physics 152 lab manual

Physics 152(154) Laboratory

General Physics II

University of Massachusetts

Fall 2013

Contents

General Instructions vOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vEnrollment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vSchedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAttendance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixMoodle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAssignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xNotekeeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiLab Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiOffice Hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcademic Honesty . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDisability Services . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiSafety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiLab Make-up Policy . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Latent Heat of Vaporization of Liquid Nitrogen (LN2) 11.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Experiment Overview/Procedure . . . . . . . . . . . . . . . . 21.4 Details of Experimental Procedure . . . . . . . . . . . . . . . 5

1.4.1 How to measure rate of vaporization dmdt

. . . . . . . . 51.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Ideal Gas Law 82.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . 9

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2.3.1 Details of Experimental Procedure . . . . . . . . . . . 102.3.2 How to measure the height (H) of the air inside the

cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 How to determine absolute pressure (P ) in the cylinder 12

2.4 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 132.4.1 Room Temperature Measurement . . . . . . . . . . . . 132.4.2 Ice Water Measurement . . . . . . . . . . . . . . . . . 13

2.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 DC Circuits 163.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Voltage Divider . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Experiment Overview/Procedure/Analysis . . . . . . . 193.2.3 Practice to Use Oscilloscope . . . . . . . . . . . . . . . 19

3.3 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.1 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Experiment Overview/Procedure/Analysis . . . . . . . 21

3.4 Resistors in Series and Parallel . . . . . . . . . . . . . . . . . . 213.4.1 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.2 Experiment Overview/Procedure/Analysis . . . . . . . 21

3.5 Light Bulb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5.1 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 RC Circuits 244.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 RC Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.2 Experimental Procedure . . . . . . . . . . . . . . . . . 274.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 Capacitors in Series and Parallel . . . . . . . . . . . . . . . . . 284.3.1 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 284.3.2 Experimental Procedure . . . . . . . . . . . . . . . . . 294.3.3 How to measure the capacitance by using a square-

wave generator . . . . . . . . . . . . . . . . . . . . . . 29

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5 Magnetic Field Mapping 315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Single Circular Coil . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Helmholtz Coils . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5 Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 RLC Circuits 366.1 Forced Oscillations, Resonance . . . . . . . . . . . . . . . . . . 36

6.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.1.2 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 396.1.3 Experiment Overview/Procedure/Analysis . . . . . . . 396.1.4 Details of Experimental Procedure . . . . . . . . . . . 40

6.2 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2.2 Lab Goal . . . . . . . . . . . . . . . . . . . . . . . . . 426.2.3 Details of Experimental Procedure . . . . . . . . . . . 43

A Safety 45A.1 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.2 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.3 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.4 High Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B Graphical Presentation of Data 47B.1 Graphing using Logger Pro . . . . . . . . . . . . . . . . . . . . 47

B.1.1 Importing Data into Logger Pro . . . . . . . . . . . . . 47B.2 Graphing using Excel . . . . . . . . . . . . . . . . . . . . . . . 47

B.2.1 Importing Data into MS Excel . . . . . . . . . . . . . . 47

C Data and Error Analysis 48C.1 Expressing Measurements . . . . . . . . . . . . . . . . . . . . 48C.2 Uncertainties in Direct Measurements . . . . . . . . . . . . . . 49

C.2.1 Estimating Uncertainties from a Scale - Interpolation . 49C.2.2 Uncertainties from a Digital Scale . . . . . . . . . . . . 50

C.3 Repeated Measurement (Statistical) Technique . . . . . . . . . 51C.3.1 Mean Value (Average Value) . . . . . . . . . . . . . . . 51C.3.2 Standard Deviation . . . . . . . . . . . . . . . . . . . . 51

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C.3.3 Standard Deviation of the Mean . . . . . . . . . . . . . 52C.4 Propagation of Uncertainties . . . . . . . . . . . . . . . . . . . 52

C.4.1 Addition and Subtraction . . . . . . . . . . . . . . . . 53C.4.2 Multiplication and Division . . . . . . . . . . . . . . . 54

C.5 Simple Rules of Uncertainty Propagation . . . . . . . . . . . . 55

D Laboratory Reports 57D.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

D.1.1 Heading . . . . . . . . . . . . . . . . . . . . . . . . . . 57D.1.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . 58D.1.3 Questions and Answers . . . . . . . . . . . . . . . . . . 58D.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 58

D.2 Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59D.2.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 59D.2.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . 59D.2.3 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

D.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60D.4 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60D.5 Sample Report . . . . . . . . . . . . . . . . . . . . . . . . . . 61

iv

General Instructions

Overview

The physics laboratory is a complementary learning process to the rest of thecourse. It is primarily designed to develop your skills in scientific observation,analysis, and writing. The subject matter is only loosely correlated with whatyou learn in lecture. Experiments in the lab will NOT necessarily be syn-chronized with the lectures, and you may often begin an experiment beforeall aspects of the experiment have been fully covered in lecture. Nonetheless,every effort is made to perform the experiments in a logical order, so that youcan build upon previous experiences and skills. Occasionally, some experi-ments may lie outside the subject matter of the lecture and vice-versa. In allcases, your section instructor will take the responsibility for introducing youto the relevant concepts for the lab. The lab section instructor is responsiblefor weekly interactions with the students, including direct interaction in thelab, office hours and individual lab report grades. In most lab matters, thesection instructor is the final authority.

Enrollment

For issues involving enrollment (adding, switching or dropping lab sections)please contact Kris Reopell ([email protected]), the physics de-partment course manager and scheduler.

Students that are special cases and wish to enroll in the lab only Physics133, 134, 153, or 154 should contact Kris for assistance.

v

mailto:[email protected]


Physics 152(154) Laboratory · General Instructions · Fall 2013 vi

Repeated Courses

For students that are repeating Physics 131, 132, 151, or 152 there is nolab exemption. The laboratory is integrated with the lecture into a single4 credit course. There is no separate grade for the laboratory. Studentsrepeating a course must complete all components of the course including lab.Scores for lab reports that were submitted in a previous semester will not beaccepted in substitution for reports during current semester.

Schedule

There are six experiments scheduled for this semester. Each experiment isperformed on a roughly two week cycle. Students will attend laboratory everyother week (Week 1 or Week 2) according the Section Schedule and WeeklySchedule. Please note that there are University recognized holidays that willchange the lab schedule from time to time. Please visit the University Regis-trar’s Academic Calendar to see when these holidays will occur. It is yourresponsibility to read and understand the times your lab sectionswill meet. Please check your schedule on SPIRE and cross reference it withthe schedule below.

http://www.umass.edu/registrar/calendars/academic-calendar

https://www.spire.umass.edu/

Section Schedule Physics 152(154) – Fall 2013

Week 1 – labs begin Sept. 9

Hour Mon Tue Wed Thu Fri

8:00AM 5 13

10:10AM 2 6 10 14

12:20PM 7 15

2:30PM 4 8 12 16

4:40PM 17 18

Week 2 – Office Hours/Discussions

Hour Mon Tue Wed Thu Fri

8:00AM

10:10AM

12:20PM

2:30PM

4:40PM

Physics 152(154) Laboratory · General Instructions · Fall 2013 vii

Section Schedule

Physics 152(154) Weekly Schedule – Fall 2013

Weekof

Week ExperimentNumber

Experiment

Sept. 2 No Lab – Classes start Sept. 3

Sept. 9 1 1 LN2

Sept. 16 2 1 Office Hours

Sept. 23 1 2 Ideal Gas Law

Sept. 30 2 2 Office Hours

Oct. 7 1 3 DC Circuits

Oct. 14 2 3 Office Hours

Oct. 21 1 4 RC Circuits

Oct. 28 2 4 Office Hours

Nov. 4 1 5 Magnetic Field Mapping

Nov. 11 2 5 Office Hours

Nov. 18 1 6 RLC Circuits

Nov. 25 No Labs – Thanksgiving Nov. 28

Dec. 3 2 6 Office Hours

No Labs - Classes end Dec. 6

Physics 152(154) Laboratory · General Instructions · Fall 2013 viii

Weekly Schedule

Physics 152(154) Laboratory · General Instructions · Fall 2013 ix

How to figure out your lab schedule

• Log onto SPIRE and look for your register lab section. In SPIRE youshould see a two digit number with and “L” in front of it. For exampleif you are enrolled in Physics 131, you may see “131-L03”. The “03”means lab section 3.

• Look at the lab Section Schedule and see where the section number isin the schedule; i.e., what Week, Day, and Time. The Day and Timeshould match the schedule in SPIRE. See which Week (1 or 2) yoursection will meet.

• Look at the lab Weekly Schedule and see the weeks of the semesteryour experiment will be perform. That will be the schedule you willfollow.

Attendance

Attendance is mandatory. Attendance will be taken at each lab. You arerequired to attend the lab section you are formally enrolled in. Given thelimited amount of space, personnel and equipment we do not permit openattendance; i. e., you may not sit in any lab section of your choosing. If youattend a lab section that your are not enrolled in and do not have approvalyou will not receive credit for your work. In the event you have a legitimatereason for missing lab (religious observances, illness, schedule conflict due toextracurricular activity, or jury duty) and need to make up an experimentplease see the Lab Make-up Policy. Additional information on attendanceand absences can be found at the Office of the University Registrar web siteand the Academic Regulations handbook.

Moodle

Moodle is the learning management system (LMS) for the laboratory portionof the course. A separate Moodle course page from the lecture portion of thecourse will be maintained for the laboratory. All material pertaining to thelaboratory (manuals, schedules, assignments, report grades) can be accessthrough Moodle. You can access these materials by logging on to Moodle(http://moodle.umass.edu/) using your UMass NetID and password.

https://www.spire.umass.edu/

http://www.umass.edu/registrar/students/policies-and-practices/class-absence-policy

http://www.umass.edu/registrar/students/policies-and-practices/academic-regulations

http://moodle.umass.edu/



Physics 152(154) Laboratory · General Instructions · Fall 2013 x

Assignments

You will perform six experiments this semester. The assignment associatedwith each experiment includes:

• Attendance and completion of the experiment,

• keeping a record of the data taken during the experiment,

• completion of in-lab questions,

• and writing a laboratory report.

Grades

Your grade is computed based on 6 experiments, 20 points per experimentfor a total of 120 points. The point distribution for each experiment is:

• In-lab assignment includes:

– 2 points (10%) for your signed notebook data sheet, and

– 4 points (20%) for the answers to the in-lab questions.

• Lab Report includes:

– 1 point (5%) for abstract of your report,

– 12 points (60%) for answers to lab questions in report, and

– 1 point (5%) for conclusion to report.

• Totaling 20 points for each experiment.

You are expected to complete all 6 experiments. At the end of the semester,the total score for all of your experiments is reported to your course instruc-tor, who will determine how your lab grade is to be figured into your coursegrade.

For students taking the lab only (Physics 133, 134, 153 or 154), withoutthe lecture, the final letter grade is determined by the lab faculty supervisor.To get a passing grade in the lab component of the course, a student MUSThave performed all 6 experiments. A minimum grade of 60% is required topass.

Physics 152(154) Laboratory · General Instructions · Fall 2013 xi

Notekeeping

There are two options permitted for keeping notes:

• Digital notes: you will have the option of downloading from Moodle aworksheet or spreadsheet that you can fill out during lab using a laptopcomputer, then resubmit as an assignment through Moodle at the endof the lab period. OR,

• Carbonless copy notes: you can purchase a carbonless copy laboratorynotebook from the University Textbook Annex. In the notebook youwill record the same information on the worksheet, then submit a copyto your lab instructor at the end of the lab period.

We strongly invite students to bring a laptop computer to lab and use thedigital notes option, but the conventional lab notebook is acceptable.

Lab Reports

You will write a lab report for each experiment you perform this semester.A report is a testimony of the work you perform in the lab, your method ofanalysis, and the results that you obtain. Therefore, the style of writing areport is very different from any other style of writing you may have done(essay, term papers, articles, poems). Details on writing a laboratory reportcan be found in the Appendix D on Laboratory Reports.

