Digital Electronics Problems

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Daytona State College

Department of Engineering Technology CET 3198 Digital Systems – Fall 2013

Homework 1 Due Date: September 29th 11:59 p.m.

Submit the homework via drop-box. For each one of the questions justify your answer and show your work.

1. Find the minimum cost implementation in SOP (Sum of Products) and POS (Product of Sums)for the function f(x1,x2,x3)  =  Σ  m(1,2,3,5).  Using  the  formula  for  determining  the  cost  of  an   implementation, calculate the cost.

2. A four variable logic function that is equal to 1 if any three or all four of its variables are equal to 1 is called majority function. Design a minimum cost SOP circuit that implements this majority function.

3. Find the minimal SOP expression for f(x1,x2,x3, x4)  =  Σ  m(1,5,7,9,11,15)   4. Find the minimum sum-of-product of the following functions:

a. F=X’Z  +  XY  +  XY’Z   b. F=A’C’D  +  B’CD  +  AC’D+BCD   c. F=WXZ’  +  WX’YZ  +  XZ   d. F=ABC’D’  +A’BC’+ABD+A’CD+BCD’

5. A circuit with two outputs has to implement the following functions: f(x1,..,x4)  =  Σ  m(0,2,4,6,7,9)  + D(10,11) g(x1,..,x4)=  Σ  m(2,4,9,10,15)+D(0,13,14)

Design the minimum-cost circuit and compare its cost with combined costs of two circuits that implements f and g separately. Assume that the input variables are available in both complemented and uncomplemented forms.

NOTE: Calculate the cost of a logic circuit as the number of gates plus the total number of inputs to all gates in the circuit. Assuming that inputs are available both true and complemented form at zero cost. If an inversion is needed inside a circuit, then the corresponding NOT gate and its input are included in the cost. (See chapter 4 in the book)

 
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Physics Conservation Of Momentum

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EQNS_E_RV.AUON OF MOMENTUM Name IN ONE DIMENSION

Mech HW-:

1. Two gliders, A aad B, collide on a level, frictionless track, as shown below,

The mass of glider A is less than the mass of gliderB (i.e., mr< ma). Thefinal speedof glider A is greater than the final speed of glider B (i.e., ztn > as). ”

Boforc collieion Affer collision

Is the magnitude of the final momentum of glider ,\ lFd, greater than, less than, or equal to the magnitude of the final momentum of glider B, lft/? Diaw a monrentum vector diagram to support your answer. (An example of a momentum vector diagram can be found on tha second page of the tutorial.)

a

2. Two gliders, c and D, collide on a level, frictionless track, as shown below. The mas er D (i.e., fitc 1mo). The initial speed of glider C D (i.e,, uci )opi). Arter the collision, Gliders C and D m final speed, ur.

Before collision After collision

Is the magnitude of the initial momentum of glider C_rlF./, greater than, less than, or equal to the magnitude of the initial momentum of glider D, lFD/? Draw a momentum vector diigram to support your answer.

Tutorials in Intoductory Physics McDermott, Shaffer, and the P.E,G., U. Wash.

‘ 2 @pearson Custom 2″0 nd., for U.CO, Boulder

 

 

Mech Conseroation of momentum in one dimension HW{0

3. Two astronauts, A and B, participate in three collision experiments in a weightless, frictionless environment. In each experimenl, astronaut B is initially at rest, and astronaut A has initial momentumi*= 20 kg-m/s to the right. (The velocities of the astronauts are measured with respect to a nearby space station.)

Before After

The astronauts push on each other in different ways so that the outcome of each experiment is different. As shown in the figure at right, astronaut A has a different final momentum in each experiment.

a. Determine the magnitude of the final momentum of astronaut B in each experiment. Explain.

b. Rank the three experiments according to the final kinetic energy of astronaut B, from largest to smallest. Explain.

c. Is the totalkinetic energy after the collision in experirfient2 greater than, less than, or equal to the total kinetiCenergy after the collision in experiment 3? (In this case, total kinetic energy means the sum of the kinetic energies of the two astronauts.) Explain.

d. Consider the following statement: “The momentum of the system is conserved in eoch experiment becouse there is no net f orce on the system. ff momentum is conserved, then kinetic ene?gY must olso be conserved, becouse both momentum ond kinetic energy ore mode up of moss ond velocity.”

One of the sentences above is completely correct. Discuss the error(s) in reasoning in the other sentence.

Tutorials in Introductory Physics McDermott, Shaffer, and the P.E.G., U. Wash.

@Peprson

Custom 2nd F1,., for U.CO, Boulder

Experirnents 1,2, and3

At rest

 

 

Consentation of momentum in one ditnenston Name

e. In the boxes below, draw the initial mompntum, the change in momentum, andthe final momentum for each astronaut in the three experiments. The initial momentum is shown for astronaut A. Draw the other vectors using the same scale.

Mech HW-61

Initial

If the net force on a system of two colliding objecls is zero, how does the change in momentum of one object compare to the change in momentum of the other object:

. in magnitude?

. in direction?

Bxplain how your answers to part f are consistent with Newton’s third law and the impulse- momentum theorem ( 4″t N = [F) for:

. each astronaut considered separately, and

. for the system of both astronauts together.

‘,

Tutorials in Introductory Physics McDermott, Shaffer, and the P.E.G., U. Wash.

@Pearson

Custom 2nd. Ed., for U.CO, Boulder

 

 

Mech Conseroation of momentum in one dimension HW-62

4. A pyrotechnician releases a 3-kg firecracker from rest. At t = 0.4 s, the firecracker is moving downward with speed 4 mls. At this same instant, the firecracker begins to explode into two pieces, “top” and “bottom,” with masses nt op 1 kg and ffibotto – 2kg. At the end of the explosion (/ = 0.6 s), the top piece is moving upward with speed 6 m/s.

The mass of the explosive substance is negligible in comparison to the mass of the two pieces.

The questions below can serve as a guide to completing the diagram below right:

a. Determine the magnitude of the net force on the firecracker system before the explosion. (Use I = 10 m/s’.) ExPlain.

b. Determine’the magnitude of the net force on the firecracker system at an instant during the explosion. (Hint: Does the net force on a system depend on forces that are internal to that system?)

Determine the magnitude and direction of the net impulse (\”rN) on the firecracker system during the explosion (i.e., over the interval from t =0.4 s until r = 0.6 s). Explain.

Use the impulse-momentum theorem to determine the magnitude and direction of the change in momentum of the firecracker system during the explosion. Enter this vector in the table using the scale set by the initial momentum of the system.

Determine the final momentum of the firecracker system and enter it in the table. (Hint: Is it the same as the initial momentum?)

Complete the vector diagram at right.

Initial l: 0.4 s

Final /: 0.6 s

Change Final /=0.6s

Initial /=0.4s

Prop

Pbuno*

d.

/rr*,”,n

12 kg-m/s

i_l_r._l:

m=3kg

lfrl = 4 mls

,onl= 6 m/s

rurot, = I kg

ffiborton – 2 kg

A -t”bottom – :

I !

i1tirt -Tj -t–j

Tutorials in Introductory Physics McDermoff, Shaffer, and the P.E.G., U. Wash.

OPearson Custom 2″d H, for U.CO. Boulder

 
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Electric Field And Electric Potential

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Look at the attachment

(expirment … read it .. it is easy but small question need to be answerd)

Lab reports should be orgnized as the followin ( most of the allready written in the lab that is attached):

1. Title.

2. A short explanation of the purpose of the experiment.

3. An introduction section to the method and physical theory behind the experiment (yes,
including the math of it).

4. A data results section, including raw data (WITH UNITS), calculations with your data (if you
don’t want to type all of the calculations, you can attach a handwritten page at the end of
the lab report, but make it look neat, and always include calculations), experimental
results (i.e. all printed data from the experiment, and the neatly ripped of pages from your
book), theoretical results, graphics (WITH ERROR BARS), questions from the lab manual,
etc. NOTE: All tables not provided by the book must be typed!

5. An observation section, where you comment on all the happenings and mishappenings of
the experiment, including why the data was not exactly in accordance with the
experiment, what mistakes you made, what could have been done better (This is
important for me, because here I know if you are really learning the physics).

6. The conclusions.

7. extra question (Bounce) if you can solve it (the picture that i uploaded)

 
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Lab 2

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TA: Keslo Estil SEC#: Name:

Lab 2. Coulombs Law

The electrostatic forces which you observed in Lab 1 were studied in detail by Coulomb in 1784. His experiments resulted in the empirical law named after him. It describes the forces that two smalls particle-like charged objects exert on each other, forces that depend on the magnitudes of the charges, q1 and q2, as well as the distance r between the charged particles. See Fig. 1. To attempt to explore Coulomb’s Law using physical apparatus is fraught with difficulty, especially

in south Florida. For this reason, you will be using a computer simulation instead.

Preliminary Questions (1 – 5)

Procedure (4, 6, 7 use )

Analysis (1 – 6)

Note: Show all equations, calculations and very clear screenshots for the graphs with fits to receive full credit.

