LABORATORY INSTRUCTIONS IN GENERAL PHYSICS

PHY 213

Electricity and Magnetism

Part III

by

Toby Dittrich

Bob Drosd

Leonid Minkin

Portland Community College

2012

2

TABLE OF CONTENTS

Exp. No Title Page

GENERAL INSTRUCTION FOR LAB REPORT 3

1 MAPPING ELECTRIC FIELDS WITH EQUIPOTENTIAL LINES 5

2 RESISTANCE MEASUREMENTS (WHEATSTONE BRIDGE) 11

3 INTERNAL RESISTANCE, EMF AND POWER TRANSFER 15

4 HEATTING EFFECT OF ELECTRIC CURRENT 18

5 KIRCHHOFF’S AND OHM’S LAWS 21

6 CHARGING AND DISCHARGING A CAPACITOR 25

7 MAGNETIC FIELD OF THE EARTH 28

8 THE MAGNETIC FIELD IN A SOLENOID 32

9 ALTERNATING CURRENT CIRCUIT 36

3

GENERAL INSTRUCTION FOR LAB REPORT

For every lab completed, each person will be responsible for handing in a lab report. In addition to

all the other things which have been presented to you regarding the development of skills in

measurements, handling errors, drawing graphs, etc., we feel it is important for you to develop skills

in organizing your data, thoughts and ideas and present them in acceptable written form.

The style and format of your reports we leave to you to develop. Keep in mind that a report should

be logically organized; include basic information as to purpose, approach, apparatus, data, analysis

of data and conclusion. In addition to all of this it should be written so that it is easily read and

understood by your lab instructor who will be grading your labs.

Items you will want to consider including in your report are:

Cover sheet:

The first page of the report is to be a title page and shall contain the experiment number, the

experiment title, date that experiment was performed, the submitter's name, and the names of the

submitter's partners.

Purpose:

Object of the experiment. This should be one or two sentences briefly describing the purpose of

experiment and what you are setting out to prove.

Apparatus:

A brief description of apparatus actually used.

Diagram:

A well labeled drawing (sketch) of the apparatus might be helpful to reader to understand.

Theory:

Summarize the basic physics of your experiment. Include equations and other principle things the

reader would need to know in order to understand the experiment.

Experimental procedure:

Describe briefly how you carried out the experiment.

Table for data:

Include all of the data you acquire in the experiment in tabular form. Always watch significant

figures. Correct significant figures should be applied in data tables, calculation tables, and sample

calculations. It saves writing to put the units in the legend on top of the column, e.g. length (cm),

rather than to follow every number with a symbol.

Graphs:

Every graph should have a title, labeled axes with properly chosen scales and units.

4

Conclusion:

Although most experiments are group efforts, the conclusion should be an individual composition on

the part of the submitter. This is your response to the purpose of the lab. Make a quantative estimate

of errors, evaluation of how well the purpose of the experiment was achieved, and conclusion of

what could be improved to provide better results. Compare your results with the accepted values and

comment on the comparison.

Be prepared; study the lab before you come to class. If you have any questions ask your lab

instructor.

5

EXPERIMENT 1

MAPPING ELECTRIC FIELDS WITH EQUIPOTENTIAL LINES

Objective

To draw lines of equal electrostatic potential and electric field lines for various electrode

configurations

Apparatus

Sheets of conductive paper with various painted-on electrode shapes, DC Power Supply, electrical

leads, probes, digital voltmeter (multimeter) (Fig. 1)

Fig. 1. Apparatus set up

Background

Two points that are at the same potential are called equipotential points. The collection of all points

at the same potential make up an equipotential surface (in three dimensions) or an equipotential line

(in two dimensions). Several properties of potential will be useful to us in this lab. Among them

are:

1. Equipotential lines cannot cross each other.

2. Equipotential lines are always perpendicular to electric field lines.

3. The surface of a conductor is an equipotential surface.

The distribution of potential V(x,y,z) allows to calculate an electric field: the magnitude of electric

field E = V/s. Here V/s is a change of the potential in the direction of its maximum change.

The electric field points in the direction of decreasing potential. Fig. 2 illustrates this statement

6

Fig. 2. Electric (dash) and equipotential (solid) lines

Under certain circumstances, such as found in this lab, the potential can be measured directly with a

voltmeter. By placing two conductors, which will be held at different potentials, on a sheet of

conducting paper, the potential on the paper between the two conductors can be measured. By

determining several equipotential points, an equipotential line can be constructed. Doing this for

several potentials will generate a set of equipotential lines for the given arrangement of conductors.

If we then apply the second property, that E is everywhere perpendicular to the equipotential lines,

we can deduce the electric field around the conductors.

Equipotential lines can be labeled (in volts) relative to the negative electrode (zero potential).

However, it is convenient to label potential in fraction (percent) relative to potential difference

between two electrodes: in this case potential difference between two electrodes corresponds 100%

and one can find potential at the equipotential line for any potential difference of the two electrodes.

Procedure

Set up the apparatus as shown in Fig. 1.

Lay the paper on a sheet of cardboard and clip two leads to the conducting surfaces. Connect the

leads to the “DC” outputs of your power supply. (The power supply should be off until you are

ready to begin making measurements!)

Clip the lead plugged into the “common” jack of your multimeter to the lead attached to the

ground or negative jack of the power supply. The other lead from the voltmeter will remain

unattached.