• Reports will be due 8 days after the experiment is perform (1 day afteroffice hours), no later than 5:00 PM.

• Reports must be typeset, no hand written reports.

• Reports submitted after the deadline are considered late and will re-ceive an automatic 20% deduction for each day late.

• Reports submitted more than 5 days after the deadline will not beaccepted and will receive no credit.

Printed copies of lab reports are to be submitted to drop boxes located inHasbrouck. For P131(133) and P132(134) the drop boxes are located in the


Physics 152(154) Laboratory · General Instructions · Fall 2013 xii

first floor lobby of Hasbrouck. For P151(153) and P152(154) the drop boxesare located on the ground level of Hasbrouck near the vending machines.Submit your report to the drop box labeled with your course, section number,and lab section instructor’s name on it.

Office Hours

Office hours are held the week after the lab is done. Office hours are usuallythe best time to have your questions answered. Office hours will be heldin the lab room at the same lab meeting time (except for certain sectionsof Physics 131(133) and 151(153) for which a different room will be madeavailable). A list of office hour locations is posted on Moodle.

Academic Honesty

Since the integrity of the academic enterprise of any institution of highereducation requires honesty in scholarship and research, academic honesty isrequired of all students at the University of Massachusetts Amherst. Aca-demic dishonesty is prohibited in all programs of the University. Academicdishonesty includes but is not limited to: cheating, fabrication, plagiarism,and facilitating dishonesty. In the specific case of writing lab reports,collaboration is explicitly forbidden. Addition information on academichonesty can be found on the Ombuds Office web site under Academic Hon-esty.

Disability Services

Some students are eligible to received accommodations as defined by Disabil-ity Services. Letters of accommodation are sent to the instructors of eachcourse a student is enrolled in. In the specific case of physics laboratoriesthe letters of accommodation are sent to the faculty supervisor of laborato-ries not the laboratory TA’s. Before accommodations in the laboratory areimplemented for an eligible student the student must:

• contact the faculty supervisor of laboratories and discuss which accom-modations are applicable to the student’s laboratory work,


http://www.umass.edu/ombuds/

http://www.umass.edu/ombuds/honesty.php/

http://www.umass.edu/ombuds/honesty.php/

http://www.umass.edu/disability/index.html

http://www.umass.edu/disability/index.html



Physics 152(154) Laboratory · General Instructions · Fall 2013 xiii

• provide the name and contact information of the eligible student’s con-sumer manager,

• agree to the accommodations specifically discussed with the facultysupervisor of laboratories.

Accommodations are not open ended. They apply only to specific coursesand during a specific semester a student is enrolled. Accommodations do notautomatically carry over from one semester to another. Accommodations donot release an eligible student from any requirements of the laboratory likecompletion of assignments, deadlines, or attendance of the laboratory. A stu-dent who is eligible for accommodations but does not meet the requirementsof the laboratory will still be subject to point deductions for late submissionsor loss of credit for work not submitted.

Students are not obligated to divulge the reasons why they areeligible for accommodations. The reasons a student is eligible may be asensitive matter and a student has a right to privacy.

Safety

Occasionally, we may be handling dangerous equipment. It is imperativethat you treat all the equipment with care. Most equipment is painful toreplace, and some items are hazardous objects that could cause physicalinjury or worse fatality. Your section instructor will give proper handlinginstructions for all the equipment. Please follow directions and keep thingsneat around the lab. Additional information on safety can be found in Ap-pendix A on Safety.

Lab Make-up Policy

Only legitimate and documented excused absences recognized bythe University will be permitted to make-up a missed lab. Excusedabsences recognized by the University include:

• Religious Observances



Physics 152(154) Laboratory · General Instructions · Fall 2013 xiv

• Required participation in athletic events. Prior notification of absenceis required.

• Health or medical reasons. A note from UHS or doctor is required.

• Field Trip. Prior notification of absence is required

• Jury duty.

• Military service.

• Personal or family crisis, emergency or bereavement. Students arestrongly urged to contact the Dean of Students Office and notify themin such cases as soon as possible.

The following are considered unexcused absences and NO make-up willbe permitted:

• Truancy - skipping lab.

• Oversleeping.

• Forgetfulness.

• Procrastination.

• Confusion of where and when lab meet.

If a student has a legitimate and documented excuse the lab must bemade up with in 2 weeks of the original scheduled lab. After thattime period there will be no make up and the student will receive a zero forthe missed lab. The procedure for making up a missed lab is:

• Contact your lab section instructor as soon as possible.

• Present documentation verifying reason for absence.

• Schedule a time for a make-up. Start with attending lab 1) duringoffice hours on alternate week, 2) attending another lab section withpermission of section instructor or 3) arrange a time with your sectioninstructor to make up lab in lab make up room (Hasbrouck 201).

http://www.umass.edu/dean_students/contact/

Physics 152(154) Laboratory · General Instructions · Fall 2013 xv

If you know a head of time that you have a schedule conflict due to anothercourse, extra curricular activity or appointment, please inform your sectioninstructor as soon as possible. It is much easier to arrange a make- up ahead of time. The lab schedule is set before the beginning of the semesterand it is the students responsibility to manage time and identify any possibleschedule conflicts.

Some absences may be due to extended illness or bereavement and a labmake-up may not be possible in the allotted 2 week period. In these casesthe student should contact the lab faculty supervisor as soon as possible. Iffaculty supervisor deems it warranted the student will receive a pass on themissed lab and the student will be graded on the remaining work he/she hascompleted.


Experiment 1

Latent Heat of Vaporization ofLiquid Nitrogen (LN2)

To start with illustration—rubber band placed in LN2 becomes rigid andbends no more loosing its elasticity even after it is unfrozen.

It is not fatal if small droplets of LN2 will fall on your skin, as they willroll over skin. But if you drink, pour on a cloth, try to keep LN2 in yourpalm or immerse a finger in LN2, then you can get a severe frostbite.

1.1 Theory

Definition: One single atom can be considered to be a monatomic molecule.So, further only the general term molecule will be used here to stand for bothterms—atom and molecule.

If one tries to adopt a microscopic approach for the description of gases,liquids, solids, etc., then she/he most probably will start with a mechan-ical description of motion of each molecule, i.e., will figure out forces be-tween molecules, will define molecules’ initial positions and velocities and,finally, will solve equations of motion. But even in 1 cm3 of air one havean extremely huge amount—about 1024—of molecules to handle. Thus suchstraightforward microscopic approach is an impractical insanity. However,there are alternatives, for example, thermodynamics, statistical physics, ki-netic theory, etc. In contrast to classical mechanics, in thermodynamics todescribe “behavior” of ∼ 1024 molecules in a given volume one introducessuch macroscopic quantities as density, pressure, temperature, etc. For ex-

1

Physics 152(154) Laboratory · LN2 · Fall 2013 2

ample, temperature of a solid corresponds to average intensity of oscillationsof molecules around their equilibrium positions.

Consider heating of some system. One increases temperature or in otherwords intensifies random activity at a microscopic scale. If at some criticaltemperature there is a qualitative change of the system state, the system issaid to undergo a phase transition. For example, a crystal melts loosing itsrigidity, or a liquid boils transforming into gas. We are going to be interestedin the latter process.

In a gas, molecules fly independently rarely bumping into one another. Ina liquid, molecules wander around, but they are kept together as a wholeby attraction forces acting among molecules. When liquid has reached itsboiling temperature its molecules are at the highest possible level of randomactivity. And additional energy one supplies to the liquid does not changeits temperature, but is used by some amount of molecules to break “bonds”with their neighbors and to evaporate/free out in a gas. Because moleculesevaporate independently they share the total supplied energy in equal por-tions. Thus the number of evaporated molecules is proportional to suppliedenergy, and, finally, at boiling point one gets

Energysupplied = L ·Massevaporated, (1.1)

where L is nothing but a coefficient of proportionality. L is called latentheat of vaporization. Physically, L is a macroscopic characteristic of boiling.Numerical value of L is specific to a particular liquid. It is sometimes helpfulto think of L as the amount of energy required per unit mass of a substanceto vaporize the substance at its boiling point.

1.2 Lab Goal

In this lab you will attempt to measure latent heat of vaporization of liquidnitrogen (LN2).

1.3 Experiment Overview/Procedure

Meditating a little on Eq. (1.1), you will understand that in order to achievegoal of § 1.2 you need a device where you can control energy input into the


Figure 1.1: A double walled styrofoam cup standing on a scale with a resistorsuspended within LN2.

liquid and at a same time measure evaporated mass. [Or, a device where youcan control rate of energy input into the liquid and at a same time measurerate of evaporated mass.]

Consider apparatus depicted at Fig. 1.1. You have a double walled styro-foam cup standing on a scale with a resistor “hanging” inside. [A resistor is adevice which converts electrical energy into heat. Familiar examples are thefilament of a toaster or of a light bulb.] In your thermos bottle you have LN2

at a boiling temperature of 77 Kelvin (about −321 F). Once you pour LN2

in a cup, heat from a much warmer atmospheric surrounding environment,namely the room, will be conducted to LN2 from walls and top of a cup andalso along wires. And if you turn on electrical circuit, then in addition theresistor will directly heat the volume of LN2. As a result you will observeLN2 boiling and its level and mass decreasing as it continuously evaporatesinto the atmosphere. [Of course, temperature of boiling LN2 is not changing!]

When heater is on—current through the resistor is switched on—energy bal-ance for some small time ∆t is:

∆Eenv + V · I ·∆t = L ·∆mon. (1.2)

Energy ∆Eenv comes from the environment. The battery of constant voltage


V creates constant current I in the electrical circuit. As a result the resistorproduces V · I ·∆t amount of energy during time ∆t.1 During time ∆t, all“supplied” energy [left side of Eq. (1.2)] is used to evaporate mass ∆mon ofLN2 [right side of Eq. (1.2)]. Equation (1.2) is time dependent; longer youobserve the system more energy will be “supplied” and more mass of LN2

will evaporate. Dividing both sides of Eq. (1.2) by ∆t you can arrive to timeindependent differential equation:2

Penv + V · I = L · dmon

dt, (1.3)

where Penv = ∆Eenv

∆tis the rate of energy supply from environment, and dmon

dt

is the rate of mass evaporation with heater on. If [and only if this is actuallytrue in your experiment!] the level of LN2 in the cup does not vary a lotthroughout your measurements, then you can assume constant environmentalconditions to be present, i.e., the rate of energy supply from environment,Penv, can be assumed to be constant. Heating power of battery, V · I, isconstant. L is a number. Therefore, according to Eq. (1.3), the rate of evap-oration dmon

dtof LN2 should be constant.

Your strategy to evaluate L from Eq. (1.3) will be:

1. measure constant V and I with voltmeter and ammeter.

2. determine constant rate of mass evaporation dmon

dt(see § 1.4).

3. “correct” for environmental heating Penv.

Note that Penv is small but it is not negligible compared with V · I. Step 3can be carried out with the heater turned off. The energy balance is

∆Eenv = L ·∆moff , (1.4)

where ∆moff is evaporated mass of LN2 during some time ∆t when heater isoff. Dividing both sides of Eq. (1.4) by ∆t you arrive to equation

Penv = L · dmoff

dt. (1.5)

1If V is in Volt and I is in Ampere then combination of units for product V · I isVolt ·Ampere = Joule/second ≡Watt.

2Choosing ∆t sufficiently small, ∆m∆t is replaced by the derivative dm

dt . Basically, weare deriving differential equations (1.3) and (1.5) for time dependant evaporated massesmoff(t) and mon(t), respectively.