 

https://www.vernier.com/downloads/logger-pro-demo/

 

Data Table

 

Table 1
Distance between charges, r = m
    Force (N)
     
     
     
     
     
     
     
     
     
     
     
     

 

Table 2
 

 
Distance, r (m) Force (N)  
     
     
     
     
     
     
     
     
     
     
     
     
 
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Physic II Lab M3A1 Experiment: DC Circuits

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PART I – Current, Electrical Potential and Resistance

· Download the following file, to your desktop by:

· right-clicking on M3-Activity-1 : File attached

· selecting Save target as…

· saving the file to your computer where you can easily access it again

This file will be used within the lab simulation.

Start the simulation “Circuit Construction Kit (DC Only) ” (if you haven’t done so already) by clicking on the image below. : http://phet.colorado.edu/sims/circuit-construction-kit/circuit-construction-kit-dc_en.jnlp

The simulation has a set of controls on the right hand side of the screen. These are divided vertically into different areas. Make the selections indicated.

· From “Circuit,” select load.

· When prompted, upload the file M3-Activity-1.

· From “Visual,” select “Life Like.”

· From “Tools,” select noncontact ammeter.

· Position the voltmeter and ammeter as shown in the image below.

http://phet.colorado.edu/sims/circuit-construction-kit/circuit-construction-kit-dc_en.jnlp

Image retrieved from the PHET “Circuit Construction Kit (DC Only)” simulation.

Inside your notebook, draw a graph with potential (V) on the vertical axis and current (I) on the horizontal axis. Use Ohm’s Law to make a numerical prediction of the shape that this graph will take. The default values are 9.0 V for the battery, and 10.0 Ohms for the resistor, but you will want to right click on both components to verify this. Plot at least 5 prediction points, and sketch in the graph in your notebook. Now use the Circuit Construction Kit to measure the relationship between battery potential and current.

· Leave the resistor value constant.

· Record the battery voltage and the resulting current in the circuit for the default values.

· Right click on the battery, and change the voltage.

· Click on the “Run” arrow to start the simulation.

· Record the new values of potential and current.

· Take at least 5 data points.

How does the graph produced through your simulation data compare to the predicted graph in your notebook? Continue your investigation. Can you find any factors that might explain any discrepancies? Try turning on the voltmeter and measuring the battery terminal voltage as shown below. Does the battery terminal voltage match the voltage you set using the simulation controls?

Describe in detail any changes that need to be made to the procedure in order to make the predicted graph, so that it will reflect results of the simulation. Use your revised procedure, and place both graphs in your lab report. Be sure to provide a discussion of the differences between a real source of EMF V.S. and an ideal source of EMF.

Part II – Kirchhoff’s Rules for Current and Potential

Your basic mathematical tools for analyzing a circuit are Ohm’s Law, and Kirchoff’s loop rule for potentials and junction rule for currents. We are going to use this activity to gain confidence in the accuracy of these latter two rules, and also to give you some practice in applying these rules.

· Download the following file, to your desktop by:

· right-clicking on M3-Activity-2 : File attached

· selecting Save target as…

· saving the file to your computer where you can easily access it again

· Load the circuit file M3-Activity2 (as you did for M3-Activity1 in Part I).

· Activate the voltmeter.

When the simulation starts, the switch in the circuit is open. There is no conductive path, so the far right left branch of the circuit is not active. Initially, we will only concern ourselves with the portion of the circuit containing batteries and the middle branch.

The loop rule states that the sum of changes in voltage around any closed path in a circuit is zero. In this activity, you are going to use the voltmeter to follow a closed path in the circuit provided. You will record the change in voltage across every component, and then take the sum of these changes.

· Draw the circuit in your laboratory notebook.

· Draw arrows to indicate a sense of direction through the circuit.

· Traverse the loop of the circuit with the voltmeter, measuring the change in potential across each component.

Tip: If you start with the red probe ahead of the black probe make sure this orientation is maintained throughout the data taking. Why might this be important? Record the results in your notebook.

When you close the switch, you will have three circuit loops. The second is obvious, but the third conductive loop may take a moment or two to identify. (Hint: A conducting loop is not required to have a battery.)

· Close the switch.

· Repeat the analysis with all three circuit loops.

How were the potentials across the components in the first loop affected by closing the switch? How might the interior lights of your car be affected by turning on the radio, or engaging the starter motor?

The junction rule states that the algebraic sum of all currents entering or leaving a junction is zero. This is actually just a restatement of the conservation of charge in that a wire junction cannot store charge. Designate the top junction as junction 1.

· Measure the magnitude of all currents entering or leaving the junction.

· Assign a “+” algebraic sign to all currents leaving a junction.

· Assign a “-” algebraic sign to all currents entering a junction.

Be sure to record your observations and analysis in your laboratory notebook. Does your analysis support or fail to support the Kirchhoff’s loop and junction rules?

PART III – The Equivalent Resistance of Series and Parallel Resistors

Use the Circuit Construction Kit to create a circuit with at least three series resistors, or load file M3-Activity-3: Download the following file, to your desktop by:

· right-clicking on M3-Activity-3 : File attached

· selecting Save target as…

· saving the file to your computer where you can easily access it again

Use different values for each resistor. Calculate the equivalent resistance of the system using the series rule for resistors.

Describe in detail how you will measure the equivalent resistance of the entire network of resistors. Where will you place the red and black probes of the voltmeter? Where will you place the noncontact ammeter?

Record your data in your laboratory notebook, and perform your analysis. Do your results support or fail to support the series rule for equivalent resistance?

 A single loop circuit with three series resistors. Note the position of the voltmeter and the ammeter. (Image retrieved from the PHET “Circuit Construction Kit (DC Only)” simulation).

Use the Circuit Construction Kit to create a circuit with at least three parallel resistors, or load file M3-Activity-4: Download the following file, to your desktop by:

· right-clicking on M3-Activity-4 : File attached

· selecting Save target as…

· saving the file to your computer where you can easily access it again

Use different values for each resistor. Calculate the equivalent resistance of the system using the parallel rule for resistors.

Describe in detail how you will measure the relevant voltages and currents. Record your data in your laboratory notebook and perform your analysis. Do your results support or fail to support the parallel rule for equivalent resistance?

 
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Physics (Physics Bidders Only)

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Physlets Chapter Fourteen

Due by Sunday at 11:59 PM PT.

 

· Illustrations 14.1, 14.2, 14.3

· Explorations 14.1-14.3

· Problems 14.1, 14.3, 14.4, 14.5

 

 

Physlets Chapter Thirty

Due by Sunday at 11:59 PM PT.

 

· Illustrations 30.1, 30.2, 30.3

· Explorations 30.1, 30.2, 30.3

· Problems 30.1, 30.2, 30.3, 30.5, 30.6

 

 

Physlets Chapter Thirty

Due by Sunday at 11:59 PM PT.

 

Physlets Chapter 30

· Illustrations 30.1, 30.2, 30.3

· Explorations 30.1, 30.2, 30.3

· Problems 30.1, 30.2, 30.3, 30.5, 30.6

 

 

 

Physlets Chapter Thirty-Four

Due by Sunday at 11:59 PM PT.

 

· Illustrations 34.1, 34.2, 34.3

· Explorations 34.1, 34.2, 34.4, 34.5

· Problems 34.1, 34.7, 34.9
 
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Outline 6

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Surname 1

Student Name

Instructor

Course Name and Code

Date

Outline: Racist Speech as the Functional Equivalent of Fighting Words

Thesis: Face to face insults which carry racially discriminatory elements should not get First Amendment protection due to the effects that they cause.

Reasons why face to face racial insults should not be protected under the First Amendment

i. Such insults have an immediate negative impact on the victim which may be regarded as injury

ii. The First Amendment promotes free speech but face to face racial insults tend to curtail more speech and therefore should not be encouraged.

· Racial insults mostly make the victim speechless and this creates a moment of personal embarrassment with the victim feeling inferior and overwhelmed by fear.

· In other cases the victim may decide to confront the racist and this is most likely to cause a breech in peaceful coexistence.

How racial insults differ with protected speech

· Racial insults lead to emotional and reputational injury which is likely to inhibit the ability to provide a sound response while protected speech does not.

· Racial insults are directed to minority in most cases and mostly demean the victim making them more powerless while free speech does not.

· Racial insults are fighting words because they cause immediate injury to those targeted at.

· Racial insults cause shock, plight and inability to face the moment or the perpetrator and therefore cannot be classified as free speech.

· Racial insults apart from causing instant injury also lead to disorientation of the victim distracting them from normal activities like work or school and majorly threatening their peaceful existence.

Conclusion

Racial insults should not be protected under the First Amendment because they fight against the same efforts that the First Amendment promotes and they cause immediate reputational injury to the victim curtailing more speech.

 

1

 
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Physics

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Chapter 19

· The displacement from equilibrium caused by a wave on a string is given by y(x, t) = (−0.00185 m)sin[(42.8 m−1)x − (744 s−1)t]. For this wave, what are the following? (a) amplitude? (b) number of waves in 1 m? (c) number of complete cycles in 1 s? (d) wavelength? (e) speed? Ans:- 0.00185 m, 6.81 waves, 118 cycles, 0.147 m, 17.4 m/s.