Using a sheet of white grid paper supplied to you (electric field mapping grid), sketch the

configuration of the conductors, indicating which conductor is attached to which jack of the

power supply.

7

Place the free lead from the voltmeter on the conducting surface that is attached to the positive

jack of the power supply. Turn on the power supply and adjust the voltage on the power supply

until the voltmeter reads 10.0 V. (Do not use the voltmeter built into the power supply. It may

give you a different potential than at the conducting surfaces.)

Move the free lead from the voltmeter around on the black paper until you find a point at a

potential of 9.0 V. Indicate that point on your white paper.

Find a dozen or so other points with a potential of 9.0 V until you can trace out the 9.0-V

equipotential line. Label this line on your grid paper (in volts and percent).

Repeat this for equipotential lines of 2.0 V, 4.0 V, 6.0 V and 8.0 V.

If there is a good bit more space between the “negative” conductor and the 2.0-V line than there

is between the “positive” conductor and the 9.0-V line, go back and find a 1.0-V equipotential

line.

Now use your equipotential lines to sketch in the electric field lines. Remember that the two are

always perpendicular. Use a different color or pattern of lines on your white grid paper to draw

the E field lines.

11. Choose two points at your diagram of electric field distribution and find electric field (E =

V/s) in these two points for potential difference between two electrodes 10 V and 200 V. For

this you need to define potential difference between two equipotential lines and measure the

distance between them in the direction of maximum change of the potential.

Repeat the described procedure for finding equipotential and electric lines for different

electrodes. Choose two points at this diagram of electric field distribution and find electric field

in these two points for potential difference between two electrodes 10 V and 200 V.

8

9

10

ELECTRIC FIELD MAPPING GRID

11

EXPERIMENT 2

RESISTANCE MEASUREMENTS (WHEATSTONE BRIDGE)

Objective

To become familiar with resistor color codes and instruments that are used to measure resistance. To

find resistivity of some metals.

Apparatus

Slide-wire bridge (Fig. 1, Wheatstone bridge), digital multimeter, galvanometer, one dual resistance

box R0, two resistors, a dry cell battery, copper and Ni-Ag wires, connecting wires, portable

Wheatstone bridge, resistors, copper and Ni-Ag wires.

Fig. 1. Physical Layout of the Wheatstone Bridge

Background

Factors affecting the Resistance of Wire (Resistivity) :

The resistance of material depends upon nature of conductor, its shape and size. Resistance, R, of

conductor is directly proportional to length, L, and inversely proportional to the area of cross-section

of conductor A:

A

LR

where is the constant of proportionality called the resistivity of the material. Its value is constant

for a given material, independent of shape and size of the conductor.

Color coding of Resistance

Commercial resistances available in market have their magnitude written in the form of color codes.

We associate a color with each digit: 0, 1, . . . . . ., 9

Black - 0, Brown - 1, Red - 2, Orange - 3, Yellow - 4, Green - 5, Blue - 6, Violet - 7, Gray - 8,

White - 9

12

The three colored bands on one side indicate its resistance. The first two bands from one end indicate

the corresponding digits while third band’s color indicate powers of ten with which number must be

multiplied to get the resistance value in ohm. In addition to three bands, fourth band gives us

tolerance with silver band implying tolerance of ± 10% and gold band with tolerance of ± 5%. No

fourth band indicates resistance of ± 20%. For example, if four bands are of yellow, red, blue and

gold then resistance will be (42x106) ± 5%

Wheatstone Bridge

The Wheatstone bridge is used to compare an unknown resistance with a known resistance. The

bridge is commonly used in control circuits. For example, a temperature sensor in an oven may have

a resistance that increases with temperature. The control circuit should turn on the oven heater until

the sensor in the oven has reached the desired resistance. The control knob (which may be labeled

with temperature readings) adjusts a variable resistor to which the sensor is compared. The heater is

turned on when the resistance is lower than the comparison value and turned off when it is higher.

A Wheatstone bridge is a network of resistances with a sensitive galvanometer, G (Fig. 2). Rx is an

unknown resistor, R0 is the known resistor, and the two resistors R1 and R2 have a known ratio R2 /R1.

A galvanometer G measures the voltage difference VAB between points A and B. Either the known

resistor R0 or the ratio R2 /R1 is adjusted until the voltage difference VAB is zero and no current flows

through G. When VAB = 0, the bridge is said to be "balanced".

Fig. 2 Wheatstone Bridge Circuit

Since VAB = 0, the voltage drop from C to A must equal the voltage drop from C to B, VCA=VCB.

Likewise, we must have VAD = VBD. So we can write,

RxI1=R1I2

R0I1=R2I2

Dividing one equation by the other results in

13

2

10

2

10

L

LR

R

RRRx

Thus, the unknown resistance Rx can be computed from the known resistance R0 and the known ratio

R2 /R1=L1/L2. Notice that the computed Rx does not depend on the voltage Vo; hence, Vo does not

have to be very stable or well-known. Another advantage of the Wheatstone bridge is that, because it

uses a null measurement, (VAB = 0), the galvanometer does not have to be calibrated.

Series and Parallel Resistors. In series circuit the total resistance is the sum of the separate

resistances. In a parallel circuit the reciprocal of the total resistance is equal to the sum of the

reciprocals of the separate resistances.

Procedure

Part A. Measuring of Resistance. Series and Parallel Connections of Resistors

Two resistors will be used in these investigations.