Note the same Penv as in Eq. (1.3). [Formally, you could have derivedEq. (1.5) by setting V and I to be 0 in Eq. (1.3).] Penv was assumed tobe constant. L is a number. Therefore, according to Eq. (1.5), the rate ofevaporation dmoff

dtof LN2 should be constant. If you measure constant rate

of evaporation, dmoff

dt, (see § 1.4), you will not be able to calculate L from

Eq. (1.5) because you do not know the value of Penv. Instead, substitute Penv

from Eq. (1.5) into Eq. (1.3) to obtain the final formula for L

L =V · I

dmon

dt− dmoff

dt

(1.6)

[You can say that the denominator in the equation above is the net rate ofevaporation solely corresponding to the heating caused by the resistor.]

To conclude, by analyzing vaporization both with, and without, heatingthrough a resistor, using Eq. (1.6), you will be able to evaluate latent heatof vaporization L of LN2.

1.4 Details of Experimental Procedure

Make sure that resistor does not touch walls of cup in the lowest and highestpossible positions of a scale, so that it will definitely not stick cup duringyour manipulations with a scale.

Making next attempt to measure dmdt

do not forget to refill a cup with LN2 tothe level of previous trial. This will ensure constant environmental conditions(i.e., one and the same Penv) in all your trials.

1.4.1 How to measure rate of vaporization dmdt

Set timer to cumulative regime and, also, choose seconds for units of a timerscale.

Fill the cup up to the brim with LN2. Set the scale to a value a littleless than the current total mass of LN2 and cup. As soon as some LN2 hasevaporated and scale becomes balanced, start timer [this first value of massyou have read from the scale at “zero” time which corresponds to “zero”evaporated mass of LN2]. Reduce the balance mass, for example, by 5 g


and record time t1 when balance of scale will be achieved again [time t1 cor-responds for evaporation of 5 g of LN2]. Reduce balance again by 5 g andrecord t2 [cumulative time t2 corresponds for evaporation of total 10g of LN2].Having filled table similar to one below

evaporated mass, g time of evaporation, s0 05 t110 t215 t3... ...

Plot evaporated mass versus time of evaporation. According to § 1.3 youwould expect a constant rate of vaporization, so you would expect to seethat your data is fitted well with a straight line. If this is the case then valueof the slope of the best fit line is equivalent to the rate of vaporization dm

dt.

Choose 5 g interval, or any other mass interval more suitable from yourpoint of view, so that you will have about 5 readings while level of LN2 goesfrom the brim of a cup down to a point when the resistor is still below thesurface of the liquid. If you choose mass interval to be too small then youwill have not enough time to adjust scale; if too big, then you will get boredwaiting and, moreover, you will not get enough points to plot.

1.5 Analysis

Before you start to carry out error propagation for L, Eq. (1.6), you need tounderstand first what were the sources of random errors (i.e., what do youestimate to be the accuracy of your measurements of time, mass, voltage andcurrent?).

Do you observe on your plots the expected constant rate of evaporation dmdt

for both parts of experiment? [In other words, does your linear fit for youdata of evaporated mass versus time of evaporation go through error barsof you measurements?] If not, then theory of § 1.3 fails and as a result youcannot use Eq. (1.6).


How does your value of L = · · · ± · · · for LN2 compares to the table value?(Table value of latent heat of vaporization of LN2 is 199.2 J/g.)

“Optional” Thinking

Can you relate the facts below with the systematic error(s) of your experi-ment [if you actually had any systematic error(s)]? Maybe there is somethingelse you can come up with!?

When the cup is not filled to the top, for example is filled to 2/3 or 1/2,there is going to be some space in the cup above liquid for cold vapor tostay. And, it will effectively acts as an insulator preventing direct contact orheating of LN2 by room air. So, was it important to carry out both parts(heater off/heater on) of the experiment with LN2 being at about the samelevel?

Should you have been careful not to breathe (or to breathe consistently)on the cup throughout your experiment? Does this even matter?

Do you need to calibrate “zero” of your scale? In other words, think abouthow scale calibration affects your measurements/data and final results.

Experiment 2

Ideal Gas Law

2.1 Theory

Definition: An ideal gas is a non-interacting gas of identical perfectly rigidpoint-like particles.

Real gases are not something trivial. Indeed, at a microscopic level onedeals with complex three dimensional molecules—usually one has a mixtureof different sorts of molecules in a gas—that, in course of collisions, startto rotate, vibrate, etc. [Even monoatomic noble gas is not a collection ofperfectly rigid billiard balls!] Despite these complications it turns out thatthe simplest ideal gas approximation is often useful.

For an ideal gas at the equilibrium state it is straightforward to derive twofacts. First, for ideal gas its absolute temperature is proportional to averagekinetic energy of particles.1 Second, pressure (P ), volume (V ) and absolutetemperature (T ) of an ideal gas can be related by the equation of state, called

1This fact, along with Eq. (2.1), is actually used to define absolute/Kelvin (K) tem-perature scale. 0 K is the smallest possible absolute temperature, as it is related toidealization/practically unachievable situation when gas molecules are not moving. And,1 K temperature change is set to be equal to 1 C, thus one has a simple conversion fromthe Kelvin temperature scale to the Celsius scale

T()Celsius = TKelvin − 273.15.

8

Physics 152(154) Laboratory · Ideal Gas · Fall 2013 9

the Ideal Gas Law:P · V = N · kB · T, (2.1)

where N is the number of particles in the volume and kB = 1.38 · 10−23 J/Kis the Boltzmann constant.2 Here, the units of P are N/m2, of V are m3 andof T are degrees Kelvin (K).

2.2 Lab Goal

Your tasks are (a) to see if the air in this Lab obeys the Ideal Gas Law,3 and(b) to estimate the temperature of absolute zero.4

2.3 Experiment Overview

In order to to check applicability of the Ideal Gas Law, Eq. (2.1), for ap-proximate description of air one needs to come up with a device where onecontrols P , V , N and T .

Consider, for example, apparatus depicted at Fig. 2.1. It facilitates mea-surements of pressure and volume as a sample of gas (air) inside a cylinder iscompressed isothermally. [A process at a constant temperature is said to beisothermal.]—Once you insert the piston and its weight has pushed the rub-ber seal below the air inlet hole, you have a fixed amount of air (N molecules)trapped inside the cylinder under the piston. The pressure (P ) of air inside

2Normalizing N by the Avogadro number NA = 6.02 · 1023 [number of atoms in 12 g ofcarbon isotope 12C] one can rewrite Eq. (2.1) in the following equivalent form

P · V =

(N

NA

)· (NAkB) · T = n ·R · T,

where R = NAkB = 8.31J/(mole ·K) is the universal gas constant and n = N/NA is callednumber of moles.

3You cannot verify directly, for example, using light microscopy, that air consists ofminute molecules [about 10−10 m in diameter]. But you can indirectly prove this fact byshowing applicability of the Ideal Gas Law for description of air. Note that existence ofpoint-like particles is the key assumption in the derivation of the Ideal Gas Law.

4This part of the lab is a good example demonstrating how in some “pathological”cases one can start with relatively precise data and, finally, after some calculations acquirevery uncertain outcome.


is varied by placing different weights on weight pan which is fitted to thetop of the piston (see § 2.3.3). The gas cylinder is mounted inside anothercontainer, which can be filled with water at any desired temperature (T ).The function of the water is to provide a constant temperature environmentfor the gas. Volume of the trapped air V = H ·A, where A is a cross sectionarea of the cylinder, and the height (H) of the air inside the cylinder can becalculated from the height (h) of the pan above the surface of the table (see§ 2.3.2).

If the air inside the cylinder was an ideal gas then its state should havebeen described by the Ideal Gas Law, Eq. (2.1). In the present case, it canbe rewritten in the equivalent form

H =N · kB · T

A· 1

P= constant · 1

P. (2.2)

Observing how close H of the trapped air follows Eq. (2.2) as P is varied, byloading the piston with different masses, at two different temperatures (roomand ice temperature) will allow you to conclude how accurately air can beapproximated to be an ideal gas.

2.3.1 Details of Experimental Procedure

• Before doing anything else, measure the mass of the piston with theweight pan.

• Learn how to disassemble the weight pan and the cover of the watercontainer without taking the piston out of the cylinder—following thisprocedure you will be adding ice in part 2.4.2.

• If the piston does not move smoothly inside the cylinder ask the TA togrease it.

• Look closely at the weights you are going to use; because they haveboundary that covers the weight pan, you will need to measure heightin a slot cut in them.


Figure 2.1: Sketch of experimental setup with a detailed depiction of “Piston–Cylinder” geometry on the right-hand side.


2.3.2 How to measure the height (H) of the air insidethe cylinder

At the very beginning of the experiment, insert the piston in the cylinderand hold it so that you see the bottom edge of the piston in the air inletopening. In such position, measure reference height h0 from the table top tothe bottom of the weight pan. It will correspond to 15.24 cm of height of theair column inside the cylinder—the numbers 15.24 cm and 1.29 cm [diame-ter of the cylinder] are the manufacturer’s specification of the experimentalsetup (see Fig. 2.1). Knowing h0 it is obvious how to later recalculate H ofthe trapped air from measurement of the height (h) for an arbitrary positionof the piston in the cylinder.

Friction is important for estimating error of measuring H. At each pres-sure you can find a range of heights, that is, a maximum and a minimumwhich depend on whether you last pushed gently the weight platform up ordown with your finger.

2.3.3 How to determine absolute pressure (P ) in thecylinder

Pressure in the cylinder P is given by the atmospheric pressure Patm =1.013 · 105 N/m2 plus the force exerted by the loaded piston per the cross-section area, A:

P = Patm +Mtot · gA

, (2.3)

where Mtot is the mass of the piston, the pan and the weight(s) and g isacceleration due to gravity. [Note you can calculate A from the informationgiven on the diagram of Fig. 2.1.]

You will know values of mass loads with much higher accuracy than heights.Thus in Eq. (2.3) it is safe to consider P to have no error.


2.4 Experimental Procedure

Note the temperature each time you begin collecting the data for parts 2.4.1and 2.4.2. [You do not need to keep the thermometer inside the water con-tainer during each set of measurements and do not hurry too much as it takesrelatively long time for such a big volume of water in the container to changeits temperature.]

2.4.1 Room Temperature Measurement

Start the experiment at the room temperature. Fill the water at approxi-mately room temperature into the water bath. Enclose a sample of roomtemperature air into the cylinder by lowering the piston carefully into thecylinder. From this time on, do not remove the piston from thecylinder or raise the rubber seal above the air inlet hole. The reasonis that you want to keep the same number of molecules of gas, N , in theapparatus throughout the experiment.

Measure height H of the gas under the piston for each of 5 to 8 differentweights (including “zero” load, when air is compressed only by the weight ofthe piston and the pan).

2.4.2 Ice Water Measurement

Drain some water and add some ice to the water bath (ice should be cov-ered by water!). Wait until temperature stability is established (at a sametime you can complete data analysis for part 2.4.1). Then repeat the samemeasurements as in the part 2.4.1 above.

2.5 Analysis

For both parts 2.4.1 and 2.4.2 plot H versus 1P

. If air is well approximatedby the Ideal Gas Law at “everyday” temperatures, such as room and ice tem-perature, then, according to Eq. (2.2), your data can be fitted by a straightline with y-intercept equal 0. [Formally, according to Eq. (2.2), H should beequal 0 when P equals ∞ (i.e., when 1

P= 0). Or, in other words, a totally

compressed ideal gas has zero volume, because ideal gas consists of point-like


particles, see the definition in § 2.1.]

Do your lines of best fit pass through the error bars of all your data points,and do they have zero y-intercept? Can you conclude that the air in the Labobeys the Ideal Gas Law? [See also Sec. ““Optional” Thinking” below.]

In addition, calculate the temperature of the absolute zero in degrees Celsius.Let’s pretend that we do not know that the temperature of the absolute zeroin degrees Celsius is T

()abs = −273.15 C, then for conversion of temperature

from Kelvin (T ) into degrees Celsius (T ()) we should write T () = T + T()abs.