· Point A in the figure is 30 cm below the ceiling. Determine how much longer it will take for a wave pulse to travel along wire 1 than along wire 2. Ans:-2.28

· Show that the function D = A ln(x + vt) is a solution of the wave equation,

(d2D/dt2) = υ2(d2D/dx2)

· A string with linear mass density of 0.1 kg/m is under a tension of 107 N. How much power must be supplied to the string to generate a sinusoidal wave of amplitude 3 cm and frequency 116 Hz? Ans:-782 W

· 15.42. A cowboy walks at a pace of about two steps per second, holding a glass of diameter 11 cm that contains milk. The milk sloshes higher and higher in the glass until it eventually starts to spill over the top. Determine the maximum speed of the waves in the milk. Ans:- 0.44 m/s

· A 49-cm-long wire with a mass of 10.8 g is under a tension of 55.5 N. Both ends of the wire are held rigidly while it is plucked. (a) What is the speed of the waves on the wire? (b) What is the fundamental frequency of the standing wave? (c) What is the frequency of the third harmonic? Ans:- 55.5 m/s, 51.2 Hz, 154 Hz

· Consider a guitar string stretching 84.6 cm between its anchored ends. The string is tuned to play middle C, with a frequency of 256 Hz, when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced 1.84 mm at its midpoint and released to produce this note, what are the wave speed, v, and the maximum speed, Vmax, of the midpoint of the string? Ans:- 433 m/s, 5.92 m/s

· The tension in a 2.9-m-long, 1.1-cm-diameter steel cable (ρ = 7800 kg/m3) is 810 N. What is the fundamental frequency of vibration of the cable? Ans:- 5.7 Hz

chapter 20

· You drop a stone down a well that is 6.65 m deep. How long is it before you hear the splash? The speed of sound in air is 343 m/s. Ans:- 1.18 s.

The sound level in decibels is typically expressed as β = 10 log(I/I0), but since sound is a pressure wave, the sound level can be expressed in terms of a pressure difference. Intensity depends on the amplitude squared, so the expression is β = 20 log(P/P0), where P0 is the smallest pressure difference noticeable by the ear: P0 = 2. 10-5 Pa. A loud rock concert has a sound level of 109 dB, find the amplitude of the pressure wave generated by this concert. Ans:- 5.64 Pa· Two sources, A and B, emit a sound of a certain wavelength. The sound emitted from both sources is detected at a point away from the sources. The sound from source A is a distance d from the observation point, whereas the sound from source B has to travel a distance of 4λ. What is the largest value of the wavelength, in terms of d, for the maximum sound intensity to be detected at the observation point? (Assume source A is farther from the observation point than source B.) If d = 12.6 m and the speed of sound is 340 m/s, what is the frequency of the emitted sound? Ans:- (d/2), 135 Hz.

· A plane flies at Mach 1.5, and its shock wave reaches a man on the ground 21 s after the plane passes directly overhead. (a) What is the Mach angle? (b) What is the altitude of the plane? The speed of the plane is 514.5 m/s. Ans:- 41.8 degree, 9.66 km

· A standing wave in a pipe with both ends open has a frequency of 444 Hz. The next higher harmonic has a frequency of 626 Hz. (a) Determine the fundamental frequency. (b) How long is the pipe? Ans:- 182 Hz, 0.942 m.

· An observer stands between two sound sources. Source A is moving away from the observer, and source B is moving toward the observer. Both sources emit sound of the same frequency. If both sources are moving with a speed, vsound/9, what is the ratio of the frequencies detected by the observer? Ans:- 0.8

· At a distance of 29 m from a sound source, the intensity of the sound is 62 dB. What is the intensity (in dB) at a point 1.9 m from the source? Assume that the sound radiates equally in all directions from the source. Ans:- 85.7 dB

· A source traveling to the right at a speed of 10.80 m/s emits a sound wave at a frequency of 110.0 Hz. The sound wave bounces off of a reflector, which is traveling to the left at a speed of 5.15 m/s. What is the frequency of the reflected sound wave detected by a listener back at the source? (Use 343 m/s as the speed of sound.) Ans:- 121 Hz

Chapter 21

1) How many electrons do 6.94 kg of water contain?

2) Two identically charged particles separated by a distance of 7.8 m repel each other with a force of 9 N. What is the magnitude of the charges?

3) In solid sodium chloride (table salt), chloride ions have one more electron than they have protons, and sodium ions have one more proton than they have electrons. These ions are separated by about 0.28 nm. Calculate the electrostatic force between a sodium ion and a chloride ion.

4) In gaseous sodium chloride, chloride ions have one more electron than they have protons, and sodium ions have one more proton than they have electrons. These ions are separated by about 0.24 nm. Suppose a free electron is located 0.86 nm above the midpoint of the sodium chloride molecule. What are the magnitude and the direction of the electrostatic force the molecule exerts on it? (Assume the chloride ion lies on the +x-axis and the sodium ion lies on the −x-axis, with the origin as the midpoint between the two ions. The free electron lies on the +y-axis.)

5) Identical point charges Q are placed at each of the four corners of a rectangle measuring 8.0 m by 9.0 m. If Q = 26 µC, what is the magnitude of the electrostatic force on any one of the charges?

6) Charge q1 = 1.1 10-8 C is placed at the origin. Charges q2 = -1.3 10-8 C and q3 = 2.6 10-8 C are placed at points (0.18 m, 0 m) and (0 m, 0.24 m), respectively, as shown in the figure. Determine the net electrostatic force (magnitude and direction) on charge

q3.

7) A small ball with a mass of 10 g and a charge of -0.2 µC is suspended from the ceiling by a string. The ball hangs at a distance of 5.0 cm above an insulating floor. If a second small ball with a mass of 60 g and a charge of 0.4 µC is rolled directly beneath the first ball, will the second ball leave the floor?

8) Four point charges, q, are fixed to the four corners of a square that is 15.0 cm on a side. An electron is suspended above a point at which its weight is balanced by the electrostatic force due to the four point charges, at a distance of 13 nm above the center of the square. (The square is horizontally flat, and the electron is suspended 13 nm vertically above the center of the square.) What is the magnitude of each fixed charge in coulombs? What is the magnitude of each fixed charge as a multiple of the electron’s charge?

Chapter 22

1) A +1.5-nC point charge is placed at one corner of a square (1.5 m on a side), and a -3.0-nC charge is placed on the corner diagonally opposite. What is the magnitude of the electric field at either of the other two corners?

2) Consider an electric dipole on the x-axis and centered at the origin. At a distance h along the positive x-axis, the magnitude of electric field due to the electric dipole is given by k(2qd)/h3. Find a distance perpendicular to the x-axis and measured from the origin at which the magnitude of the electric field stays the same. (Use the following as necessary: h and d.)

3) A charge per unit length +λ is uniformly distributed along the positive y-axis from y = 0 to y = +a. A charge per unit length −λ is uniformly distributed along the negative y-axis from y = 0 to y = −a. Write an expression for the electric field at a point on the x-axis a distance x from the origin. (Use the following as necessary: k, λ, x, and a.)

4) An electron is observed traveling at a speed of 1.67 107 m/s parallel to an electric field of magnitude 11,360 N/C. How far will the electron travel before coming to a stop? (The mass of electron is 9.109 10-31 kg, and the charge on the electron is 1.602 10-19 C. The electron is moving in the direction opposite to the electric field.)

5) A -7-nC point charge is located at the center of a conducting spherical shell. The shell has an inner radius of 2 m, an outer radius of 4 m, and a charge of +8 nC. (Let the radially outward direction be positive.) (a) What is the electric field at r = 1 m? (Indicate the direction with the sign of your answer.) (b) What is the electric field at r = 3 m? (Indicate the direction with the sign of your answer.) (c) What is the electric field at r = 7 m? (Indicate the direction with the sign of your answer.) (d) What is the surface charge distribution, σ, on the outside surface of the shell? (Indicate the direction with the sign of your answer.)

Chapter 23

1- A metal ball with a mass of 7. 10-6 kg and a charge of +6 mC has a kinetic energy of 2. 108 J. It is traveling directly at an infinite plane of charge with a charge distribution of +5 C/m2. If it is currently 1 m away from the plane of charge, how close will it come to the plane before stopping? Ans:- 0.882 m

2- A proton gun fires a proton from midway between two plates, A and B, which are separated by a distance of 83.0 cm; the proton initially moves at a speed of 135.0 km/s toward plate B. Plate A is kept at zero potential, and plate B at a potential of 400.0 V. (a) Will the proton reach plate B? (b) If not, where will it turn around? (If the proton reaches plate B, enter 0.830 m, the distance between the plates.) (c) With what speed will it hit plate A? (If the proton reaches plate B, enter 0.) Ans:- No. 0.612 m, 2.38e+05 m/s.

3- Consider a dipole with charge q and separation d. What is the potential a distance x from the center of this dipole at an angle θ with respect to the dipole axis, as shown in the figure? (Do not assume that x » d. Use any variable or symbol stated above along with the following as necessary: k.)