1. Using color code (CC), determine the resistance of each resistor together with their

respective tolerance.

2. Measure and record the resistance of each resistor using digital multimeter (DMM).

3. Measure the same resistors with the Wheatstone Bridge (WB). If the slider is too near one

end, change the resistance in the box (R0) until the slider is near the middle of the wire when

bridge balanced.

4. Connect these two resistors in series and parallel. Use the digital multimeter and Wheatstone

Bridge to measure the resistance of combinations (DMMexp, WBexp).

5. Calculate the total resistance for connections of two resistors in series (Rs) and parallel (Rp).

Use these formulas for connection in series and parallel: Rs=R1+R2, 1/Rp =1/R1+1/R2. Make

these calculations using data for measuring by DMM and WB. Find the percent discrepancy

between the results of calculations and measuring.

6. Fill up the table provided.

CC DMMexp WBexp DMMcal WBcal DMM

Discrepancy

WB

Discrepancy

R1(Ω)

R2 (Ω)

Connection in

Series (Ω)

Connection in

Parallel (Ω)

Part B. Resistivity of Metals

In this part, the resistivity of copper (Cu) and Ni-Cu alloy will be found. The unknown resistor Rx is

one of 5 coils of wire mounted on a board and in the boxes. The lengths of the wires, their

composition, and their diameters (in inches) are printed on the board or on the resistor. There are 22

14

and 28 gauge wires (22 gauge wire has a diameter of 0.643mm; 28 gauge wire has a diameter of

0.321mm). Use a multimeter to measure unknown resistor. Knowing Rx, length of the wire L and its

diameter D allows you to find resistivity:

L

RD x

4

2

Accepted values of resistivity for Cu and Ni-Ag alloy are 1.72·10-8

and 31·10-8

Ω ·m. Find percent

discrepancy between accepted values of resistivity, acc, and found from experiment exp. Fill the

table provided

Rx (Ω) L (m) D(m) exp (Ωm) acc (Ωm) %Discrepancy

Cu

Ni-Ag

15

EXPERIMENT 3

INTERNAL RESISTANCE, EMF AND POWER TRANSFER

Objective To find internal resistance and EMF of battery and the power transfer to the load.

Apparatus Decade resistance box, Digital Multimeter (2), dry cell battery, resistor (approximately 20 Ω), and

connecting wires.

Background The resistance of an object depends on the material out which the object is made, the length of the

object, and the cross-section area of the object. Batteries are also made of some sort of material and

must have inherent resistance. This resistance is called internal resistance Ri. The chemical reaction

in the battery produces by-products, which along with the original chemicals, are the cause of this

internal resistance. As the reactions chemical are used up, the resistance inside the battery increases

(new batteries have small internal resistance). This resistance is inside the battery and cannot be

measured directly. If an external resistor, Re, were connected across the battery terminals, a current

would flow which is proportional to the electric potential produced by chemical reaction

(electromotive force, E) in the battery and the total resistance of the complete circuit (Fig. 1a),

including the internal resistance of the battery (electromotive force is equal the voltage drop on the

internal and external (Ve) resistors)

eiei VIRRRIE )( (1)

As an internal resistance can be small enough, we will add resistance Ri2 ≈ 25 Ω to

simplifymeasurements (Fig. 1b). We now call Ri=Ri2+Ri1 internal resistance where Ri1 is internal

resistance of the battery.

a b

Fig. 1.A model of the real battery connected to the external resistance (a) and experimental set up (b)

16

The power, P, delivered to the external resistor will change as the external resistance change

2

22

)( ei

e

RR

RERIP

(2)

The maximum power occurs when dP/dRe=0. Given the expression for P above, this yields an

equation

0)(

2

)( 3

2

2

2

ei

e

ei RR

RE

RR

E

and therefore the maximum power transfer to the external load, Re, when Re=Ri. This case is a

simple example of a general happening of great importance in power transfer in circuit theory.

Efficiency is the ratio of power released on the load to the power of battery Pb

E

V

P

PEff e

b

(3)

A 50% efficiency is involved during the maximum power transfer when Re=Ri .When Re equals Ri,

half the developed power goes to the external load; the other half of the power heats the electrolyte

in the battery and Ri1 resistor.

Procedure

1. Measure and record Ri2. Connect the apparatus in the circuit as shown in Fig 1a. Set the

decade resistance box at 5 Ω. Take voltmeter and ammeter reading. Repeat this procedure for

values of Re in the range from 10 to 110 Ω with the increment 5 Ω. If power P does not have

a maximum increase the range of the measurement.

2. For every load calculate power on the load (P=IVe) and efficiency (equation (3)). Fill up the

following table

Re (Ω) I(A) Ve(V) P (W) Eff %

5

10

110

3. Plot a graph of Ve versus I. According to equation (1) it is a linear function

IREV ie

Find Ri (–Ri is the slope of this graph) and electromotive force of the battery (E is the y-

intercept).

4. Find an internal resistance of the battery (Ri1=Ri-Ri2).

5. Plot a second graph of P versus Re. Use a fit function (equation 2)

17

2

2

)( xB

xAy

to find electromotive force of the battery (A=E) and resistance Ri (Ri=B).

Compare these values to the E and Ri found from linear graph Ve versus Re (paragraph 4).

Find the maximum of the graph of P versus Re and record the load, providing maximum of

power transfer.