And, if [and only if!] air is well approximated by the ideal gas idealization,then, according to Eq. (2.2), slopes (S) of the linear fits of the graphs H ver-sus 1

Pfor two temperatures of parts 2.4.1 and 2.4.2 [note that in Eq. (2.2) T

is in Kelvin] should be

S1 =N · kB · (T ()

1 − T ()abs)

A, (2.4)

S2 =N · kB · (T ()

2 − T ()abs)

A(2.5)

To solve this system of equations for T()abs we need to get rid of the second

unknown N . So, we form ratio of slopes and after some math arrive to thefinal formula:

S1

S2

=T

()1 − T ()

abs

T()2 − T ()

abs

⇒ T()abs =

S1 · T ()2 − S2 · T ()

1

S1 − S2

. (2.6)

Calculate your experimental value of T()abs = · · ·± · · · [do error propagation!].

Compare it with the table value T()abs = −273.15 C. What are your conclu-

sions?

When you do the error propagation for T()abs note a dramatic increase of error

from the denominator of Eq. (2.6). What are the original physical reason(s)for high uncertainty of the denominator, and, as a result, of your experimen-tal value of T

()abs? How might be improved experiment to determine more

precisely value of T()abs?



Friction has the most pronounced effect when air is compressed only by rela-tively small weight of just the piston with the pan (no load).—Piston can getstuck at the “wrong” height.—How have you taken this into considerationduring your data analysis?

There is a possibility that your height measurements will be systematicallyoffset. What could be the reason(s)?

Estimate what will be the actual H of liquified air in this experiment atvery high [“infinite”] pressure. The density of air in the gaseous phase is1.29 kg/m3 and the density of liquid air is 1125 kg/m3.

An alternative route to find T()abs would be to plot H versus T () for a given

pressure and fixed amount of gas. According to Eq. (2.2), graph should come

out to be a straight line, with x-intercept (when H = 0) equal to T()abs. For

a given P (or in other words weight load), you have only two different Hmeasured (one at room and another at ice temperature). Thus you have

only two points to do a linear fit of H versus T () in order to obtain T()abs,

and this it is not “nice”. Why?—Try to follow this alternative route.

Experiment 3

DC Circuits

3.1 Theory

Consider a metal at a microscopic scale. Positive ions of metal form a lattice,there are some neutral impurities, and electrons randomly move around. Inorder to have an organized flow of electrons one creates some favorable condi-tions for their motion. For example, one can use a battery to set the potentialenergy of electrons to vary along the wire. As a result, electrons travel alongcomplicated zigzag paths scattering from the ion lattice and impurities (seeFig. 3.1). The corresponding average electron drifting for common householdwiring happens with speeds of about 1 mm/s. There is a negligible kineticenergy of electrons associated with this drifting flow. The dominating effectis the transformation of the potential energy of electrons into vibration (i.e.,heat) of metal ions and atoms of impurities during collisions with them.

Voltage and current are the main physical quantities that describe the elec-tron flow. Voltage is in one to one correspondence with the electron potentialenergy. Current is the rate of charge flow (its numerical value equals to theamount of charge that passes through the cross-section of a conductor perunit of time).

It is convenient to discuss how fictitious positive charges—to be called furthercharge—would have flown in a circuit rather then talk about the negativecharge flow, i.e., the electron flow, because ...

16

Physics 152(154) Laboratory · DC Circuits · Fall 2013 17

Figure 3.1: Current through a piece of a metal conductor (flow of manyelectrons). Long dotted curve depicts the detailed path of one particularelectron.

Positive charge moves or flows from higher to lower voltage. In contrast,negative charge moves from lower to higher voltage, so one would say thatnegative charge rises from lower voltage to higher voltage, which is inconve-nient. If historically positive charge has been associated with electrons, thenwording for metallic systems would have been simplier. [If you are interestedin details see your text book, or ask your TA to describe in terms of electronsflow what happens in a circuit you have build.]

For a wide class of objects, the value of current I is directly related to theapplied voltage V . Ohm’s law is a mathematical representation of this fact

V = I ·R, (3.1)

where the coefficient of proportionality R is called resistance. The Interna-tional System of Units [abbreviated SI] uses Ohm (Ω) for units of resistance,and also Ohm = Volt/Ampere. The value of resistance is specific for a givenconductor. For example, consider the circuit in Fig. 3.3. The charge runsthrough the copper wires more freely, than through a carbon resistor R, sincethe later has higher concentration of impurities at microscopical scale. ThusRwire R, and the charge flowing through this circuit losses its potentialenergy (the voltage drops) solely at the carbon resistor.

The circuits, more complicated compared to the case of a single resistor


(see Fig. 3.3), will include at least two resistors either in parallel or in series(see Fig. 3.4). In the case of resistors in series, the effect resistance Reff canbe calculated by adding the individual resistances together:

Reff = R1 +R2 (3.2)

In the case a resistors in parallel, the Reff is:

1

Reff

=1

R1

+1

R2

(3.3)

For simplicity, all resistors in these circuits are considered to be identical andthe applied voltages to be the same.

According to Ohm’s law, Eq. (3.1), the current in a single resistor is

Isingle =V

R. (3.4)

Nonohmic behavior, for example, is exhibited by a light bulb. Although, thebulb filament is a thin metal wire, its voltage-current characteristic does notfollow Ohm’s law, Eq. (3.1). An increase of current corresponds to a dramaticincrease of the filament temperature (observed as its brightness variation),and this affects conducting properties of the metal.—Higher temperaturemanifests itself at the microscopic level in a higher intensity of oscillationsof metal ions around their equilibrium positions, which makes the crawlingof electrons harder.

3.2 Voltage Divider

Consider the circuit in Fig. 3.2. [When working with analog voltmeters andammeters, pay attention to the polarity positive (+) or negative (-) of theterminals (see Fig. 3.2). The (-) terminal is often called COM (common).When dealing with light bulbs and resistors, polarity is not important.] Thepower supply delivers charge at constant (DC) voltage of 5 Volt to a point(a). Charge goes through the slide wire resistor losing its potential energyand eventually arriving to the common ground of 0 Volt. Positioning thesliding contact at some distance x from the bottom of the slide wire resistor,one can pick charge at voltages ranging from 0 Volt to 5 Volt at the point(b). In other words, this construction serves as a voltage divider. On the


Figure 3.2: Voltage source.

way to the output, from point (b) to (c), charge goes through the ammeter.Ammeter practically does not disturb the charge flow, but simply counts thetotal charge q that passes through in time t and calculates the rate of flow,i.e., the current, I = dq

dt. At the point (c) some tiny fraction of charge is

taken by the voltmeter and is conducted to the common ground. This allowsthe voltmeter to determine the potential difference between the two points.Thus the voltmeter measures the output voltage supplied to load.

3.2.1 Lab Goal

Build the voltage source depicted in Fig. 3.2. Test its output characteristics.

3.2.2 Experiment Overview/Procedure/Analysis

Put together the voltage source (see Fig. 3.2). You will use it in later partsof this lab. Without any load, i.e., the switch is off, measure and plot theoutput voltage as a function of the sliding contact position (x), measuredfrom the bottom end of the slide wire resistor. Your TA may explain why alinear relationship should be expected.

3.2.3 Practice to Use Oscilloscope

The oscilloscope is a fancy voltmeter that shows graphically how the voltageof a signal varies with time. You could have achieved the same results with


a simple voltmeter by measuring and plotting voltage versus time.

1. Connect the oscilloscopes channel 1 in parallel to the voltmeter (seeFig. 3.2). [Connect the ground (-) to the black jack and (+) to redjack.]

2. Set the switch above the input of channel 1 to DC. On the screen, youwill see a horizontal line, as the voltage (y-axis on the screen) in yourcircuit does not change with time (x-axis on the screen). You can offsetthe voltage by the knob POSITION.

3. Set the switch above the input of channel 1 to GND.—This discon-nects the oscilloscope from the circuit and allows you to access theoscilloscope ground.—Center the level of the oscilloscope ground, forexample, in the middle of the screen by the POSITION knob (i.e., setthe 0 Volt level for the y-axis in the middle of the screen). Then switchback to DC.

4. Set appropriate units for the vertical scale by VOLTS/DIV knob [thered calibration knob (CAL) in its center should be clicked fully clock-wise].

For different positions of the sliding contact compare the voltage readings ofthe oscilloscope with the ones of the voltmeter.

Figure 3.3: Single carbon resistor R.


3.3 Ohm’s Law

3.3.1 Lab Goal

Verify the applicability of the Ohm’s law for a carbon resistor and evaluateits resistance.


You can burn the fuse of your ammeter if the current that you are try-ing to measure exceeds the ammeter’s range. Therefore before turning onthe double-pole single-throw switch (see Fig. 3.2), set the ammeter to the200 mAmp range.

Load the voltage source that you have build in § 3.2 with a single smallcarbon resistor (see Fig. 3.3). Measure voltage-current characteristic. [Notethat 0 Volt / 0 Amp is a perfectly reasonable measurement.] Plot V as afunction of I. If the carbon resistor can be described by the Ohm’s law,Eq. (3.1), then your data can be fitted by a straight line that passes througherror bars with the y-intercept that equals to 0 Volt. What do you observe?

If the resistor is ohmic, determine its resistance R from the slope of thelinear voltage-current characteristic. Also figure out R using the color code(see Fig. 3.5). And, also, measure R using digital multimeter as an ohm-meter. Compare all the values obtained above with each other, and make astatement about their mutual agreement!

3.4 Resistors in Series and Parallel

3.4.1 Lab Goal

Study properties of resistors in parallel and in series.


You will be given two identical carbon resistors in a sense that their resis-tance R will be the same according to their color coding. You can find outR directly from their color codes [or measure resistance of each resistor (R1,


and R2) according to the procedures of § 3.3]. This allows you to make pre-diction for the expected value of the effective total resistance of the paralleland series connections to be R

2and 2R, respectively [see Eqs. 3.2 and 3.3].

Do not forget to carry out error propagation!

Using the procedures of § 3.3 measure the effective total resistances of theparallel and series connections [also note uncertainty of your measurements].Are your measurements in the agreement with your expectations?

Optional

For circuits with single resistor (see Fig. 3.3), two resistors in parallel and tworesistors in series (see Fig. 3.4), measure the total current in these circuits ata fixed voltage, e.g., at 3Volt. (Check to be sure the voltage has not changedbetween trials; adjust it if necessary.) Substitute Eqs. 3.2 and 3.3 for R inEq. 3.4 and calculate Iseries and Iparallel How do your findings for the totalcurrent in these circuits compare with your calculation?

3.5 Light Bulb

3.5.1 Lab Goal

Observe nonohmic voltage-current characteristic of a light bulb.

Figure 3.4: Circuits with two resistors R in parallel and in series.


Figure 3.5: Color code scheme for labelling resistors.

Colors for M , N , P :

0 Black 5 Green1 Brown 6 Blue2 Red 7 Violet3 Orange 8 Gray4 Yellow 9 White

Examples:a) Brown, Black, Black, Colorless:→ R = 10 · 100 Ω± 20% = 10± 2 Ωb) Yellow, Violet, Black, Gold:→ R = 47 · 100 Ω± 5% = 47 Ω± 5%c) Gray, Orange, Brown, Silver:→ R = 83 · 101 Ω± 10% = 830 Ω± 10%

Experiment Overview/Procedure/Analysis

Similar to § 3.3 measure voltage-current characteristic for a light bulb overas wide range as possible, for example, going from 0Volt to 2Volt in 0.25Voltincrements, then 2Volt to 5Volt in 0.5Volt increments. Plot V as a function ofI. Why can (or maybe cannot) you claim that your data represents nonohmicbehavior?