4- Consider an electron in the ground state of the hydrogen atom, separated from the proton by a distance of 0.0529 nm. (a) Viewing the electron as a satellite orbiting the proton in the electrostatic potential, calculate the speed of the electron in its orbit. (b) Calculate an effective escape speed for the electron. (c) Calculate the energy of an electron having this speed. Using this energy, determine the energy that must be given to the electron to ionize the hydrogen atom. Ans:- 2.19e+06 m/s; 3.10e+06 m/s; 4.36e-18 J; 13.6 eV

5- An electric field varies in space according to this equation: E = E0xe−x. (a) For what value of x does the electric field have its largest value, xmax? (Use the following as necessary: E0.) (b) What is the potential difference between the points at x = 0 and x = xmax? (State an expression for the difference V(xmax) – V(0). Use the following as necessary: E0.) Ans:- 1.

6- An infinite plane of charge has a uniform charge distribution of +5 nC/m2 and is located in the yz-plane at x = 0. A +24 nC fixed point charge is located at x = +1 m. (a) Find the electric potential V(x) on the x-axis from 0 < x < +1 m. (Use the following as necessary: x, k, σ for the surface charge distribution, and q and x0 for the charge and position of the point charge, respectively. Do not substitute numerical values; use variables only.) (b) At what position(s) on the x-axis between x = 0 and x = +1 m is the electric potential a minimum? (Enter your answers from smallest to largest starting with the first answer blank. Enter NONE in any remaining answer blanks.)

7- The electric field, (), and the electric potential, V(), are calculated from the charge distribution, ρ(), by integrating Coulomb’s Law and then the electric field. In the other direction, the field and the charge distribution are determined from the potential by suitably differentiating. Suppose the electric potential in a large region of space is given by V(r) = V0 exp (−r2/a2), where

 

 

V0 and a are constants and r = 

 x2 + y2 + z2

is the distance from the origin. (a) Find the electric field () in this region. (Use the following as necessary: V0, r, a, and ε0.) (b) Determine the charge density ρ() in this region, which gives rise to the potential and field. (Use the following as necessary: V0, r, a, and ε0.) (c) Find the total charge in this region. (Use the following as necessary: V0, r, a, and ε0.) (d) Roughly sketch the charge distribution that could give rise to such an electric field.

8- Two metal balls of mass m1 = 6 g (diameter = 6 mm) and m2 = 7 g (diameter = 7 mm) have positive charges of q1 = 1 nC and q2 = 8 nC, respectively. A force holds them in place so that their centers are separated by 8 mm. What will their velocities be after the force is removed and a large distance separates them? (Enter the magnitudes of the velocities.) Ans:- 0.0402 m/s; 0.0344 m/s

9- A charge of 0.520 nC is placed at x = 0. Another charge of 0.140 nC is placed at x1 = 10.9 cm on the x-axis. (a) What is the combined electrostatic potential of these two charges at x = 20.1 cm, also on the x-axis? (b) At which point(s) on the x-axis does this potential have a (local) minimum? (Enter your answers from smallest to largest starting with the first answer blank. Enter NONE in any remaining answer blanks. If you need to use or –, enter INFINITY or –INFINITY, respectively.) Ans:- 36.9 V; -Infinity; 7.18 cm; Infinity.

Chapter 24

1- A spherical capacitor is made from two thin concentric conducting shells. The inner shell has radius r1, and the outer shell has radius r2. What is the fractional difference in the capacitances of this spherical capacitor and a parallel plate capacitor made from plates that have the same area as the inner sphere and the same separation d = r2 − r1 between plates? (Use the following as necessary: r1, r2, and ε0.)

2- Four capacitors with capacitances C1 = 3.4 nF, C2 = 2.2 nF, C3 = 1.6 nF, and C4 = 5.3 nF are wired to a battery with V = 10.3 V, as shown in the figure. What is the equivalent capacitance of this set of capacitors?

3- A potential difference of V = 80.0 V is applied across a circuit with capacitances C1 = 14.5 nF, C2 = 4.00 nF, and C3= 26.5 nF, as shown in the figure. What is the magnitude and sign of q3l, the charge on the left plate of C3 (marked by point A)? What is the electric potential, V3, across C3? What is the magnitude and sign of the charge q2r, on the right plate of C2 (marked by point B)?

4- Fifty-two parallel plate capacitors are connected in series. The distance between the plates is d for the first capacitor, 2d for the second capacitor, 3d for the third capacitor, and so on. The area of the plates is the same for all the capacitors. Express the equivalent capacitance of the whole set in terms of C1 (the capacitance of the first capacitor).

5- A 7000-nF parallel plate capacitor is connected to a 2.4-V battery and charged. (a) What is the charge Q on the positive plate of the capacitor? (b) What is the electric potential energy stored in the capacitor? The 7000-nF capacitor is then disconnected from the 2.4-V battery and used to charge three uncharged capacitors, a 100-nF capacitor, a 200-nF capacitor, and a 300-nF capacitor, connected in series. (c) After charging, what is the potential difference across each of the four capacitors? d) How much of the electrical energy stored in the 7000-nF capacitor was transferred to the other three capacitors?

24.48. The Earth is held together by its own gravity. But it is also a charge-bearing conductor. (a) The Earth can be regarded as a conducting sphere of radius 6371 km, with electric field E = (−150. V/m) at its surface, where is a unit vector directed radially outward. Calculate the total electrostatic potential energy associated with the Earth’s electric charge and field. (b) The Earth has gravitational potential energy, akin to the electrostatic potential energy. Calculate this energy, treating the Earth as a uniform solid sphere.(Hint: dU = −(GM/r)dM. The mass of the Earth is 5.97 1024 kg. Assume that the density of the Earth is uniform.) (c) Use the results of parts (a) and (b) to address this question: To what extent do electrostatic forces affect the structure of the Earth?6- A 2.1-nF parallel plate capacitor with a sheet of Mylar (κ = 3.1) filling the space between the plates is charged to a potential difference of 150 V and is then disconnected. (The initial capacitance including the dielectric is 2.1 nF.) (a) How much work is required to completely remove the sheet of Mylar from the space between the two plates? (b) What is the potential difference between the plates of the capacitor once the Mylar is completely removed?

7- A proton traveling along the x-axis at a speed of 5.0 106 m/s enters the gap between the plates of a 4.0-cm-wide parallel plate capacitor. The surface charge distributions on the plates are given by σ = ±1.0 10-6 C/m2. How far has the proton been deflected sideways (Δy) when it reaches the far edge of the capacitor? Assume that the electric field is uniform inside the capacitor and zero outside. (The plates of the capacitor are parallel to the x-axis. The charge of the proton is +1.602 10-19 C and the mass of the proton is 1.67 10-27 kg.)

Chapter 25

1- A copper wire has a diameter dCu = 0.0470 cm, is 3.00 m long, and has a density of charge carriers of 8.50 x1028 electrons/m3. As shown in the figure, the copper wire is attached to an equal length of aluminum wire with a diameter dAl = 0.0110 cm and density of charge carriers of 6.02 x1028 electrons/m3. A current of 0.470 A flows through the copper wire. (a) What is the ratio of the current densities in the two wires, JCu/JAl? (b) What is the ratio of the drift velocities in the two wires, vd-Cu/vd-Al?

2- As illustrated in the figure, a current, i, flows through the junction of two materials with the same cross-sectional area and with conductivities σ1 and σ2. Show that the total amount of charge at the junction is ε0i(1/σ2 − 1/σ1).

3- A 10-gauge copper wire, with a constant potential difference of 0.4 V applied across its 1 m length at room temperature (20° C), is cooled to liquid nitrogen temperature (77 K = −196 °C). (The resistivity and temperature coefficient of resistivity of copper are 1.72 10-8 m Ω and 3.9 10-3 K−1, respectively.) (a) Determine the percentage change in the wire’s resistance during the drop in temperature. (b) Determine the percentage change in current flowing in the wire. (c) Compare the drift speeds of the electrons at the two temperatures. (Copper has 1 conducting electron per atom.)

4- What is the equivalent resistance of the five resistors in the circuit in the figure? Let R1 = 28 Ω, R2 = 56 Ω, R3 = 84 Ω, R4 = 112 Ω, R5 = 140 Ω. (When entering units, use ohm for Ω.)

5- A potential difference of V = 0.550 V is applied across a block of silicon with resistivity 8.70 10-4 Ω m. As indicated in the figure, the dimensions of the silicon block are width a = 2.00 mm and length L = 18.5 cm. The resistance of the silicon block is 50.0 Ω, and the density of charge carriers is 1.23 1023 m-3. Assume that the current density in the block is uniform and that current flows in silicon according to Ohm’s Law. The total length of 0.500-mm-diameter copper wire in the circuit is 78.0 cm, and the resistivity of copper is 1.69 10-8 Ω m. (a) What is the resistance, Rw, of the copper wire? (b) What are the direction and the magnitude of the electric current, i, in the block? (Indicate the direction in the block as displayed in the figure above.) (c) What is the thickness, b, of the block? (d) On average, how long does it take an electron to pass from one end of the block to the other? (e) How much power, P, is dissipated by the block? (f) In what form of energy does this dissipated power appear?