6. Make a graph of efficiency versus load (equation 3). Choose a fit function asdx

xy

100 and

find d=Ri (ie

e

RR

REff

). Find the load corresponding to 50 % efficiency. Compare this

load to the load providing maximum power transfer.

7. Fill the table provided

Ve vs I (graph 1) P vs Re (graph 2) Eff vs Re (graph 3) Average

E(v)

Ri (Ω)

Ri1=(Ri-Ri2) (Ω)

18

EXPERIMENT 4

HEATTING EFFECT OF ELECTRIC CURRENT

Objective

To determine the relation between joule and calorie.

Apparatus

Electro calorimeter (immersion heater and calorimeter), DC power supply, voltmeter (digital

multimeter), connecting wires, thermometer (PASCO probe), interface, stopwatch, laboratory

balance, and ice. The circuit diagram and electric calorimeter are in the Fig. 1a,b.

(a) (b)

Fig.1. An electric calorimeter (a) and circuit diagram (b)

Background

The work done, W (or energy expended) per unit charge in moving a charge q from one point to

another in the potential difference or voltage V is,

,IVtqVW (1)

where I is a current and t is time.

19

The electrical energy expended is manifested as heat energy and is commonly called joule heat. The

energy expended in an electrical circuit as given by equation (1) has unit of joule (I in Amperes, V in

volts, and t in seconds). The relationship (conversion factor) between joules and heat units in

calories was established by Prescott Joule (1818-1889) from mechanical considerations. The result

was 1 cal = 4.186 J. Experimentally, the amount of electrical joule heat generated in a circuit

element of resistance R is measured by calorimetric methods. If a current is passed through a

resistance (immersion heater) in a calorimeter with water in an arrangement as illustrated in Fig 1,

then by the conservation of energy, the electrical energy expended in the resistance, W, is equal to

the heat (joule heat), Q, gained by system (electrical energy expended equals heat gained). Heat

received by the calorimeter, water, and coil is equal

Q=(mwcw+mcalccal+ Ccoil)(Tf-Ti) (2)

where m’s and c’s are the masses and specific heat of the water and calorimeter cup, respectively, Tf

and Ti are the final and initial temperatures of the system, and Ccoil is the heat capacity of the coil.

Procedure

1. Determine and record the masses of the inner calorimeter cup (without ring).

2. Fill the calorimeter cup about two-third full of cool water (add some ice to the water in

the calorimeter cup. When the ice has melted, measure and record the equipment

temperature Ti. This should be 10-12 Co below room temperature.) The helix of heater

must be completely immersed in water. Determine and record the mass of the calorimeter

cup and water.

3. Place the immersion heater in the calorimeter cup. Make certain that the heating coil is

completely immersed.

4. Turn on power supply and adjust the rheostat until there is a constant current between

2.5-3 A. Then unplug the power supply. This procedure should be done as quickly as

possible to avoid heating the water.

5. Then plug in the power supply and at the same time start the stopwatch. When the

temperature of the water (and the calorimeter system) is 10-12 Co above the room

temperature, simultaneously unplug the power supply and stop the timer. Continue

stirring until a maximum temperature is reached and record this temperature, Tf.

6. Compute the electrical energy, W, expended in the coil (in joules) from the electrical and

time readings.

7. Compute the heat energy (in calories) gained by calorimeter system. The calorimeter

cups are made from aluminum (ccal=0.215 cal/(g Co). The specific heat of the water

cw=1.00 cal/(g Co) and Ccoil = 8.50 cal/C

o.

8. Take the ratio of the electrical to heat energy results (W/Q) to find the “electrical

equivalent of heat” (equations 1,2). This ratio has unit J/cal. Compare this to the value of

the mechanical equivalent of heat by computing the percent error. Determine W/Q ratio

by a statistical method. You will work in groups. We are going to use information from

all class’ groups. Every group must feel the corresponding column which represents their

result for W/Q ratio (number of groups can be different). Find the mean of W/Q (W/Qav)

and standard deviation of W/Q (SW/Q ) :

20

n

QW

QW

n

k

k

av

1

)/(

)/(1

])/()/[(1

2

2

/

n

QWQW

S

n

k

avk

QW

n is the number of groups (sample size). Fill up the table provided

Group 1 Group 2 Group 3 Group 4 Group 5 W/Qav SW/Q

W/Q (J/cal)

9. Find the discrepancy between accepted value of W/Q=4.19 J/cal and found

experimentally W/Qav.

10. Find the confident interval for 95% confident level

(W/Q)av2SW/Q

21

EXPERIMENT 6

KIRCHHOFF’S AND OHM’S LAWS

Objective

To verify Kirchhoff’s and Ohm’s Laws

Apparatus

Simpson 373 mA Ammeter, 2 digital multimiters (DMM), three batteries, three 100 Ω resistors and

connecting wires (Part A). 100 Ω resistor, two cables with alligator clips, 6-V light bulb, voltage

sensor (PASCO CI-6503), Science Workshop 750 interface, computer (Part B).

Background

There are two Kirchoff's laws for electrical circuits.

The first is the current or nodal law:

At any point (node) in the circuit, the sum of the currents must equal 0. (current into the node

is positive, current leaving the node is negative)

The second is the mesh loop law:

Around any closed loop in the circuit (called a mesh loop) the sum of the potential

differences must equal zero (the potential difference across a dissipative device, such as a

resistor, is negative if it is in the direction of the current passing through the device; the

potential difference across an energy supply is positive if the current is traveling from the -

terminal to the + terminal.)