Experiment 4

RC Circuits

4.1 Theory

Charge can be stored on two conductors placed near each other; such a com-bination is called a capacitor. Usually, but not necessarily, the conductorsare in the form of parallel plates. One way of charging a capacitor is to con-nect the two plates to a battery. In Fig. 4.1 the battery causes a momentaryflow of electrons from the top plate through the battery to the bottom plate.One can prove that amount of charge Q on either plate is directly propor-tional to applied voltage V (potential difference across capacitor created bythe battery), thus

Q = C · V, (4.1)

where C is the coefficient of proportionality called capacitance. Details of acapacitor construction determine the numerical value of C, for example, forthe case of a pair of parallel plates close together C ∝ A/d, where A is thearea of each plate, and d is their separation. The International System ofUnits [abbreviated SI] uses Farad (F) for units of capacitance.

If a charged capacitor is connected across a resistor R as shown in Fig. 4.2(a),current will begin to flow through the resistor. Solving corresponding differ-ential equation it can be shown that the voltage across the capacitor willdecay exponentially from the initial voltage Vi toward zero. The equationgoverning the time dependence of the voltage decay is

V (t) = Vie−t/RC = Vie

−t/τ , (4.2)

24

Physics 152(154) Laboratory · RC Circuits · Fall 2013 25

+ + + +

- - - -V

+Q

-Q

Figure 4.1: A parallel plate capacitor connected to a battery.

Vf

t=0

+ + + +

- - - -C

RV V(t)

+

-

CR

V V(t)+

-

t=0

(a)

(b)

Figure 4.2: (a) The upper sketch shows process of a capacitor discharge, and(b) the lower one depicts charging of a capacitor. In both cases the voltmeteron the right side of each diagram is used to monitor the voltage V (t) acrossthe capacitor.


+

-C

RV V(t)

+

-

DC

power

supply switch

0 to 15

Volt

1 kOmh

+

-

Figure 4.3: Circuit to be used to observe slow RC decay.

where τ = RC is the time constant of the decay. SI units of τ are seconds.[Using Eq. (4.1) one can rewrite Eq. (4.2) to describe exponentially decreas-ing charge Q(t) on the plates of the capacitor.]

This exponential function has some rather remarkable properties. Note thatit takes forever for the voltage to reach zero. Note also that at the end ofeach time interval τ , the voltage is less than it was at the beginning of thetime interval by a factor of e−1 no matter when the time interval began. [eis the base of the natural logarithm, and e−1 = 0.3678 . . . or approximately37%.] Thus when one decay time has passed (t = 1τ), the voltage is approx-imately 37% of its initial value. After two decay times (t = 2τ) the voltageis about 13.5%, after 3τ about 5%, etc. After five decay times (t = 5τ), thevoltage is less than one percent of its initial value [e−5 = 0.0067 . . .] and formost practical purposes it is zero.

For charge buildup the same time constant applies. If an uncharged capacitoris connected to a battery of voltage Vf through a resistor R, as shown onFig. 4.2(b), the voltage across the capacitor grows exponentially from zeroto the battery voltage according to the following equation

V (t) = Vf (1− e−t/RC) = Vf (1− e−t/τ ). (4.3)


4.2 RC Decay

4.2.1 Lab Goal

Study the decay of charge on the capacitor in the Resistor-Capacitor (RC)circuit by using a digital voltmeter. Obtain the value of capacitance.

4.2.2 Experimental Procedure

Slow voltage decay across a capacitor can be monitored directly with a digi-tal voltmeter (DVM). Wire the circuit shown in Fig. 4.3. Be sure to observethe polarity for both the capacitor and the DVM. Select the 20− Volt scaleof the DVM. With switch open and R set on the decade resistance box at100 Ω, or less, check the DVM reading. It should be zero, or no more than 1in the last digit. If it is not, put a short circuit (a wire) across the capacitorterminals, see that the voltage falls to zero, then remove the short.

Set R to 300 kΩ. With the variable DC power supply on, but turned down,close the switch. Now turn up the voltage to about 10 Volt. Measure thevoltage across the discharging capacitor as a function of time by openingthe switch and recording V (t). Obtain at least 10 to 15 values. Plan yourexperiment so that your final measurement of voltage will at least ten timessmaller than your initial one. [The actual initial voltage does not matter,but it will be more convenient to make the first measurement at 10 Volt.]

4.2.3 Analysis

Check that the measured voltage across the capacitor indeed decays expo-nentially according to Eq. (4.2). [Plot your data of V versus t to see howit looks. You cannot yet claim the curve to be exponential, unless you havespecial software, like Origin, to carry out an exponential fit.] Note that tak-ing a logarithm of both sides of Eq. (4.2) you arrive at a linear relationshipbetween lnV and t,

lnV (t) = lnVi − t/τ. (4.4)

Thus it is convenient to plot voltage versus time using the logarithmic scalefor the voltage axis, or, equivalently, you can plot lnV versus t. If in thisrepresentation your data points are close to a straight line, then you haveverified that the discharge of the capacitor is described by Eq. (4.4) and,


consequently, by the original Eq. (4.2). [Moreover, if this is true, then youcan also state that you have used in your RC circuit good capacitor andgood resistor, because only two basic Eqs. (3.1,4.1) where used to deriveEqs. (4.3,4.2) describing charging and decay.]

Once the functional form of V (t) is established, find the time constant τ fromthe slope of the linear fit of lnV versus t. Finally, calculate the value of C. Donot forget to take into account the uncertainty of your measurements! Whatis the uncertainty of C after error propagation? Does your result agree withthe nominal value of the capacitance?—Look at the capacitor actually usedin your circuit. Record the value printed on it; it should be 220 µF± 20%.

Optional

In anticipation of the next part of the lab, make a “spot” measurement ofthe time constant by adjusting the initial voltage (Vi) to 10 Volt and timingthe fall to 3.7 Volt. Alternatively you could time the drop from an arbitraryVi to 0.37× Vi. Compare this result for τ with that in the previous part.


Actually, the effective resistance is not just R that you have set on the decaderesistance box, but is instead R in parallel with 10 MΩ input resistance ofthe DVM. Thus even if you remove R from the circuit, the capacitor will stilldischarge very slowly through the DVM. To see this, repeat the experimentalprocedure without R in the circuit, except this time measure only a fewvoltages at intervals of one minute. Find the time constant. Is it what youexpect?

4.3 Capacitors in Series and Parallel

4.3.1 Lab Goal

Experimentally establish formulas for the effective capacitance of two capac-itors connected in series and parallel.


CR

Function

Generator

To scope

Channel 1

To scope

Channel 2

To scope

Ground

Figure 4.4: Circuit to be used to observe fast charging and discharging ofcapacitor.

4.3.2 Experimental Procedure

You will be given two capacitors of nominal capacitance 0.5 µF ± 10%, and1 µF± 10%, and also a decade capacitance box. Come up with a strategy toinvent formulas for effective capacitance of two capacitors connected in seriesand parallel. For example, carry out some meaningful measurements of theeffective capacitance of parallel and series connections (see § 4.3.3), think,put forward a hypothesis, test it, if it does not hold true, think more, comeup with some other bright idea...

4.3.3 How to measure the capacitance by using a square-wave generator

Wire the circuit shown in Fig. 4.4, using R = 2000 Ω and C = 0.5 µF. Thevoltage source is a square-wave generator [actually a function generator, setto give square-wave output], and is displayed on channel 1 of the oscillo-scope. In a square wave, the voltage jumps from +V0 to −V0 at a presetfrequency. This simulates the switch in Fig. 4.3 being opened and closedrepetitively, alternately charging and discharging the capacitor many times


V

0

V

t t0

(a) (b)

Figure 4.5: Both channel 1 and 2 are displayed simultaneously on the screenof the oscilloscope. Solid curves are the time-dependent voltage across thecapacitor for different frequencies of the square-wave generator. Dotted linesrepresent output of the square-wave generator. In the (a) case frequency ofthe square-wave generator is adjusted correctly, in the (b) case one shoulddecrease frequency.

each second. This repetition enables you to observe the capacitor voltagedecay and buildup on channel 2 as a stationary pattern on the oscilloscopescreen. [The instructor will help you with the operation of the oscilloscopeas needed.]

Set the generator on the square-wave output at the frequency ≈ 100 Hz.Adjust the TIME/DIV on the scope to about 1 ms/DIV. You should see awave form that looks like Fig. 4.5(a). If the wave form looks like Fig. 4.5(b),reduce the frequency of the square-wave generator until the wave looks likeFig. 4.5(a).

Measure the time constant using either decreasing or increasing capacitorvoltage. The way to do this is to find the time it takes the voltage to dropto 37% from its maximum value, or to find the time it takes the voltage torise to 63% of the way up to the final value. [Note that from Eq. (4.3) itfollows that V (τ) = Vf (1− e−1) ≈ 0.63Vf .] Estimate the uncertainty in yourmeasurement of the time constant. Calculate the capacitance. Does yourvalue agrees with the nominal value?

Experiment 5

Magnetic Field Mapping

5.1 Introduction

A magnetic field ~B is a vector quantity which exists in space in the vicinityof magnetic poles and is analogous to the electric field ~E which exists in thevicinity of electric charges. Whereas E-fields always exert forces on electriccharges, B-fields only exert forces on a charge that is moving. In additionthe velocity ~v of the charge must have a component perpendicular to thedirection of ~B. The relation ship between the force ~F , the charge q , ~v and~B is given by ~F = q~v × ~B. This is a vector cross product so the magnitudeof the force is F = qvB sin θ, where θ is the angle between ~v and ~B. Thisequation defines the magnetic field ~B and its units are Teslas (T). We shallnot be concerned with magnetic forces in this lab.

Another relationship between electricity and magnetism is that a wire car-rying an electric current I produces a magnetic field ~B in its vicinity. Themagnitude of B is proportional to I and the distribution of ~B in space de-pends on the geometry of the wire arrangement. We shall be studying thefield due to a circular coil, and Helmholtz coils, which is two coils of radius Rseparated along their axes by a distance equal to R. Finally we shall studya solenoid which is a coil helically wound and cylindrical in shape.

A third relationship between electricity and magnetism is that a changingmagnetic field ~B passing through a coil induces a voltage across the termi-nals of the coil. The magnitude of the induced voltage is proportional to

31

Physics 152(154) Laboratory · Magnetic Field Mapping · Fall 2013 32

the rate of change of ~B. This is known as Faraday‘s Law and will be statedin a generalized mathematical form when you do it in the lecture course.This is used in the lab to observe an oscillating B-field as a voltage on theoscilloscope.

5.2 Apparatus

Fig. 5.1 illustrates the field measuring device which is called a pick-up coil orprobe. The coil is inserted in an oscillating magnetic field Bmax sinωt. Thechanging magnetic field induces an oscillating voltage Vmax cosωt in the coil.The amplitude of the voltage Vmax is proportional to Bmax.

Figure 5.1: Pick-up coil.

5.3 Single Circular Coil

The entire experimental arrangement is illustrated in Fig. 5.2. The exampleshown is to measure the field along the axis of a single circular coil of radiusR. The pick-up coil is shown at a distance x from the center of the coil.Measure Vmax on the oscilloscope as a function of x starting at x = 0 (thecenter of the coil) to a distance in excess of the radius of the coil. Thevariation of B(x) along the axis of the coil is given by the ratio:


B(x)

B(0)=

1(1 + ( x

R)2)3/2

(5.1)

1. Make a plot of V (x)/V (0) versus (1 + (x/R)2)−3/2 and fit the beststraight line.

2. Measure and make a plot of Vmax vs r at x = 0. There is no simple forthese data, but you will make a qualitaive comparison with your plotfrom section 5.4, part 2.

Figure 5.2: Experimental setup for magnetic field mapping.

5.4 Helmholtz Coils

A Helmholtz pair of coils consists of co-axial coils separated by a distanceequal to their radius R as illustrated in Fig. 5.3. This arrangement givesalmost a uniform (constant) magnetic field in the region near the axis andbetween the coils. Measure Vmax along the axis vs the distance x from thecenter. Estimate the fractional variation of the field over the entire region.Also measure Vmax vs r at the center of the coils (x = 0).