Chapter 26

1- Find the equivalent resistance for the circuit in the figure. (Use the following as necessary: Vemf, R.)

2- The dead battery of your car provides a potential difference of 9.630 V and has an internal resistance of 1.150 Ω. You charge it by connecting it with jumper cables to the live battery of another car. The live battery provides a potential difference of 12.00 V and has an internal resistance of 0.0100 Ω, and the starter resistance is 0.0700 Ω. (a) Draw the circuit diagram for the connected batteries. (The starter is in parallel with the live battery.) (Do this on paper. Your instructor may ask you to turn in this work.) (b) Determine the current in the live battery, in the dead battery, and in the starter immediately after you closed the circuit.

3- The circuit shown in the figure consists of two batteries with VA = 6.3 V and VB = 12.1 V and three light bulbs with resistances R1, R2, and R3. Calculate the magnitudes of the currents i1, i2, and i3 flowing through the bulbs.

4- A circuit consists of two 3.45-kΩ resistors in series with an ideal 12.0-V battery. (a) Calculate the current flowing through each resistor. (b) A student trying to measure the current flowing through one of the resistors inadvertently connects an ammeter in parallel with that resistor rather than in series with it. How much current will flow through the ammeter, assuming that it has an internal resistance of 2.5 Ω?

5- What is the time constant for the discharging of the capacitors in the circuit shown in the figure? If the 2-µF capacitor initially has a potential difference of 63 V across its plates, how much charge is left on it after the switch has been closed for a time equal to half of the time constant?

6- A 6.0-V battery is attached to a 6-mF capacitor and a 440-Ω resistor. Once the capacitor is fully charged, what is the energy stored in it? What is the energy dissipated as heat by the resistor as the capacitor is charging?

 

Chapter 27

· Determine the force on a semicircular wire with current I in the presence of a field B.

 

·

An electron with a speed of 3.8 105 m/s enters a uniform magnetic field of magnitude 0.053 T at an angle of 35° to the magnetic field lines. The electron will follow a helical path. (a) Determine the radius of the helical path. (b) How far forward (parallel to the magnetic field lines) will the electron have moved after completing one circle? 

 

 

· A conducting rod of length L slides freely down an inclined plane, as shown in the figure. The plane is inclined at an angle θ from the horizontal. A uniform magnetic field of strength B acts in the positive y-direction. Determine the magnitude and the direction of the current that would have to be passed through the rod to hold it in position on the inclined plane. (Use any variable or symbol stated above along with the following as necessary: g. Indicate your direction based on the side view.)

A copper wire with density ρ = 8960 kg/m3 is formed into a circular loop of radius 47.0 cm. The cross-sectional area of the wire is 1.00 10-5 m2, and a potential difference of 0.020 V is applied to the wire. What is the maximum angular acceleration of the loop when it is placed in a magnetic field of magnitude 0.20 T? The loop rotates about an axis through a diameter. (Use 1.72 10-8 Ω m for the resistivity of copper.)

A semicircular loop of wire of radius R is in the xy-plane, centered about the origin. The wire carries a current, i, counterclockwise around the semicircle, from x = −R to x = +R on the x-axis. A magnetic field, B, is pointing out of the plane, in the positive z-direction. Calculate the net force on the semicircular loop. (Use the following as necessary: R, i, and B.) 

 

Chapter 28

 

·

Suppose that the magnetic field of the Earth were due to a single current moving in a circle of radius 2960 km through the Earth’s molten core. The strength of the Earth’s magnetic field on the surface near a magnetic pole is about 6. 10-5 T. About how large a current would be required to produce such a field? (The radius of the Earth is 6.39 103 km.)· A square loop, with sides of length L, carries current i. Find the magnitude of the magnetic field from the loop at the center of the loop, as a function of i and L. (Use any variable or symbol stated above along with the following as necessary: μ0.)

· The figure below shows a Helmholtz coil used to generate uniform magnetic fields. Suppose the Helmholtz coil consists of two sets of coaxial wire loops with 13 turns of radius R = 71.0 cm, which are separated by R, and each coil carries a current of 0.134 A flowing in the same direction. Calculate the magnitude and the direction of magnetic field in the center between the coils.

· What is the magnitude of the magnetic field inside a long, straight tungsten wire of circular cross section with diameter 3.6 mm and carrying a current of 2.7 A, at a distance of 0.5 mm from its central axis? (The magnetic suspect

Chapter 29 Class Problem

· A circular conducting loop with radius a and resistance R2 is concentric with a circular conducting loop with radius b » a (b much greater than a) and resistance R1. A time-dependent voltage is applied to the larger loop, having a slow sinusoidal variation in time given by V(t) = V0 sin ωt, where V0 and ω are constants with dimensions of voltage and inverse time, respectively. Assuming that the magnetic field throughout the inner loop is uniform (constant in space) and equal to the field at the center of the loop, derive expressions for the potential difference induced in the inner loop and the current i through that loop. (Use the following as necessary: a, b, R1, R2, V0, ω, t, and μ0.)

· The conducting loop in the shape of a quartercircle shown in the figure has a radius of 20 cm and resistance of 0.8 Ω. The magnetic field strength within the dotted circle of radius 6 cm is initially 5 T. The magnetic field strength then decreases from 5 T to 1 T in 7 s. (a) Find the magnitude of the induced current in the loop. (b) Find the direction of the induced current in the loop.

· A short coil with radius R = 18 cm contains N = 30 turns and surrounds a long solenoid with radius r = 3.0 cm containing n = 60 turns per centimeter. The current in the short coil is increased at a constant rate from zero to i = 1.5 A in a time of t = 16 s. Calculate the induced potential difference in the long solenoid while the current is increasing in the short coil.

· In the circuit in the figure, a battery supplies Vemf = 10 V and R1 = 4.0 Ω, R2 = 4.0 Ω, and L = 2.0 H. Calculate each of the following a long time after the switch is closed. (a) the current flowing out of the battery. (b) the current through R1; (c) the current through R2; (d) the potential difference across R1; (e) the potential difference across R2; (f) the potential difference across L; (g) the rate of current change across R1.

· An emf of 30 V is applied to a coil with an inductance of 34 mH and a resistance of 0.70 Ω. (a) Determine the energy stored in the magnetic field when the current reaches one fourth of its maximum value. (b) How long does it take for the current to reach this value?

Q. An emf of 24.0 V is applied to a coil with an inductance of 58.0 mH and a resistance of 0.41 Ω.

a) Determine the energy stored in the magnetic field when the current reaches one fourth of its maximum value.

b) How long does it take for the current to reach this value?

Answer: a)The maximum current, after a long time has elapsed, will be 24/0.41 = 58.5A. A quarter of this is 14.63A. Energy stored in an inductor is 0.5*L*I^2 = 6.21J

b) you can use I(t)/ Imax =1 – exp(-t*R/L) where I(t) is the current at time t (14.63A), Imax is the maximum current (58.5A), R is the resistance, L the inductance

0.75 = exp(-t*7.07)

t*7.07= 0.288

t = 40.7ms

 

· A motor has a single loop inside a magnetic field of magnitude 0.95 T. If the area of the loop is 200. cm2, find the maximum angular speed possible for this motor when connected to a source of emf providing 210 V.

Chapter 30

· 30.27. A 18-µF capacitor is fully charged by a 12-V battery and is then disconnected from the battery and allowed to discharge through a 0.9-H inductor. Find the first three times when (the magnitude of) the charge on the capacitor is 78-µC, taking t = 0 as the instant when the capacitor is connected to the inductor.

· 30.31. A 2.10-µF capacitor was fully charged by being connected to a 12.0-V battery. The fully charged capacitor is then connected in series with a resistor and an inductor: R = 53 Ω and L = 0.240 H. Calculate the damped frequency of the resulting circuit.

30.34. A capacitor with capacitance C = 6. 10-6 F is connected to an AC power source having a peak value of 13 V and f = 120 Hz. Find the reactance of the capacitor and the maximum current in the circuit. (When entering units, use ohm for Ω.)· 30.44. Design an RC high-pass filter that passes a signal with frequency 9 kHz, has a ratio Vout/Vin = 0.5, and has an impedance of 4.9 kΩ at very high frequencies. (When entering units, use ohm for Ω.) (a) What components will you use? (b) What is the phase of Vout relative to Vin at the frequency of 9 kHz?

· 30.50. A circuit contains a 100-Ω resistor, a 0.0510-H inductor, a 0.450-µF capacitor, and a source of time-varying emf connected in series. The time-varying emf is 55.0 V at a frequency of 2000 Hz. (a) Determine the current in the circuit. (b) Determine the voltage drop across each component of the circuit. (c) How much power is drawn from the source of emf?

