conductive materials, we will find that the current is not the same. This is referred to as the wire's

electrical resistance. The larger the electrical resistance, the smaller the current:

We measure resistance in ohms, . A device that is meant to provide a certain resistance is called a resistor.A property of materials is its resistivity, , which can be determined by considering the electric field andcurrent density through a point:

Its reciprocal is referred to the conductivity of a material:

The unit for resistivity is ohm-meter, m, and thus for conductivity, m.The resistivity of a material will vary with temperature according to the following formula:

The quantity is called the temperature coefficient of resistivity.Provided that we make a cylindrical wire or device of some conducting material with length,, and crosssectional area,, the resistance of that wire or device can be found using:

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CHAPTER 27C I R C U I T S

We now return to the analysis of circuits: several electrical devices connected by wires (which we take to havenegligible resistance). We use an emf device, such as a battery, to maintain a potential across a circuit so

that charge can flow. We can classify them as ideal or realemfdevices, the latter having an internalresistance and the former without. In other words, a real emf device has a potential difference across its

terminals different from its emf when a current runs through it.

27.7 RESISTORS IN CIRCUITS

Using the same principles as those mentioned in Chapter 25: The current along the original wire will split

across the subsequent wires, not necessarily equally. However, the voltage across each of those wires

remains the same as the original wire.

Resistors in parallel can be reduced to an equivalent capacitor using:

Further, the current is the same for each resistor connected in series but, the potential difference across

the ends of the wires is split between the connected resistors. Again, it is not necessarily divided

equally.

Resistors in series can be reduced to an equivalent capacitor using:

27.7 KIRCHHOFFS CIRCUIT LAWS

Kirchhoffs two circuits also help us to analyze circuits. For the use of these laws, we must make an educated

assumption as to the directions of the current.

KIRCHHOFFS VOLTAGE LAW (KVL) or LOOP RULE

The algebraic sum of the changes in potential encountered in a complete transversal of any loop of any

circuit must be zero. This is used in combination with:

RESISTANCE RULE: For a move through a resistance in the direction of the current, the change in potential is ; in the opposite direction it is . Also, for a move

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EMF RULE: For a move through an ideal emf device in the direction of the emf arrow, the change in potential

is +; in the opposite direction it is .KIRCHHOFFS CURRENT LAW (KCL) or JUNCTION RULE

The sum of the currents entering any junction must be equal to the sum of the circuits leaving that

junction.

27.9 RC CIRCUITS

When an emf is applied to a resistor of resistance, , and capacitor of capacitance, , in series, thecapacitor is being charged. The charge on the capacitor increases according to:

The quantity is the final charge of the capacitor. The quantity is also called the capacitive timeconstant and is sometimes denoted by . If there is an initial charge on the capacitor , then:

During the charge, the current through the capacitor is:

When the emf device is removed, it may discharge through the resistor and the charge decreases according

to:

In this case, , does not necessarily refer to the final charge , but instead to the initial charge on thecapacitor when 0. During the discharging, the current through the capacitor is:

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POWER

The power, or rate of energy transfer, in an electrical device across which a potential difference of ismaintained is given by:

For resistors, the rate at which energy is dissipated (electrical potential energy being transferred into thermal

energy) is given by:

The rate at which the chemical energy in the battery changes is:

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