1. Show that the field along the axis is just the sum of the fields due tothe 2 displaced single coils.

2. Plot Vmax vs r at the center of the coils (x = 0). There is no formulafor these data, but the plot can be compared the plot from section 5.3,step 2. What qualitative differences do you notice between the twoplots?

Figure 5.3: Helmoltz pair and solenoid.

5.5 Solenoid

A solenoid is a helically wound coil in the shape of a cylinder.It is illustratedin Fig. 5.3. Near the axis, inside the coil, and at a distance from the endslarger than the radius the field is nearly uniform. The field along the axis ofa solenoid is:

B =µ0NI

2L(sinφ2 − sinφ1) (5.2)

1. Measure Vmax along the axis vs distance x (Let x = 0 be the center ofthe solenoid) for distances inside AND outside the solenoid. Make atplot of Vmax vs x for distances inside and outside the solenoid. Howdoes the magnet field compare to the single coil and the Helmholtzcoils?


2. Measure Vmax vs distance r along diameter of inside (x = 0) ANDoutside of solenoid. Discuss your results.

Experiment 6

RLC Circuits

6.1 Forced Oscillations, Resonance

6.1.1 Theory

The term “oscillatory motion” in physics has a wider meaning than just adisplacement of an object back and forth. Oscillatory motion is defined tobe any periodic or almost periodic process, i.e., when one or several valuesof some physical quantities repeat themselves exactly or almost exactly afterequal or almost equal intervals of time.

We are going to discuss oscillatory circuit (see Fig. 6.1) under both freeand forced oscillation conditions. It consists of an inductor L, capacitor Cand resistor R connected in series. Oscillations of the charge Q(t) on theplates of the capacitor (and the corresponding current I(t) = dQ

dtin the

circuit) are equivalent to a one-dimensional oscillations of the displacementx(t) (and the corresponding velocity v(t) = dx

dt) of an object attached to a

spring.1 More specifically, in the oscillatory circuit, L is equivalent to themass m of the object in the mechanical system, C−1 is equivalent to theforce constant k of the spring, R is equivalent to the damping coefficient,and the time-varying voltage output V (t) of the generator is equivalent tothe external driving force. Such systems are called driven damped harmonicoscillators.

1Both situations are described by one and the same differential equation. Moreover,many other physical systems have equivalent behavior, for example, small oscillations ofa pendulum, vibrations of molecules and so on. Sound waves and electromagnetic wavesare also closely related phenomena.

36

Physics 152(154) Laboratory · RLC Circuits · Fall 2013 37

In an oscillatory circuit, the capacitor accumulates energy in the form ofelectric field produced by the charge Q(t) on its plates. The inductor, e.g.,a solenoid, accumulates energy in the form of a magnetic field produced bythe current/charge flow I(t) = dQ

dt.

In the case of an object attached to a spring, the spring “holds” potentialenergy when an object is displaced to x(t) from its equilibrium position. Theobject has a kinetic energy when it is moving with the velocity v(t) = dx

dt.

It is very important that the period T of free oscillations (i.e., no externalvoltage/force) of the harmonic oscillator (i.e., no damping) does not dependon the initial conditions,2 and, as a consequence, T does not depend on theamplitude of oscillations. For the oscillatory circuit T = 2π

√LC and for the

mechanical system T = 2π√

mk

. In other words, T depends on the parameters

of the system that determine the “potential” and the “kinetic” energies. Theperiod T determines the natural frequency f0 of the free oscillations and thecorresponding natural angular frequency ω0:

f0 ≡1

T=

1

2π√LC

and ω0 ≡ 2πf0 =1√LC

. (6.1)

Once the alternating with the angular frequency ω driving voltage/forceof the amplitude V0 has been applied,

V (t) = V0 · sin (ωt), (6.2)

it can be shown that the current in the circuit, I(t), varies at the sameangular frequency ω, but with some phase shift δ:

I(t) =V0 · sin (ωt+ δ)√R2 +

(ωL− 1

ωC

)2. (6.3)

Measured voltage VR(t) across the resistor (see Fig. 6.1) divided by R is the cur-

rent I(t) flowing in the loop, I(t) = VR(t)R , since, according to the Omh’s Law,

VR(t) = I(t) ·R.

The amplitude of oscillations of I(t) in Eq. (6.3) (see Fig. 6.2) has its max-imum as a function of ω at so-called resonant angular frequency ωres, which

2Initial conditions of oscillator are, for example, initial capacitor charge/pendulumdisplacement and initial current in the circuit/pendulum velocity.


function

generator

V(t)

I(t) L

C

R

To Common

Ground

V(t)

To Oscilloscope

Channel 1

V (t)

To Oscilloscope

Channel 2

R

+Q(t)

-Q(t)

Figure 6.1: RLC circuit.

minimizes the denominator of Eq. (6.3),

ωresL−1

ωresC= 0 ⇒ ωres =

1√LC

. (6.4)

Apparently, the resonant angular frequency is equal to the natural angularfrequency, ωres = ω0, and fres = f0. Your TA will explain in detail the physicsof this “coincidence” common to most oscillating systems.3

Note that the resonant frequency f0 does not depend on the damping R,but R limits the amplitude of the current at resonance.

R → Omh(Ω) L → Henry (H)C → Farad(F) f , ω → Hertz (Hz)

Table 6.1: International System of Units

3To have a child-swing oscillate at maximum amplitude you push it once it comes, forexample, to the most right position. This happens about once per period T . So, for theresonance to occur, the frequency of your pushing should be about the natural frequency,f0 = 1

T , of the child-swing.During such forced oscillations, friction limits the amplitude preventing the child-swing

from overturning (and, correspondingly, circuit from burning the fuse, spring from break-ing, etc.).


6.1.2 Lab Goal

Study the resonance in the RLC circuit. Obtain the value of the inductance.


To build the RLC circuit you will be given an inductor, the decade capaci-tance box, and the decade resistance box. Suggested range for C is 10-10000nF and 500-5000 Ω for R.

[1]: Check the theoretical prediction that the resonant frequency f0 at thefixed L and C does not depend on the variations of R in the RLC circuit.Choose some value of C (for example, 75 nF) and some value of R (forexample, 0.5 kΩ). Achieve resonance and measure f0 (see § 6.1.4). Then, forexample, increase R twofold, achieve resonance and measure f0. Try someother values of R.

[2]: Once the irrelevance of R to resonant frequency f0 is established, checkthat f0 follows the functional form of Eq. (6.1) by measuring f0 at differentvalues of C over as wide range of capacitance as possible.In order to have a pronounced response of the RLC circuit to a change offrequency of the driving voltage near the resonant frequency f0, choose a“small” value for R, for example, R ≈ 100 Ω.

One possible kind of analysis is to plot f0 versus C− 12 . [If you think a

little, then you will realize that for such an analysis it would have been niceto choose C = 10, 16, 28, 59, 204, . . . nF.] Do the data lie on a straight linethat passes through the origin? Deduce the value for L and its uncertaintyfrom the slope and compare it to the value of inductance written on the topof your inductor (about 130mH).

A better way to carry out the analysis of presumable power-law-dependantdata (see Eq. (6.1)) is to plot f0 versus C using logarithmic scales on bothaxes. This is equivalent to studying the dependance of log f0 on logC. And,from Eq. (6.1) it follows that

log f0 = log

(1

2π√LC

)= log

(1

2π√L

)− 1

2· logC.

Do the data lie on a straight line with the slope −12? Deduce L and its

uncertainty from the y-intercept and compare it to the provided value.


The advantage of the second method is that you do not need to thinkabout the good choice of values of C for measuring f0, since for any geomet-rical progression of values of C, for example, for C = 10, 100, 1000, 10000 nF,the values of logC are evenly spaced. Do not hesitate to ask your TA aboutthis approach, it will take only 5 minutes. [Note that you cannot actuallyset 10000 nF on the decade capacitance box, but setting 9999.9 nF will dothe trick.]


Assamble the RLC circuit shown on Fig. 6.1. Channel 1 will display thevoltage V (t) driving the circuit. Channel 2 will display the response of thecircuit VR(t).

Set the function generator to produce a SINE-wave.Connect the frequency counter to the SYNC output of the function generator.This way you will know frequency f of the output signal more accurately thanfrom the reading on the frequency dial of the function generator.Set INTERNAL/CHANNEL 1/PEAK-PEAK AUTO for the oscilloscopetrigger mode.Adjusting the frequency to some specific value, do not forget to set the cor-responding frequency range on the function generator. For example, if youhave found out that resonant frequency is about 5 kHz, then set functiongenerator to the closest range of 10 kHz.Reminder: by the oscilloscope SEC/DIV knob you can adjust the horizontaltime scale.

How to achieve a resonance (method A): Look at the amplitude V ampR

of the voltage across the resistor VR(t) (channel 2) as a function of the sourcefrequency (i.e., keep the amplitude of the source voltage constant and varythe source frequency). Having hit the resonant frequency f0 you will observethe maximum value of V amp

R .

How to achieve a resonance (method B): Investigate the relative phaseshift of a driving signal (channel 1) and voltage across the resistance (chan-nel 2). The phase shift will vanish at resonance and both signals will almost


match each other.4 The best way to see this is to adjust ground levels ofchannel 1 and 2 to coincide at the middle of the oscilloscope screen and thenswitch channel 1 and 2 to AC mode.

Optional

Choose some particular values for R and C. Select different shapes of drivingvoltage, i.e., TRIANGLE or SQUARE. Do you see resonance response asyou vary the source frequency? Does the resonant frequency differs from theone obtained with SINE-wave? Ask your TA for explanation.

6.2 Impedance

6.2.1 Theory

For a circuit consisting only of a battery and a resistor, V = I · R accord-ing to the Omh’s law. In our case, we have a resistor, an inductor and acapacitor in the circuit, and the circuit is driven by time-varying voltage,Eq. (6.2). Nevertheless, one can similarly introduce an effective resistance(called impedance) by

V (t) = I(t) ·Reff . (6.5)

For amplitudes V0 and Iamp of V (t) and I(t), respectively, this equation yields

V0 = Iamp ·Reff . (6.6)

Making use of Eq. (6.3) we obtain

1

Reff(ω)=Iamp(ω)

V0

=1√

R2 +(ωL− 1

ωC

)2. (6.7)

Here 1Reff(ω)

is nothing but frequency-dependent effective conductance of the

RLC circuit. Curve of Iamp

V0versus ω (or f ≡ ω

2π) for fixed R, L and C is

called the resonance curve (see Fig. 6.2).

4It is known that, at resonance, the phase shift δ in Eq. (6.3) is 0, so VR(t) ≡ V (t)!

Thus I(t) = V (t)R at resonance, meaning that oscillatory and damping parts of the system

are effectively decoupled.—The LC circuit experiences oscillations at its natural frequency,while the external driving source perfectly takes care of damping by supplying the neces-sary power to be dissipated in the resistor.


At resonance, according to Eq. (6.7), the effective resistance of the RLCcircuit, Reff(ω0), is minimal and its value is equal to R. Thus you can makethe following statement: driving RLC circuit by an external voltage of reso-nant frequency f0 results in the maximum amplitude of the oscillating current(divergent in the limit R→ 0).

An RLC circuit can be interpreted as a frequency filter. Indeed, relation-ship between amplitude of the output voltage across resistor and amplitudeof the input signal from the function generator

V ampR (ω) = Iamp(ω) ·R = V0 ·

R

Reff(ω). (6.8)

shows angular frequency dependant scaling of magnitude RReff(ω)

. Figure 6.2depicts this characteristics of a filter across a wide range of angular frequencyω. Width/broadness/sharpness of the resonance peak determines selectivityof the filter.5

6.2.2 Lab Goal

Measure resonance curve for a given RLC circuit and compare it to thetheoretical prediction of Eq. (6.7).

Experiment Overview/Procedure/Analysis

Measure how the amplitude V ampR (ω) changes as you detune f from f0 for

the given RLC circuit. Calculate ω = 2πf , and also calculateV ampR (ω)

R·V0.