· 30.54. A transformer has 820 turns in the primary coil and 34 turns in the secondary coil. (a) What happens if an AC voltage of 100 V is across the primary coil? (What is the voltage across the secondary coil?) (b) If the initial AC current is 2 A, what is the output current? (c) What happens if a DC current at 100 V flows into the primary coil? (What is the voltage across the secondary coil?) (d) If the initial DC current is 2 A, what is the output current?

Chapter 31

· A wire of radius 1.0 mm carries a current of 21.0 A. The wire is connected to a parallel plate capacitor with circular plates of radius R = 3.1 cm and a separation between the plates of s = 2.0 mm. What is the magnitude of the magnetic field due to the changing electric field at a point that is a radial distance of r = 1.0 cm from the center of the parallel plates? Neglect edge effects. (Assume the current has just begun charging the capacitor.) Ans:- 4.37e-05 T

Three FM radio stations covering the same geographical area broadcast at frequencies 94.9, 95.1, and 95.3 MHz, respectively. What is the maximum allowable wavelength width of the band-pass filter in a radio receiver so that the FM station 95.1 can be played free of interference from FM 94.9 or FM 95.3? Use c = 3.0 108 m/s, and calculate the wavelength to an uncertainty of 1 mm. (State your answer to two significant digits.) Ans:- 13 mm· A voltage, V, is applied across a cylindrical conductor of radius r, length L, and resistance R. As a result, a current, i, is flowing through the conductor, which gives rise to a magnetic field, B. The conductor is placed along the y-axis, and the current is flowing in the positive y-direction. Assume that the electric field is uniform throughout the conductor. (a) Find the magnitude and the direction of the Poynting vector at the surface of the conductor. (Use the following as necessary: V, r, L, R, and μ0.) Show that = i2R

· A laser produces light that is polarized in the vertical direction. The light travels in the positive y-direction and passes through two polarizers, which have polarizing angles of 35° and 55° from the vertical, as shown in the figure. The laser beam is collimated (neither converging nor expanding), has a circular cross section with a diameter of 1.70 mm, and has an average power of 16 mW at point A. At point C, what are the magnitudes of the electric and magnetic fields, and what is the intensity of the laser light? Ans:- 1770 V/m, 5.92e-06 T, 4180 W/m^2

· A microwave operates at 565 W. Assuming that the waves emerge from a point source emitter on one side of the oven, how long does it take to melt an ice cube 2 cm on a side that is 10 cm away from the emitter if 10% of the photons are absorbed by the cube? Assume a cube density of 0.96 g/cm3. (Also assume that the ice cube is initially at 0°C but has not yet started melting, and that the ice cube maintains its shape until it absorbs enough energy to completely melt.) How many photons of wavelength 10 cm hit the ice cube per second? Ans:- 3.97 hr, 9.05e+23 photon/sec

Chapter 32 practice Problems

· A single concave spherical mirror is used to create an image of a source 6.80 cm tall that is located at position x = 0 cm which is xo = 18.0 cm to the left of Point C, the center of curvature of the mirror, as shown in the figure. The magnitude of the radius of curvature for the mirror is R = 13.5 cm. Calculate the position xi where the image is formed. Use the coordinate system given in the drawing. What is the height hi of the image? Is the image upright or inverted (upright = pointing up, inverted = pointing down)? Is it real or virtual? Ans:- 22.9 cm, 1.85 cm, inverted, real

· A ray of light is incident on an equilateral triangular prism with an index of refraction of 1.29. The ray is parallel to the base of the prism when it approaches the prism. The ray enters the prism at the midpoint of one of its sides, as shown in the figure. What is the direction of the ray when it emerges from the triangular prism? Ans:- 21.2 degree

· In a step-index fiber, the index of refraction undergoes a discontinuity (jump) at the core-cladding interface, as shown in the figure. Infrared light with wavelength 1530 nm propagates through such a step-index fiber through total internal reflection at the core-cladding interface. The index of refraction for the core at 1530 nm is ncore = 1.48. If the maximum angle αmax at which light can be coupled into the fiber such that no light will leak into the cladding is αmax = 13.832°, calculate the percent difference between the index of refraction of the core and the index of refraction of the cladding. (Let dcore = 9 µm and dcladding = 125 µm.) Ans:- 1.31. %

· Refer to the figure below and prove that the arc of the primary rainbow represents the 42° angle from the direction of the sunlight.

· A concave mirror forms a real image twice as large as the object. The object is then moved such that the new real image produced is three times the size of the object. If the image was moved 69.0 cm from its initial position, how far was the object moved? What is the focal length of the mirror? Ans:- 11.5 cn, 69 cm

· How deep does a point in the middle of the floor of a 3-m-deep pool appear to a person standing 3.3 meters horizontally from the point? Take the refractive index for the pool to be 1.3 and for air to be 1. (Assume the person is standing in the pool with their eyes just above the level of the water.) Ans:-0.938 m

Chapter 33 practice Problems

· Demonstrate that the minimum distance possible between a real object and its real image through a thin convex lens is 4f, where f is the focal length of the lens.

· An air-filled cavity bound by two spherical surfaces is created inside a glass block. The two spherical surfaces have radii R1 = 34.4 cm and R2 = 21.0 cm, respectively, and the thickness of the cavity is d = 41.2 cm (see diagram below). A light-emitting diode (LED) is embedded inside the block a distance of 60.0 cm in front of the cavity. Given nglass = 1.50 and nair = 1.00, and using only paraxial light rays (i.e., in the paraxial approximation) do the following. (The paraxial approximation is a small angle approximation so sin(θ) ≈ θ. This applies to both the incident angles and refracted angles.) (a) Calculate the final position of the image of the LED through the air-filled cavity. (b) Draw a ray diagram showing how the image is formed. Ans:- 48.2 cm, to the left of the second surface.

· Two lenses are used to create an image for a source 10.0 cm tall that is located at do = 31.0 cm to the left of the first lens, as shown in the figure below. Lens L1 is a biconcave lens made of crown glass (index of refraction n = 1.55) and has a radius of curvature of 20.0 cm for both surface 1 and 2. Lens L2 is d = 39.0 cm to the right of the first lens L1. Lens L2 is a converging lens with a focal length f = 30.5 cm. At what distance relative to the object does the image get formed? Determine this position by sketching rays and calculating algebraically. Ans:- 77.1 cm, to the right of lens L2

· Some reflecting telescope mirrors utilize a rotating tub of mercury to produce a large parabolic surface. If the tub is rotating on its axis with an angular frequency ω, show that the focal length of the resulting mirror is: f = g/2ω2.

· Jack has a near point of 30.2 cm and uses a magnifier of 27.0 diopter. (a) What is the magnification if the final image is at infinity? (b) What is the magnification if the final image is at the near point? Ans:- 8.15, 9.15.

· A diverging lens with f = −28.0 cm is placed 16.1 cm behind a converging lens with f = 21.6 cm. Where will an object at infinity in front of the converging lens be focused? Ans:- 6.84 cm

Chapter 34

1- Coherent monochromatic light with wavelength λ = 530 nm is incident on two slits that are separated by a distance d = 0.500 mm. The intensity of the radiation at a screen 2.50 m away from each slit is 168.0 W/cm2. Determine the position y1/3 at which the intensity of the central peak (at y = 0) drops to Imax/3. Ans:- 0.00108 m

· White light (400 nm < λ < 750 nm) shines onto a puddle of water (n = 1.33). There is a thin (109.7 nm thick) layer of oil (n = 1.47) on top of the water. What wavelengths of light would you see reflected? (Enter your answers from smallest to largest starting with the first answer blank. Enter NONE in any remaining answer blanks.) Ans:- 645 nm, none

· Some mirrors for infrared lasers are constructed with alternating layers of hafnia and silica. Suppose you want to produce constructive interference from a thin film of hafnia (n = 1.90) on BK-7 glass (n = 1.51) when infrared radiation of wavelength 1.39 µm is used. What is the smallest film thickness that would be appropriate, assuming the laser beam is oriented at right angles to the film? Ans:- 1.83e-07 m

· Sometimes thin films are used as filters to prevent certain colors from entering a lens. Consider an infrared filter, designed to prevent 847.0-nm light from entering a lens. Find the minimum film thickness for a layer of MgF2 (n = 1.38) to prevent this light from entering the lens. (Assume that nlens < 1.38.) Ans:- 153 nm

· White light shines on a sheet of a material that has a uniform thickness of 1.44 µm. When the reflected light is viewed using a spectrometer, it is noted that light with wavelengths of 499.2 nm, 561.6 nm, 641.8 nm, 748.8 nm, and 898.6 nm is not present in the reflected light. What is the index of refraction of the material? Ans:- 1.56

· A common interference setup consists of a plano-convex lens placed on a plane mirror and illuminated from above at normal incidence with monochromatic light. The pattern of circular interference fringes (fringes of equal thickness)—bright and dark circles—formed due to the air wedge defined by the two glass surfaces, is known as Newton’s rings. In an experiment using a plano-convex lens with focal length f = 80.50 cm and index of refraction nl = 1.500, the radius of the third bright circle is found to be 0.8525 mm. Determine the wavelength of the monochromatic light. Ans:- 7.22e-07 m