For many materials the resistance remains constant over a range of voltage. Such resistors are called

“ohmic” and they obey Ohm’s Law

V=IR

For such a material, a graph of voltage versus current is a straight line, the slope of which is the

value of the resistance. If the resistance is non-ohmic a graph voltage versus current is not linear

function.

Procedure

Part A. Kirchhoff’s Laws

To illustrate Kirchhoff’s two laws, two examples will be given. You will use one Simpson 373

ammeter and two digital multimeters (DMM). The internal resistance of Simpson 373 mA depends

on the scale used (Table 1); internal resistance of ammeter (DMM) is 5 Ω.

22

Scale (mA) Resistance (Ω)

0-50 3

0-25 7.5

0-10 15

0-5 35

0-1 100

Table 1. Internal resistance of Simpson 373 mA

1. Measure the resistance of three resistors (approximately 100 Ω each).

2. Assemble the circuit which is given in Fig. 1. (circuit 1).

Fig. 1. Circuit 1

I1-I2+I3=0

(r1+R1)I1+(r2+R3)I2+0•I3=E1

0•I1+(r2+R3)I2+(r3+R2)•I3=E2

3. Solve this system of linear algebraic equation to find I1, I2, I3. We neglected the internal

resistances of batteries although for the old batteries these resistances can be large enough.

Measure the voltage differences on the batteries terminals for the assembled circuit (these are

E1 and E2) to exclude the error connected with this negligence.

4. Record the currents I1, I2, I3 and find percent discrepancy between experimental and

calculated currents. Fill the table provided

23

I1(mA) I2(mA) I3(mA)

calculation

measurement

% discrepancy

5. Verify the first and second Kirchhoff’s Laws:

0

03

1

k

k

k

k

V

I

To verify the second Kirchoff’s law, measure the voltage drops around any chosen loop. For

example, for the circuit in the Fig. 1 measure the voltages V12, V23 , V34 V45 , V51 and find

their sum.

6. Repeat the procedures described in paragraphs 1-5 for the second circuit (Fig. 2)

Fig.2. Circuit 2.

24

Part B. Ohm’s Law

For this part of activity you will make graphs of current versus voltage for two different loads –

ohmic and non-ohmic. The first graph is linear and the second non-linear. You will use Science

Workshop 750 interface to detect the current and voltage and make the graphs. Use a signal

generator (Science Workshop 750 interface) to produce triangle wave voltage of 0.200 Hz. Set the

amplitude 6.00 V.

1. Assemble the circuit given in Fig. 3.

Fig. 3. The load (ohmic and non-ohmic) is connected to the signal generator. The voltage sensor

measures the voltage drop across the load.

2. Use 100 Ω resistor as a load

3. On the signal generator window choose to measure the output current

4. Press the START button. After a few second, press the STOP button.

5. Create a graph Voltage versus Current.

6. Find a slope of this linear function. The slope is the resistance of the resistor. Compare this

resistance with the measurement done by DMM.

7. Change the 100 Ω resistor for a small 6 V light bulb and repeat the described procedure. Find

the resistance of the bulb for three different values of current.

8. Repeat the above procedure for high frequency of generator (60 Hz) and low frequency

generator (0.01 Hz). Explain the difference in the graphs of voltage versus current.

25

EXPERIMENT 7

CHARGING AND DISCHARGING A CAPACITOR

Objective

To investigate the properties of an RC circuit. To analyze the charging and discharging of two

capacitors (separately, in series, and in parallel)

Apparatus

Two capacitors:330 μF and 470 μF (NOTE: The stated value of a capacitor may vary by as much as

30% from the actual value), 2 voltage PASCO sensors, 10 kΩ resistor, digital multimeter (DMM),

PASCO interface 750, computer.

Background

Consider the resistor-capacitor circuit in the figure below

Fig. 1.

If the capacitor is initially uncharged, and the switch is placed in position 1, then current is able to

flow around the outer loop of the circuit, and the capacitor charges up. The rate at which the

charging occurs is not constant, but rather decreases as time passes. If the switch is connected at time

t=0, the voltage across a capacitor in the process of being charging is given by

)e1(VV /t

max

where τ=RC is a “time constant”, associated with the charging process, R is resistance, C is

capacitance of capacitor, and Vmax is the maximum voltage across the capacitor when it is fully

charged.

It he capacitor is charged, and the switch is changed to position 2 (Fig. 1), then the circuit branch

containing the battery is effectively removed, and charge is free to flow through the middle branch

of the circuit. Consequently, the capacitor will discharge. The voltage across the capacitor will be /t

maxeVV

Procedure

In this experiment, you will use “Output” features of the PASCO interface unit to supply a “positive

square-wave” voltage to the resistor-capacitor circuit (Fig.2.). This simply means that a constant

voltage is turned on and off at regular time interval. Effectively, the emulates

26

the effect the changing the switch in the circuit in Fig. 1 at regular interval. The voltage

Fig. 2.

sensors will be used to measure the voltage across the capacitors and resistor as a function of time.

Using the appropriate fit function for voltage on the capacitor (inverse exponential function) you will

be able to determine the time constant of the RC circuit. Using the known value of resistor, you can

calculate the capacitance of the capacitor. This calculated value of C can be compared to the stated

value of C for the capacitor.