Plot the theoretical expectation Eq. (6.7) for resonance curve

1

Reff(ω)=Iamp(ω)

V0

=V ampR (ω)

R · V0

versus ω and results of your measurements on the same graph. Do you seean agreement?

5Note that the circuit that consists of just a single resistor R has Reff(ω) ≡ R, thusV ampR (ω) ≡ V0 and filter has no selectivity and corresponding resonance curve, Fig. 6.2,

will be a horizontal line.



Fix value of C from 10 to 100 nF and R to be in the range 1000-50000 Ω.Find resonant frequency f0. Measure the amplitude of the voltage across theresistor V amp

R (ω0) on the channel 2 at resonance.Measure the amplitude of the driving voltage V0 on the channel 1. It shouldbe equal to V amp

R (ω0)!Measure the amplitude of the voltage across the resistor V amp

R (ω) on thechannel 2 at frequencies ω above and below the resonance, for example,when V amp

R (ω) is 75%, 50% and 25% of its resonance value, V ampR (ω0). If the

amplitude of the driving voltage varies as you take measurement, then informyour TA about this fact!

Optional

Measure and plot on the same graph several resonance curves for several val-ues of R at some fixed L and C. Observe that the position of resonance peakf0 does not change, and that the curves are getting broader and shallowerwith the increase of R.


Iamp

V0

Figure 6.2: Resonance curve.—Amplitude of the current in the circuit Iamp

versus angular frequency ω of the driving voltage.

Appendix A

Safety

The laboratory is a controlled environment; one in which you may encounterhazards that are not common in other environments. It is therefore importantthat you observe all of the safety rules that pertain to the laboratory. Failureto do so could result in harming yourself or others in the laboratory.

A.1 Lasers

We typically use Helium-Neon (He-Ne) lasers in the introductory physicslaboratories. These kind of lasers are low power (< 1mW) lasers. Thegeneral rule is never look directly into the laser or shine the laser in anotherperson’s eye. Additional information on laser safety can be found on theUMass Environmental Health & Safety web page.

A.2 Radiation

The radioisotopes used in lab are low activities but precautions should beobserved. The general rules for handling radioisotopes and minimizing ex-posure are:

• Minimize time of exposure.

• Maximize distance between radioisotope and yourself.

• Use shielding whenever possible.

45

http://www.ehs.umass.edu/laser-safety-manual

Physics 152(154) Laboratory · Safety · Fall 2013 46

• Do not handle radioisotopes directly. Use tweezers and wear protectiveclothing (gloves, goggles).

Additional information on radiation safety can be found on the UMassEnvironmental Health & Safety web page.

A.3 Cryogenics

Some of the experiments use inert cryogenic liquids like liquid nitrogen (LN2).The boiling point of LN2 is 77K (−195C or −312F). Exposure to LN2

could cause cryogenic burns (blistering), frostbite or hypothermia. Rapidexpansion in a closed space could cause asphyxiation due to oxygen beingdisplace. In most cases the quantities of LN2 that students will use during thelab is just a few fluid ounces. Student will be required to wear eye protectionand aprons during labs with LN2. Additional information on radiation safetycan be found on the UMass Environmental Health & Safety web page.

A.4 High Voltage

There is no specific definition of high voltage. In general any voltage that cancause a shock is consider high. The general rule is to avoid any devices thatcan cause a shock. Observe all signs that warn of high voltage. The specificdanger is when your body becomes a part of a circuit ; especially acrossyour heart. The rule of thumb is not to touch a high voltage source withboth hands. Your heart lies between your arms and if there is a sufficientlyhigh voltage between your arms, then current could pass through your heartcausing fibrillation and your heart could stop beating. Some less dangers areelectrical burns and muscle spasms or seizures.

http://www.ehs.umass.edu/radiation-safety-manual

http://www.ehs.umass.edu/cryogenic-liquids-guidelines

Appendix B

Graphical Presentation of Data

Graphs are pictorial representations of data. In a graph it is much easier tosee trends in data and infer from the trends the physics of the experimentthe data comes from.

B.1 Graphing using Logger Pro

B.1.1 Importing Data into Logger Pro

Start Logger Pro. From the menu select File > Import From > Text

File.... From the file manager window select the file that you exportedfrom DataStudio and click Open. In most cases Logger Pro will automati-cally plot the data from the file you selected.

B.2 Graphing using Excel

B.2.1 Importing Data into MS Excel

Start Excel. From the menu select File > Open... From the file managerwindow select the file that you exported from DataStudio. If you do notsee the file listed go to “Files of types” and select either All Files or TextFiles. Select the data file and click Open. The Text Import Wizard willopen. Select the Delimited radio button, then click Finish.

47

Appendix C

Data and Error Analysis

The basic activity of any experiment is to make measurements. Data (pluralfor datum) are the collection of measurements and quantitative observationsmade during an experiment. There is no such thing as perfect data! No mat-ter how well an experiment is designed, no matter how well the instrumentsused are calibrated there will always be some limitation in how well the dataare known. That limitation is called error.

Once an experiment is completed the data are analyzed to determine a result.Since the data have errors, the result has error too. Therefore, the analysisof data and error go hand in hand. You cannot do one without the other.

An important note: many times we will use the word “uncertainty” insteadof error. In the context of experimental science, the two words mean thesame thing.

C.1 Expressing Measurements

Every measurement has two parts: a best value and an error. There aretwo common ways of expressing a measurement. The first is in terms of theabsolute error :

xmeasured = xbest ±∆x (C.1)

The ∆ symbol is typically used to indicate the quantity that represents theerror in the measurement. The ± symbol is use to indicate that the mea-

48

Physics 152(154) Laboratory · Data and Error Analysis · Fall 2013 49

surement is really a range of values between xbest + ∆x and xbest−∆x. Notethat ∆x has the same units as the measurement. For example, if x is themeasurement of an objects position in meters, then ∆x is also in meters.

The second way to express a measurement is in terms of the fractional error :

fx =∆x

xbest(C.2)

the advantage of the fractional error is it can also be easily express as apercent error too. Consider this example: a student weights a golf ball andgets a values of mball = (46.4± 0.1)g. Then the fractional error would be:

fm =0.1 g

46.4 g

= 0.002 or 0.2% (C.3)

Now the mass of the ball can be express in terms of the percent error:

mball = 46.4 g ± 0.2% (C.4)

Note that the fractional or percent error have no units.

C.2 Uncertainties in Direct Measurements

C.2.1 Estimating Uncertainties from a Scale - Inter-polation

The simplest measurement one can make is comparing an object to a scalelike a ruler. Look at Figure C.1 below:

If we assume the edge of the arrow is aligned with the edge of the ruler,and the edge of the ruler represents the zero of the ruler’s scale, then whatis the length of the arrow? Since the tip of the arrow lies between themarkings on the ruler, we have to interpolate the position of the arrow tip.A reasonable value for the length of the arrow may be 5.5 cm. How do wechoose a reasonable estimate of the uncertainty? I stress the word estimatebecause uncertainties are just that, estimated values. One way to estimate


Figure C.1: Ruler measuring length of arrow

the uncertainty is by considering what is not a reasonable value of the lengthof the arrow. Most people would agree that the length of the arrow is lessthan 5.9 cm and greater than 5.1 cm. Can we make a better estimate of thelength of the arrow? Perhaps less than 5.8 cm and great than 5.2 cm. Wecontinue this process until we reach a point where we are uncertain what thelength is. Our first estimated was 5.5 cm, but someone else may see 5.6 cmor 5.4 cm. The result is there is an uncertainty in the actual value of thelength of the arrow. The range of values that represent a reasonable estimateof the length is the uncertainty in the length. In this particular case ±0.1cm is our estimate. The length of the arrow would be recorded as:

larrow = (5.5± 0.1)cm (C.5)

From the example of the length of the arrow we can express our measurementas:

larrow = 5.5cm± 2% (C.6)

C.2.2 Uncertainties from a Digital Scale

Throughout the semester we will use digital devices to make measurementslike a digital balance to measure masses or a stopwatch to measure time.What is the uncertainty in a digital measurement? Unlike a scale on a rule,there is no way to interpolate the value from a digital display. What you seeis what you get. The rule of thumb for digital display is an uncertainty of±1 in the last digit.

In Fig. C.2 the digital balance displays a mass of 45.03 grams. We assumesthe least certain digit is the “3”. There is no way for us to tell how the


Figure C.2: Measuring a mass with a digital balance

balance chose 3 as the last digit, so the best we can say is the uncertainty is±0.01 grams. The measurement is recorded as:

mmeasured = (45.03± 0.01)grams (C.7)

C.3 Repeated Measurement (Statistical) Tech-

nique

If a measurement is repeated in independent and unbiased ways, the resultsof the measurements will be slightly different each time. A statistical analysisof these results then, it is generally agreed, gives the “best” value of the mea-sured quantity and the “best” estimate of the uncertainty to be associatedwith that result.

C.3.1 Mean Value (Average Value)

The usual method of determining the best value for the result is to computethe “mean value” of the results: If x1, x2, ..., xN are the N measurements ofthe quantity x, then the mean value of x, usually denoted by x, is defined as

x =x1 + x2 + ....+ xN

N=

1

N

N∑i=1

xi (C.8)

C.3.2 Standard Deviation

The uncertainty in the result is usually expressed as the “root-mean-squareddeviation” (also called the “standard deviation”), usually denoted as σx(Greek letter sigma). Formally, the standard deviation is defined as:


σx =

√(x1 − x)2 + ...+ (xN − x)2

N − 1(C.9)

Let’s decipher Eq. C.9 in words. Eq. C.9 tells us to take the difference be-tween each of the measured values (x1, x2, ...) and the mean value x. Wethen square each of the differences (so we count plus and minus differencesequally). Next we find the average of the squared differences by adding themup and dividing by the number of measured values. (The −1 in the denomi-nator of Eq. C.9 is a mathematical refinement to reflect the fact that we haveused the N values once to calculate the mean. In practice, the numerical dif-ference between using N and using N − 1 in the denominator is insignificantfor our purposes.) Finally, we take the square-root of that result to give aquantity which has the same units as the original x values.

C.3.3 Standard Deviation of the Mean

Now Eq. C.9 gives the deviation for any one of the individual measurementsxi, but what about the deviation of the average x itself? There is a simpleway of calculating it. Simply divide the standard deviation of x by the rootof the number of measurements N :

∆x =σx√N

(C.10)

It is this quantity ∆x that is often called the Standard Error.

Note that in the case of possibly hundreds of measurements, calculatingEq. C.8 and Eq. C.9 by hand or with a calculator would be a very long andtedious exercise. Fortunately most calculators and spreadsheet programs likeExcel have built-in functions to calculate the average (AVERAGE) and stan-dard deviation (STDEV) quickly. We strongly encourage you to use theseprograms and functions.

C.4 Propagation of Uncertainties

Often the physical quantity of interest is not one that can be measureddirectly but is calculated from other measured quantities. Since all measured


quantities have associated uncertainties, how do those uncertainties affect thecalculated quantity? There are various rules for propagating the measureduncertainties into the calculated uncertainties.

C.4.1 Addition and Subtraction

Figure C.3: Dimensions of brass block

Imagine we have a brass block with dimensions illustrated in Fig. C.3. Let’simagine we measure the blocks dimensions l, w, and t and estimate the un-certainties in these measurements using techniques from Section C.2.1.

Let’s calculate the perimeter around the face of the brass block. Both lengthand width of the block have uncertainties. it’s not to hard to figure out whatthe uncertainty of the perimeter of the block is (Eq. C.22):

p = 2l + 2w

(p±∆p) = 2(l ±∆l) + 2(w ±∆w)

= 2(l + w)± 2(∆l + ∆w)

∆p = 2∆l + 2∆w (C.11)

Actually the form of C.11 over estimates the uncertainty in the perimeter.We have to think of l and w as independent measurements. To properlypropagate the uncertainty of independent measurements, we add the uncer-tainties in quadrature.