A single beam of coherent light (λ = 633 · 10−9 m) is incident on two glass slides, which are touching at one end and are separated by a t = 0.0275-mm thick sheet of paper on the other end, as shown in the figure below. Beam 1 reflects off the bottom surface of the top slide, and Beam 2 reflects of the top surface of the bottom slide. Assume that all the beams are perfectly vertical and that they are perpendicular to both slides, i.e., the slides are nearly parallel (the angle is exaggerated in the figure); the beams are shown at angles in the figure so that they are easier to identify. Beams 1 and 2 recombine at the location of the eye in the figure below. The slides are L = 7.20 cm long. Starting from the left end (x = 0) at what positions xbright do bright bands appear to the observer above the slides? (Enter your answer as a function of m. Assume x is measured in meters.) Ans:-
 
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Astronomy Assignment

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Name:

NAAP – Blackbody Curves & UBV Filters 1/7

Lab – Blackbody Curves & UBV Filters

Background Material

Thoroughly review the “Spectra” and “Filters” background pages. The color index page may also

be helpful to review. Here the links:

http://astro.unl.edu/naap/blackbody/spectra.html

http://astro.unl.edu/naap/blackbody/filters.html

Filters Simulator Overview

The filters simulator allows one to observe light from various sources passing through multiple

filters and the resulting light that passes through to some detector. An “optical bench” shows the

source, slots for filters, and the detected light. The wavelengths of light involved range from 380

nm to 825 nm which more than encompass the range of wavelengths detected by the human eye.

The upper half of the simulator graphically displays the source-filter-detector process. A graph of

intensity versus wavelength for the source is shown in the leftmost graph. The middle graph

displays the combined filter transmittance – the percentage of light the filters allow to pass for

each wavelength. The rightmost graph displays a graph of intensity versus wavelength for the

light that actually gets through the filter and could travel on to some detector such as your eye or

a CCD. Color swatches at the far left and right demonstrate the effective color of the source and

detector profile respectively.

The lower portion of the simulator contains tools for controlling both the light source and the

filter transmittance.

 In the source panel perform the following actions to gain familiarity.

o Create a blackbody source distribution – the spectrum produced by a light bulb which is a continuous spectrum. Practice using the temperature and peak height

controls to control the source spectrum.

o Create a bell-shaped spectrum. This distribution is symmetric about a peak wavelength. Practice using the peak wavelength, spread, and peak height

controls to vary the source spectrum.

o Practice creating piecewise linear sources. In this mode the user has complete control over the shape of the spectrum as control points can be dragged to any

value of intensity.

 Additional control points are created whenever a piecewise segment is clicked at that location.

 Control points may be deleted by holding down the Delete key and clicking them.

 Control points can be dragged to any location as long as they don’t pass the wavelength value of another control point.

 In the filters panel perform the following actions to gain familiarity.

 

 

NAAP – Blackbody Curves & UBV Filters 2/7

o Review the shapes of the preset filters (the B, V, and R filters) in the filters list. Clicking on them selects them and displays them in the graph in the filters panel.

o Click the add button below the filters list.

 Rename the filter from the default (“filter 4”).

 Shape the piecewise linear function to something other than a flat line.

o Click the add button below the filters list.

 Select bell-shaped from the distribution type pull down menu.

 Alter the features of the default and rename the filter.

o If desired, click the remove button below the filters list. This removes the actively selected filter (can’t remove the preset B, V, and R filters). Filters are not

saved anywhere. Refreshing the flash file deletes the filters.

Click (or copy/paste) here for simulator:

http://astro.unl.edu/naap/blackbody/animations/filters.html

Filters Simulator Questions

 Use the piecewise linear mode of the source panel to create a “flat white light” source at maximum intensity. This source will have all wavelengths with equal intensity.

 Drag the V filter to a slot in the beam path (i.e. place them in the filter rack).

 Try the B and the R filter one at a time as well. Dragging a filter anywhere away from the filter rack will remove it from the beam path.

Sketch the graphs for the flat white light and V filter in the boxes below. What is Question 1:

the effective color of the detected distribution?

 

With the flat white light source, what is the relationship between the filter Question 2:

transmittance and the detected distribution?

 

 

 

source distribution

combined filter transmittance

 

detected distribution

 

 

 

NAAP – Blackbody Curves & UBV Filters 3/7

 Add a new piecewise linear filter.

 Adjust the filter so that only large amounts of green light pass. This will require that addition of points.

Use this green filter with the flat white light source and sketch the graphs below. Question 3:

ACME Source FILTER RACK ACME Detector

 

Use the blackbody option in the source panel to create a blackbody spectrum that Question 4:

mimics white light. What is the temperature of this blackbody you created?

 Add a new piecewise linear filter to the filter list.

 Modify the new filter to create a 40% “neutral density filter”. That is, create a filter which allows approximately 40% of the light to pass through at all wavelengths

(transmittance)

 Set up the simulator so that light from the “blackbody white light” source passes through this filter.

Sketch the graphs created above in the boxes below. (This situation crudely Question 5:

approximates what sunglasses do on a bright summer day.)

 

Remove all filters in the filters rack. Place a B filter in the beam path with the flat Question 6:

white light source (about 75% intensity). Then add a second B filter and then a third. Describe

and explain what happens when you add more than one of a specific filter.

 

 

 

source distribution

combined filter transmittance

 

detected distribution

 

source distribution

combined filter transmittance

 

detected distribution

 

 

 

NAAP – Blackbody Curves & UBV Filters 4/7

Place a B filter in the beam path together with the 40% neutral density filter. Then Question 7:

add a V filter into the beam path. Describe and explain what happens when you add more than

one filter to the filter rack.

 

 

 

Create a piecewise linear filter that when used with the Question 8:

flat white light source would allow red and blue wavelengths to pass

and thus effectively allowing purple light to pass. Draw the filter in

the box to the right.

FILTER RACK

 Remove all filters from the filters rack.

 Create a very narrow bell-shaped source distribution that is peaked at green wavelengths (somewhere close to 550 nm). Notice the color!

 Expand the spread of the source distribution to maximum. Notice how the color changes.

 Change the distribution source to a blackbody source peaked at green wavelengths (a temperature close to 5270 K). Again notice the color.

 

Using observations from the above actions, explain why we don’t observe “green Question 9:

stars” in nature, though there are indeed stars which emit more green light than other

wavelengths.

Close your applet. We are going to the next level, ready?

purple filter profile

NAAP – Blackbody Curves & UBV Filters 5/7

Blackbody – Curves Mode Familiarization

The Blackbody Curve Simulator has two main modes – the curves mode and the filters mode.

The curves mode allows the exploration of blackbody curves including their peak wavelength

and the area under the curve which is related to their total energy production.

 Learn how to add and remove curves and change their temperatures.

o Click the add curve button one or more times.

o Change the temperature slider. Notice which curve changes.

o Select a second curve and change temperature.

o Remove all but 1 or 2 extra curves.

 Learn the vertical scale options. Have 2 or 3 curves in the explorer.

o Change temperature with the auto scale all curves mode.

o Change temperature with the auto scale to selected curve mode.

o Change temperature with the lock scales mode.

 Learn the horizontal scale options. Select the horizontal scale tab.

o Note how changing the rightmost limit changes the view.

 Use, if desired, the indicate peak wavelength and highlight area under curve options.

Here the link:

http://media.wwnorton.com/college/astronomy/animations/interactive/bbexplorer.html

 

Create a blackbody curve of temperature Question 10:

6000 K and draw the shape in the box to the right.

Does it have a peak? Is it symmetric about this peak?

_____________________________________________

Create a second curve using the add curve button and use the temperature slider to Question 11:

vary its temperature, chose one with 7000 K then move down to 5000 K. Can you find a

blackbody curve of another temperature that intersects the 6000 K curve at some wavelength?

_______________________________________________________________________

NAAP – Blackbody Curves & UBV Filters 6/7

Make sure that there is only one curve and check indicate peak wavelength. Vary Question 12:

the temperature of the curve and note how the peak wavelength changes. Formulate a general

statement relating the peak wavelength to temperature. Then compare this statement with Wien’s

Law discussed in your textbook.

___ _ ______

Select the highlight area under curve option and lock the vertical scale. Vary the Question 13:

temperature of the curve and note how the area under the curve changes. Formulate a general

statement relating the area under curve to temperature.

(Calculator Required) Complete the following table below. The “Area Ratio” is Question 14:

the area for the curve divided by the area for the curve in the row above. This will tell you many

times greater the new ratio is compared to the previous one.

Can you specify a more precise statement relating the area under curve to temperature? Is this

consistent with what was referred to as the Stefan-Boltzmann Law in the background pages?

Curve Temperature Area Under Curve (W/m 2 ) Area Ratio

3000 K

 

6000 K

 

12000 K

 

24000 K

NAAP – Blackbody Curves & UBV Filters 7/7

Blackbody Explorer – Filters Mode

The second mode is the filters mode and explores the use of UBVR filters with blackbody

curves.

 Unselect highlight area under curve and indicate peak wavelength. Select the filters tab.