1. Using DMM measure the resistance of resistor R.

2. Connect the PASCO interface unit to the computer.

3. Assemble the RC circuit (Fig. 2.). Keep in mind that the capacitors have + and – polarities.

4. Connect the voltage sensors to the capacitor and resistor.

5. Turn on the interface

6. Turn on the computer

7. Select “Open Activity”

8. Set the “Signal Generator” to 1 V positive square wave and frequency 0.010-0.015 Hz

9. To start acquiring data, click “Start” the start button.

10. Collect voltage on the capacitor versus time data and then click “Stop”

For charging and discharging the capacitor find 1/τ and τ by choosing inverse exponential

function as a fit function (exponent of the inverse fit function is 1/τ). Calculate the

capacitance of capacitor.

11. Use Smart Tool to find 0.63Vmax and 0.37Vmax voltage for discharging and charging the

capacitor to find time constant τ.

12. Repeat procedure described in 9,10 for the second capacitor and for two capacitors connected

in series and in parallel

13. Fill the tables of experimental data

Table 1. Time constants of RC circuits

C1 C2 C1 and C2 in series C1 and C2 in parallel

Method used τ (s) τ (s) τ (s) τ (s)

Fit function (charging capacitor)

Fit function (discharging capacitor)

Smart Tool (charging capacitor)

Smart Tool (discharging capacitor)

27

Table 2. Capacitance of capacitors C1, C2 and their connection in series and parallel

C1 C2 C1 and C2 in series C1 and C2 in parallel

Units

Fit function (charging a capacitor)

Fit function (discharging a capacitor)

Smart Tool (charging capacitor)

Smart Tool (discharging capacitor)

14. Measure the capacitance of two capacitors using DMM. Find the percent discrepancy

between capacitances found by using “Fit Function” method (for charging the capacitor), nominal

values, and measuring using DMM.

15. Using data for “Fit Function Method”(discharging the capacitor) calculate total capacitance

of capacitors connected in series and in parallel:

Cparallel=C1+C2

Cseries=C1C2/(C1+C2)

and compare these capacitances with the experimental data found by the same method .

16. For C1 capacitor make a graph of output voltage (Voutput) and voltages on the capacitor, VC1,

and the resistor, VR, as a function of time. Arbitrary choose two times (first for charging and

second for discharging capacitor) and using “Smart Tool” find voltage on the capacitor and

the resistor. Confirm the second Kirchhoff’s Law (Voutput=VC1+VR).

28

EXPERIMENT 8

MAGNETIC FIELD OF THE EARTH

OBJECTIVE

To determine the strength of the earth's magnetic field in the lab room by tangent galvanometer and

by Hall’s sensor.

Apparatus

Tangent galvanometer and dipping needle (Fig. 1), DC Power supply, Vernier caliper, leads,

magnetic compass, magnetic field sensor, rotary motion sensor, WorkShop 750 interface, computer.

Fig. 1. Tangent galvanometer and dipping needle

Background

The Earth's magnetic field resembles that of a huge bar magnet. Navigators have used the Earth's

field for centuries. Compass needles are light bar magnets which align themselves with the Earth's

field when they are free to rotate. The Earth's field varies from place to place and therefore must be

determined experimentally. In general, the magnetic field lines enter the Earth's surface at an angle

and so can be resolved into horizontal and vertical components, as shown in the following figure

(Fig. 2) drawn in a vertical plane. The dip angle () can be measured with a compass in such a

vertical plane.

29

Fig. 2. Dip Angle

The instrument used in this experiment is a tangent galvanometer that consists of circular coils of

wire oriented in a vertical plane that produce a horizontal magnetic field, Bc, at its center of

magnitude (square coils)

D

iNB o

c

where 0 is the permeability of free space (0=4π·10-7

Tm/A), N is the number of turns of wire, i is

the current in the wires, D is the diameter of the coils. If the coils of the galvanometer are oriented so

that the field due to the coils Bc is perpendicular to the horizontal component of the earth's magnetic

field, Bex, the net field B is the vector sum of the two (Fig. 3)

Fig. 3. Set up for measuring of horizontal component of magnetic field of the earth

30

Because a compass aligns itself with the lines of force of the magnetic field within which it is

placed, a compass can be used to find the angle between Bex and B. If the compass is first aligned

with the magnetic field Bex and current is then supplied to the coils, the compass needle will undergo

an angular deflection. This angular deflection is The horizontal component of the earth's field can

now be found knowing the field due to the coils and the direction of the net magnetic field relative to

the direction of the earth's field

DBe

Ni

Be

B

x

o

x

c tan (1)

One can see that tanθ is a linear function of current.

PASCO’s Magnetic Field Sensor is sensitive enough to detect the sensor’s direction change in the

earth’s magnetic field.

Procedure

Part A. Determination of the earth magnetic field using tangent galvanometer

1. Set up magnetic compass on the stand inside the center of the multiloop coil. Turn the

multiloop coil (tangent galvanometer) on the table until its plane is parallel to the earth’s

magnetic field (horizontal component) evident by the direction of the magnetic compass

needle. Tape the corner of the tangent galvanometer on the table so that its direction will not

change during your measurements. Make electric connections as shown in Fig. 1.