∆p =√

(2∆l)2 + (2∆w)2 (C.12)


Another example is the difference (d) between the l and w:

d = l − w(d±∆d) = (l ±∆l)− (w ±∆w)

= (l − w)± (∆l + ∆w)

∆d = ∆l + ∆w (C.13)

Note that uncertainties ∆l and ∆w are added together even when taking thedifference of two numbers (Eq. C.23). The reason is the measurements of land w are independent of each other.

Again, simply adding the uncertainties would over estimate the uncertaintyin the difference so we have to add the uncertainties in quadrature:

∆p =√

(∆l)2 + (∆w)2 (C.14)

Note the absence of the extra factor “2”.

C.4.2 Multiplication and Division

Now let’s calculate another quantity, the surface area A = lw of the block.Here is how we would determine the uncertainty in A (Eq. C.24):

A = lw

(A±∆A) = (l ±∆l)(w ±∆w)

= (lw ± w∆l ± l∆w ±∆l∆w)

A(1± ∆A

A) = lw(1± ∆l

l± ∆w

w± ∆l

l

∆w

w) (C.15)

At this point let’s make an approximation. If the fractional uncertainties∆ll

and ∆ww

are small (much less than 1), then the product of the fractionaluncertainties ∆l

l∆ww

is even smaller and can be neglected. The result for thefractional uncertainty in A is:

∆A

A=

∆l

l+

∆w

w(C.16)


Now add the fractional uncertainties in quadrature:

∆A

A=

√√√√(∆l

l

)2

+(

∆w

w

)2

(C.17)

There is a similar derivation of the fractional uncertainty for the ratio R =l/w. The fractional uncertainty of R (Eq. C.25) is the same as the area A:

R =l

w∆R

R=

∆l

l+

∆w

w(C.18)

Now add the fractional uncertainties in quadrature:

∆R

R=

√√√√(∆l

l

)2

+(

∆w

w

)2

(C.19)

C.5 Simple Rules of Uncertainty Propagation

1. Adding a constant k to measurement B:

A = B + k

∆A = ∆B (C.20)

2. Multiply a measurement B by a constant k:

A = kB

∆A = k∆B (C.21)

The uncertainty is multiplied by the constant.

3. Adding two measurements A and B together:

C = A+B

∆C =√

(∆A)2 + (∆B)2 (C.22)

The uncertainties add in quadrature.


4. Subracting two measurements A and B:

C = A−B∆C =

√(∆A)2 + (∆B)2 (C.23)

The uncertainties still add in quadrature.

5. Multiplying two measurements A and B:

C = AB

∆C

C=

√(∆A

A

)2

+(

∆B

B

)2

(C.24)

The fractional uncertainties add in quadrature.

6. Dividing two measurements A and B:

C =A

B

∆C

C=

√(∆A

A

)2

+(

∆B

B

)2

(C.25)

The fractional uncertainties still add in quadrature.

7. Measurement B raised to power n:

A = Bn

∆A

A= |n|∆B

B(C.26)

The fractional uncertainty multiplied by absolute power n.

Appendix D

Laboratory Reports

A lab report is the formal document that conveys to the reader the experi-ment you perform and the results you found. A report is a factual record ofwhat was done during the experiment. It addresses all the who, what, where,when, and why.

D.1 Structure

The basic structure of a lab report consists of four parts: a heading, anabstract, answers to lab report questions, and a conclusion. The details ofwriting these parts are given below.

D.1.1 Heading

The heading begins with the title of the report. The title should be a de-scriptive statement of the experiment positioned at the top-center of the firstpage. The heading also includes:

• Author (your name).

• Course and section number.

• Lab room.

• Date.

57

Physics 152(154) Laboratory · Lab Reports · Fall 2013 58

These should be listed along the left margin of the report. The heading is apart of the report; no cover page.

D.1.2 Abstract

The abstract is a synopsis of the experiment. Abstracts are typically oneparagraph long (4 to 6 sentences). The abstract should highlight the essentialparts of the experiment but not a lot of details. Even though the abstract isthe first paragraph of the report it is typically written last. Once all of thedata are analyzed, questions are answered, and the conclusion is made, thenthe abstract is written.

D.1.3 Questions and Answers

The main body of a report is a series of answers to questions. The an-swers could be as simple as stating your result or more conceptual answersdescribing the underlying physics to the experiment. Every question has animplicity Why? Explain. A terse answer with out explanation will not receivefull credit. Some questions may be computations; ie., you must perform somekind of calculation. Show your work! Numerical results without derivationwill not receive full credit.

D.1.4 Conclusion

The conclusion is a brief summary of the experiment, usually a single para-graph long. The conclusion should clearly state what are the experimentalresults; for example, what is your experimental value for free fall accelera-tion g. How well does your experimental results agree (within error) withthe expected value. Also, if there are assertions that you are trying to testin the experiment, you want to state why those assertions are true and whatexperimental evidence do you have that shows the assertions are true.

On a rare occasion if your results do not agree with expected value, yourresults may be inconclusive. In cases like this you want to explain if there issome wrong with the measurements made or ways to correct or improve theexperiment you performed.


D.2 Format

Lab reports are to be typeset, not hand written. Typically a word processorlike Microsoft Word or LibreOffice Writer is used to write a report. Reportsincorporate a variety of components like tables, figures, and equations. Thedetails on how to format these components are given below. Some standardformatting for the text of the report are:

• Pages are letter (8.5in by 11in) portrait.

• Times New Roman 12pt. font.

• Left and right margins set to 1in.

• Lines are double-spaced.

D.2.1 Tables

Here are some standard formatting rules for tables:

• Tables are lists of measurements arranged in columns.

• The top of each column is a heading with the name of the physicalquantity and the UNITS the quantity is measured in.

• A spreadsheet program like Microsoft Excel or LibreOffice Calc is thebest way to make tables. Once you produce a table, you can copy/pastethe table into the document of your report.

D.2.2 Equations

Physics is a heavily mathematical science, and equations are a short handfor communicating physics principles and concepts. Think of an equation asa mathematical sentence.

D.2.3 Figures

The typical figure in a physics laboratory report is a graph plotting onephysical quantity vs. another physical quantity. Some standard formattingfor graphs include:


• A title at the top of the graph. The title should be of the form “Y-variable (vertical axis) vs. X-variable (horizontal axis)”. For examplea graph of “Position vs. Time” would have the values of position onthe vertical axis and time on the horizontal axis.

• the axes should be labeled with the physical quantities and correctUNITS!

• the scales of the axes should be multiples of 1, 2, 5, or 10.

D.3 Composition

Good writing skills matter. Poor writing can alter the content (meaning) ofan answer or even contradict the correct answer. Physics is a logical, self-consistent science, so the writing should be logical and self-consistent too.Reports will be graded as submitted, so any mistakes that alter the contentof an answer will be marked down. The reader (your lab instructor) cannotsecond guess what you intended the answer to a question to be, so make sureyour answers mean what they say and say what they mean.

• Use spellchecker

• Proofread!

• Revise and rewrite as need.

• Use definite, specific, concrete language

• Be clear!

D.4 Content

The content of the report addresses the subject matter; the principles, ideasand concepts the report is about. Since we believe physics is a logical,self-consistent science, the content of the report should be logical and self-consistent as well.

- Clarity: the underlying principles are clearly articulated, all relevantterminology is defined. You should be specific in the language used.Avoid vague or ambiguous statements.


- Completeness: all elements of the report are present. Missing or omit-ted content will mislead or confuse the reader.

- Conciseness: specific and to the point. The writer should avoid redun-dant, irrelevant or circuitous statements. Stay on topic.

- Consistency: all elements of the report direct the reader to a single, log-ical conclusion. Avoid illogical, erroneous, unsubstantiated, specious,irrelevant statements and contradictions.

D.5 Sample Report

Below is an example of how a lab report is written. The sample report illus-trates all of the structure,formatting, and composition for writing a properreport.

Measurement of π

Jane SmithPhysics 131 – Section 14Hasbrouck 210Sept. 18, 2013

AbstractIn this experiment we measure the number pi ( π ). The dimensions of a set of plastic disks of various sizes are measured. Our objective is to measuret pi o a precision of 0.01%. Our measurement of pi isπ=3.1414±0.0003 .

Questions & Answers1. In your own words, define pi ( π ). What quantities do you need to measure? What uncertainties are associated with the measured quantites?

Pi ( π ) is an irrational number; a number that cannot be expressed as a fraction of two integers. Geometrically pi is defined as the ratio of a circle's circumference (C) to its diameter (d) (Eq. 1).

π=Cd

(1)

Figure 1 illustrates the dimensions of a circle.

C is measured to ±0.1mm and d is measured to ±0.01mm.

2. What are the units of pi?

Pi has no units; it is a dimensionless number. Both the circumference and diameter are measured in the same units. Since pi is the ratio of circumference to diameter, the units cancel.

3. How did you measure the diameter (d) of the disk? What is the average d? Give table of measurements.

Figure 1: Dimensions of a circle.

TitleHeading

Variables italicized to distinguish from text.

Equations positioned incenter of page with referencenumber.

Figures positioned in center of page with captions and reference number at bottom.

Abstract give very short description of experiment. 4 to 6 sentences only.

diameter d (±0.05mm)

99.97

99.97

99.99

100.02

99.99

100.00

99.98

The diameter of the disk is measured using a dial caliper. The diameter is measured seven times across seven randomly chosen diameters. The average diameter is d = (100.00±0.01)mm.

4. How did you measure the circumference (C) of the disk? Plot the distance traveled by the disk vs. the number of of rotations.

The disk's circumference was measured by rolling the disk along the edge of the a meter stick. The distance (s) the disk traveled was measured for each rotation (n). The distance travel is related to the number of rotations by:

s=Cn (2)

A plot of s vs. n is a straight line with the best fit slope equal to the circumference of the disk.

The best fit slope to data is C=(314.170±0.005)mm

Column headingwith variable and UNITS!

Data are listed in columns.

Figure 2: Plot of Distance vs. Rotations

0 2 4 6 8 10 120

500

1000

1500

2000

2500

3000

3500

Distance (s) vs. Rotations (n)

Number of Rotations n

Dis

tanc

e tr

avel

e d s

(m

m)

Plot of data dependent vs. independent variable.Axes labeled andnumbered in multiplesof 1, 2, 5, or 10.

5. Calculate your experimental value of pi. What is the precision of your value?

π = C / d= 314.170 mm/100.01mm= 3.1414

Our experimental value of pi is π=3.1414±0.0003 . The precision of our value is 0.01%.

6. Does the value of pi depend on the size of the disk? Why or why not?

The value of pi is independent of the size of the disk. We measured pi using three different size disks.

diameter d (mm) Circumference C (mm) pi

30 94.238 3.1413

50 157.078 3.1416

100 314.155 3.1415

All three values of pi are consist within 0.01% of the accepted value. There is no difference in our measured value of pi for different size disks.

7. What is the accuracy of your experimental value of pi? Calculate the percent difference compared to the true value of pi.

%diff = ∣π−3.1414∣/π= 0.00006= 0.006%

The experimental value is within 0.03% uncertainty of the true value and therefore is consistent with true value.

8. Which measurement has the larger experimental uncertainty? C or d?

f C = 0.005 mm /314.170 mm f d = 0.01 mm /100.01 mm= 0.00002 = 0.0001= 0.002 % = 0.01 %

The percent uncertainty in d (0.01%) is five times than the percent uncertainty in C (0.002%). One wayto decrease the percent uncertainty in d is to measure a larger disk. The absolute uncertainty of 0.01mmwould remain the same, but the percent uncertainty would decrease as the diameter of the disk increases.

ConclusionOur measurement of pi is, π=3.1414±0.0003 and is consistent with the true value within the 0.03% experimental uncertainty. The largest source of uncertainty is the measurement of the circumference d. We found no depends of the value of pi on the size of disk measured.

Calculations shows work.

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