The light from a blackbody curve that passes through the UBVR filters are shown as colored

areas under the curve. It is this area which is later translated into a number for color magnitude.

Remember that a magnitude is a logarithmic version of the flux (i.e. the amount of light) that

passes through a filter and that lower numbers reflect larger fluxes. Note that this area depends

on both the source and the filter. What is listed as a V value is the apparent magnitude of a star

(assumed to blackbody which isn’t exactly true) through the V filter.

 

Vary the temperature and in the table to the right Question 15:

note the temperature in at which each filter peaks. Where are

the filters most sensitive, i.e. which temperature will give the

strongest response in a detector?

______________________________________________________________________________________________________________________________________________________

Use the color index feature to create a B-V index. This will compare the apparent Question 16:

magnitude of a star through the B filter to that through the V filter. Plot temperature on the

vertical axis and B-V on the horizontal.

Temperature B-V

3000 K

4000 K

5000 K

6000 K

8000 K

10,000 K

15,000 K

20,000 K

25,000 K

 

Use your graph to estimate the B-V value of a 12,000 K blackbody: ___________ Question 17:

Curve Peak Temperature

U

B

V

R

 
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Physics Lab

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temple university physics

Mapping the Electrostatic Potential and Electric Field

The objective of this experiment is to study the potentials, equipotential curves and electric fields produced by various two-dimensional electrostatic charge distributions. In practice, direct measurement of the electric field turns out to be quite difficult. Instead, we exploit the fact that the electric force is a conservative force, and thus can be considered to be associated with a potential – the electric potential , where the components of the electric field vector are given by the change of the electric potential in that direction,

    (1)

One consequence of Equation 1 is that if one can identify a line (or surface) along which the potential has a constant value then the electric field is necessarily perpendicular to that line at all points, see Figure 1. Therefore, in order to map the electric field for a charge configuration, it is sufficient to map out the equipotential lines.

There is a technical difficulty, however, with setting up and controlling static charge distributions: it is not easy to fix charges at precise locations. For this reason we will simulate static charge distributions using a small direct current flowing through electrodes, drawn to look like our static charge distributions, and conducting paper. The electric field shapes, potential and equipotential lines will be identical to those for the simulated static charge configurations.

Figure 1. Equipotentials and electric field lines for a positive point charge (circle at center).

 

NOTE: You may want to reference your text or other sources to confirm your results and aid in mapping the fields accurately and expeditiously.

 

 

 

Learning Goals for this Laboratory:

· Practice visualizing electric fields and electric potentials around conductors of many shapes.

· Practice graphing and analyzing nonlinear relations.

· Practice connecting simple circuits.

 

Apparatus

Pasco field mapping board, digital voltage meter with point probes, D.C. power supply, several sheets of conducting paper with different electrode configurations, push pins

 

Figure 2. Setup for Part IV of this lab showing the cork board with the parallel plate electrode configuration on conducting paper, voltmeter and D.C. power supply.

 

 

Part I. Point Source and Guard Ring

Figure 3. Point source and ring guard configuration. Voltmeter probes not shown.

 

1. The electrodes in this experiment are made with conducting silver paint on conductive paper. Locate the conductive paper with the point source and guard ring electrode configuration (Figure 3) and pin the corners of the paper to the cork board.

2. Take a cable with a banana plug on one end and a ring terminal on the other and plug the banana end into the positive of the power supply, then pin the ring terminal end of the cable into the central point electrode using a metal pushpin as in Figure 3. Make sure there is good contact between the ring terminal and the painted electrode. Try to avoid making new holes in the electrode with the pushpin, it is sufficient to have physical contact between the terminal of the wire lead and the silver of the electrode.

3. Similarly, connect the negative of the power supply with the circle-shaped guard ring electrode.

4. Turn on the power supply and set it to 5 V by first increasing the current limit knob, then increasing the voltage to 5 V. This voltage provides a continuous source of charge to the electrodes, creating the electric field and potential we will measure.

5. Before measuring the field and potential, let’s check the electrodes for proper conductivity (a damaged electrode could skew your results).

If the electrode is a good conductor, all points on the electrode should have nearly the same potential. For our purposes, the maximum potential between any two points on a single electrode should not be more than a few mV. Use the voltmeter to probe the potential between different points along an electrode. Please do not push the voltmeter probes through the paper; simply placing the probes on the paper should give you a good reading. If you see a potential larger than a few mV between different points on an electrode, first check that the power supply wires are firmly connected to the electrode via the pushpin. If you still see a potential where there shouldn’t be one the electrodes could be damaged, so ask your lab instructor to double-check your setup and get a replacement electrode if necessary.

6. Once you’ve checked for good conductivity, you’re ready to measure the potential of the point source. Place the black (common) voltmeter probe on the guard ring so that it is the reference, and use the red probe to measure the voltage starting at the point 2 mm from the edge of the point source and ending 20 mm from the point source at intervals of 2 mm. In this way, you will have 10 data points: 2, 4, 6, 8, and 10 mm from the point source. If it helps, you can use a pencil to mark the 2 mm points. The voltage at 2 mm should be somewhere in the 3 – 5 V range.

7. Make a graph of the potential as a function of distance from the point source (the edge of the point source is at = 0.0 cm, and = 5 V).

8. Make a second graph of potential vs. (do this by having Excel calculate in new column).

Question 1. According to theory, how does the potential of a point charge vary with distance as you move away from the point charge? Is this what your graph shows? For an ideal point source, the graph of potential vs. should be a line, is yours a line? Our point source is not ideal because the negative electrode is not infinitely far away. However, very near the point source the negative electrode is sufficiently far away as to have no influence, thus the first few points at higher voltage in the graph should follow a linear trend.

9. Mapping equipotential lines. Use the voltmeter to find equipotentials by keeping the black probe on the negative electrode and placing the red probe at a point somewhere inside the circle. Watch the voltmeter voltage reading while you slide the red probe along the paper while also trying to keep the voltage constant. This line of constant potential is an equipotential. Try doing this for one or two other values of potential nearer and farther from the point source.

Question 2. What are the shapes of the equipotentials in the region around a point source? Do your results agree with the equipotentials show in Figure 1?

10. Briefly survey the potential in the region outside the guard ring.

Question 3. Does the potential vary much from one point to another outside of the guard ring? What does this imply about the electric field outside the guard ring? Refer to equation 1 for help with your answer.

Part II. Electric Dipole

1. Replace the point source electrode with the electric dipole configuration and connect the leads to the 5 V source as in Fig. 4, making sure the pushpins provide good contact between the ring terminal and the silver paint.

Figure 4. Dipole configuration. Voltmeter probes not shown.

 

2. Place the black reference voltage probe halfway between the two electrodes, we’ll keep it fixed at this location. Place the red probe somewhere on the paper. Then slide the red probe along the paper while watching the voltage reading. Use this method to map out several equipotential curves nearer and farther from the point charges. Record your results on the white grid paper provided. To see what the shape should be in theory, refer to online or text sources for the potential of a dipole.

3. After drawing several equipotentials on your grid paper, use a different color to draw in the corresponding electric field lines. To do this, start at the positive charge and draw a line moving outward such that it crosses any equipotential lines at right angles. The electric field lines should terminate at the negative charge. Be sure to indicate the direction of your e-field with arrows along the lines. To see what the shape of these field lines should be, refer to online or text sources for the electric field of a dipole. Once you have several field lines drawn, take a picture of the resulting map of electric field and potential for your lab report.

Question 4. Note all but one of the equipotential lines for a dipole are curved. Where is the uniquely straight equipotential line for a dipole?

Part III. Like Charges in a Box

1. Set up the configuration shown in Fig. 5 and apply 5 V. Note that the two point charges are positive, you can use a wire to daisy-chain them together so they are both charged. Also note that the box electrode surrounding the point charges is negative. As in Part II, place your reference voltage probe halfway between the two electrodes.

Question 5. Relative to this halfway point, where on the conducting paper is the potential highest? Where is it lowest?

Figure 5. Like Charges in a Box. Voltmeter probes not shown.

 

2. As you did with the dipole, map out a few equipotential lines near the point charges and also near the walls of the box. Make a rough map of these equipotentials and their corresponding E-field and include it with your lab report.

Part IV. Parallel Plates

Figure 6. Parallel plate electrodes (blue lines) connected to power supply. Voltmeter probes not shown.

The potential and field in between charged parallel plates differs significantly from that of a point source (Part I).

1. Connect the parallel plate electrodes to a power supply as shown. Measure the potential every 0.5 cm along a line from the midpoint of the negative electrode to the midpoint of the positive electrode (i.e. along the dotted line in the figure at right). Use the negative electrode as the fixed reference for your measurement.

2. Make a graph of the potential as a function of distance from one plate (the reference probe is at = 0.0 cm).

Question 6. Referring to your graph, describe in how the potential changes with distance from the electrode. How does this contrast to the potential vs distance for a point source you found in Part I?

For Your Lab Report:

Refer to the lab syllabus and grading rubric for what to include in the report. It may be helpful to use the answers to the questions to aid in writing the discussion.

 

 

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