2. Throw the reversing switch on one side and adjust the power supply for the current of 0.10

A, take the compass angle reading and note if it is a left or a right deflection. Reverse the

switch and take another reading. Repeat for nine other values of current in the increment of

0.1 A ending with 1.0 A. Find the average of each set of deflections. Fill the following table

I (A) θ (deg) right deflection θ (deg) left deflection θ (deg) av tanθ

0 0 0 0 0

0.10

0.20

1.0

3. Make a graph tanθ versus current in the multiloop. Slope of this graph is (Equation 1)

DBe

Nslope

x

o

Measure the diameter of the coils. Find the horizontal component of magnetic field.

31

4. Using the dip angle apparatus for your table’s location, determine the dip angle

(Fig. 2). Use the horizontal component, Bex with the dip angle to determine the earth

magnetic field, Be (Be=Bex/cosθ)

The magnetic field of the earth in this lab room is about 40 μT.

Part B. Determination of the earth magnetic field using PASCO’s magnetic field sensor

1. Assemble Magnetic Field Sensor (MFS) to the Rotary Motion Sensor (RMS).

2. Connect MFS and RMS to WorkShop 750 interface.

3. Choose the 100x range for MFS

4. Put the MFS in the position of axial direction

5. Rotate RMS 360 degrees in the horizontal plane.

6. Make a graph of horizontal component of magnetic field of the earth versus angular position

of RMS. This is a harmonic function (sinusoidal function). Choose sinusoidal function as a

fit function and find its amplitude. It is Bex. Calculate Be=Bex/cosθ.

7. Compare this magnetic field with the earth magnetic field found by using tangent

galvanometer.

32

EXPERIMENT 9

THE MAGNETIC FIELD IN A SOLENOID

Objective

To determine the magnetic field strength in solenoid using a current balance and the Hall’s sensor.

Apparatus

Air core solenoid and current balance (Fig. 1), two DC current power supplies, mass balance, thread,

wires, WorkShop 750 interface, magnetic field sensor, and computer.

Fig. 1. Solenoid and current balance

Background

Solenoid is constructed by winding wire in a helical coil around a cylinder. The windings are very

close to each other and usually consist of many layers. When a current is carried by the wire, a

magnetic field is generated by solenoid. If the length of a solenoid is large compared with its

diameter, the magnetic field created inside the solenoid is uniform and parallel to the axis. The

magnetic field outside the solenoid is very small and decays quickly with the distance. The

magnitude of the magnetic field in the center of the solenoid is proportional to the number of turns

per unit length of the solenoid, n=N/L, and the magnitude of the current, I

nIB 0 (1)

where μ0=4π•10-7

Weber/A m, N number of turns, and L is the length. For solenoids that are not very

long, more accurate formulae should be used

22

0

4RL

NIB

(2)

where R is the radius of the solenoid. When a wire carrying a current is placed in a uniform magnetic

field a force is exerted on the wire. This force depends on the magnitude of the current, the length of

the wire, d, and on the relative orientation of the wire with regard to the magnitude magnetic field

33

F=IdxB

When the wire is perpendicular to magnetic field the last equation can be written in the form

Id

FB

Vectors F, B, and d are perpendicular to each other.

Magnetic field strength can be measured by using current balance and Hall’s sensor. A Hall device

consists of a piece of semiconductor material which is doped to create a considerable amount of free

charge carriers. The device is placed in a magnetic field, so the direction of electrical current and

magnetic field are perpendicular to each other. A charged particle moving through a magnetic field

experiences a force described by the Lorentz equation: F=qvxB, where F is the force, q is the

charge, v is the velocity, and B is the magnetic field. When the electrons move in the sensor, a

potential difference has developed across the strip, and by measuring this voltage, the strength of the

magnetic field may be determined.

Procedure

The air core solenoid is made of enameled copper wire wound on a phenolic core. The ends of the

wire are brought out to the brass binding posts on the rigid end plates. There are five layers of turns.

Measure and record the length of the loop current perpendicular to the field, d (Fig. 1), the length, L,

and radius, R, of solenoid. Count the number of turns, N, and calculate number of turns per unit

length, n.

1. Calculate the magnetic field in the center of the solenoid for 3A current using equations (1)

and (2).

2. Connect the current loop, solenoid, variable rheostats, and ammeters to the source of current

in two-branch circuit (Fig. 2)

34

Fig. 2. A schematic of the experimental set up

With no current flowing in the apparatus, move the end of the current loop into the center of

the coil and make sure that it can freely oscillate on support brackets attached to the end plate

of the solenoid. If necessary level the current loop placing masking tape underneath if needed

for horizontal balance. Make sure that the loop is in horizontal position before turning the

current on.

3. Measure the length and the mass of the thread and find its linear mass density, μ (mass per

unit length).

4. With a current 3A establish a magnetic field in the center of the solenoid. Maintain this

current in the solenoid for all the trails. Using the current balance loop pass a current 3.0 A

through it. The end of the beam balance (out of the solenoid) should move upward. If it

moved downward you will have to reverse the direction of the current in either the solenoid

or the loop by switching the wires. When the loop current is turned on the balance is changed

because the magnetic force is exerted on the loop. Note that only the part of the loop, which

is perpendicular to the magnetic field, produces this force. The current flowing through the

two conductive strips parallel to the symmetry axis of the solenoid does not interact with the

magnetic field and can be ignored. The plastic beam should have the length adjusted in such

a way that the end of the beam with the conductive strip is right in the center of the solenoid.

But usually it is not the case.

5. Balance the loop with a long piece of thread. Use scissors or fingernail clippers to shorten the

thread, the amount needed. Measure the length of this thread, l. The torque of the pull of the

earth on the thread (weight of the thread) is equal to the magnitude of the torque of the

magnetic force on the opposite end

BidL2=mgL1 (3)

where m is the mass of the thread (m= μl), L1 and L2 are lever arms of force of gravity and

magnetic force (Fig. 3 ). Measure L1 and L2.

35

Fig. 3. Current balance in equilibrium (BidL2=mgL1)

6. Repeat for two more trails with the increment through the loop approximately of 2.0 A, 1.00

A (it is easier to change current than the length of the thread; you do not need to keep

increment of the current change constant, equal 1.0 A). The descending order of current is

preferred to conserve the thread used. The thread is cut shorter for each mass required for

balance. Fill in the table provided

i(A) l x10-2

(m) mx10-3

(kg)

3.0

2.0

1.0

0 0 0

7. Make a graph of the mass of the thread versus current in the loop (Do not forget to include

the point for which i=0). It is a linear function. Find the slope of the graph and magnetic field

strength in the solenoid (equation 3)

2

1

dL

gLslopeB

8. Use a WorkShop interface 750 and Magnetic Field Sensor (which uses Hall’s effect) to

measure magnetic field of the solenoid. Insert this sensor inside the solenoid and find its

position for the maximum axial magnetic field. Record this magnetic field.

9. Present your calculations and measurements of the magnetic field in the solenoid in the table

form (1 T=10000 Gauss).

B (Gauss) B (Gauss)

Calculations Measurement

Equation 1 Equation 2 Current Balance Hall’s Sensor

36

EXPERIMENT 10

ALTERNATING CURRENT CIRCUIT

Objective

To measure the reactance of a capacitor and inductor over a range of frequencies. To measure a

capacitance of a capacitor and inductance of an inductor. To predict and measure resonance

frequency of RLC circuit.

Apparatus

PASCO AC power supply, two voltage sensors, 47μF capacitor, DMM, wire coil inductor,

computer.

Background

The reactance of a capacitor, XC, and inductor, XL, are functions of frequency f:

1

2

2

C

C

C

L

L

L

VX

I fC

VX fL

I

where C is capacitance of a capacitor, L is inductance of an inductor, VC, VL, IC, IL are amplitudes of

voltages and currents on the capacitor and the inductor. For RLC circuit (R, L, and C are connected

in series) (Fig. 1), IC=IL=I and impedance Z is equal:

2 2( )L C

VZ R X X

I

where V is amplitude of generator voltage (output voltage), R is the resistance of the circuit.

Fig. 1. RLC circuit

Phase shift between the voltage at RLC circuit and current is

)(R

XXtan CL1

37

At the resonance frequency 1

2f

LC the current is maximum (I=Imax), impedance is minimum

(Z=Zmin), the following equations are valid

L C

L C

X X

V V

Z R

and phase shift, , between current and voltage of the generator equals zero.

PROCEDURE:

1. Set up a circuit as shown in the Fig. 2 with the 47μF capacitor and inductor connected

to the AC power supply (PASCO interface box). Connect voltage sensors with the

capacitor and inductor.

Fig. 2

2. Set the “Signal Generator” to 0.1 volt and frequency of 200 Hz

3. In the Signal Generator window, click on “measurements” and choose “output

current”

4. Under “displays” choose “scope” and select “output current”

5. You should now have the oscilloscope display on the screen along with the signal

generator window.

6. With the signal generator “on”, (click “auto” off first) click “start”, you should see a

sine wave of the supply current which is going through the circuit.

7. You will have to adjust voltage and time scales on the Scope display

8. Record the peaks of current level (current must be less than 350 mA), voltages on

the capacitor, inductor, and output voltage as the frequency is stepped from 200 Hz to

1200 Hz (100 Hz is a step increment). Use the “Smart tool” for measurements.

38

9. Use the following table to keep track of the data

f I VC VL XC XL Z

Hz A V V rad

200

300

400

1200

10. Make a plot of XC, XL, and Z versus f on the same graph.

11. For the graph XC versus f choose an inverse function as a fit function (y=a/x) and find

the capacitance of capacitor Cexp=1/(2a). Compare this capacitance with the nominal

capacitance 47μF and measurement done by DMM.

12. For the graph XL versus f choose a proportional function as a fit function (y=ax) and

find the inductance of the inductor Lexp=a/(2).

13. For the graph Z versus f choose 5.02 ])/([ xcbxay as a fit function.

14. Calculate the resonance frequency

exp exp

1

2calf

L C

15. Find the resonance frequency by finding the interception of the graphs XC and XL

versus f (at resonance XC = XL).

16. Find the resonance frequency by finding the abscissa of the minimum of the

impedance. Find the resistance R=Zmin. The circuit resistance could be due to several

factors such as wiring resistance, contact resistance, and power supply resistance. The

first two possibilities can be measured with an ohmmeter. (Do not try to measure

the power supply resistance this way, it will damage the instrument.)

17. Make a graph of current versus frequency and find the resonance frequency. For this

graph choose 5.02 ))/([ xcbxaAy as a fit function.

18. Make a graph of phase shift versus frequency and find resonance frequency

( )a

x/cbx(tany 1

is a fit function).

19. Make a table of the resonance frequencies found in paragraphs 14, 15, 16, 17and 18.

20. Make phaser diagrams for 400Hz frequency. Find V and from these diagrams and

compare with your experimental data.