Lab Book.docx · Web viewThe electrical potential energy of one coulomb of test charge at a certain location in an electric field is defined as the potential energy per coulomb, or

PHYSICS LAB NOTES

FOR

ELECTRICITY, MAGNETISM

AND

GEOMETRIC OPTICS

EXPERIMENTS

PHYSICS 38

Los Angeles Harbor College

TABLE OF CONTENTS

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1. Mapping the Electric Field .... 5 2. The Oscilloscope .............. 9 3. Ohm's Law ..................... 15 4. Joule Heat .................... 19 5. Resistance and Resistivity .... 23 6. The Wheatstone Bridge ......... 25 7. The RC Time Constant .......... 29 8. Series and Parallel Circuits .. 33 9. Circuit Analysis .............. 3910. Mapping The Magnetic Field .... 4311. The Tangent Galvanometer ...... 4912. The Ballistic Galvanometer .... 5313. RL and RC Circuits ............ 5714. The RLC Circuit ............... 6115. The Visible Spectrum .......... 6516. Reflection and Refraction ..... 6917. The Thin Lens ................. 75

The Statistics of MeasurementThe Least-Squares Fit to Data

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1. MAPPING THE ELECTRIC FIELDPURPOSE: To study the nature of electric fields by:

a) plotting equipotential points.b) sketching equipotential lines and electric field

lines.c) calculating the field strength near charged surfaces.d) calculating the charge on a dipole.

INTRODUCTION:

The visualization of an electric field by the use of lines of force is a useful concept introduced by Michael Faraday. The electric field lines indicate the direction and magnitude of electrical forces acting on a positive electrical test charge placed in this region of space. The electric field strength can be represented by a vector tangent to the electric field lines. The work done to bring a positive test charge from infinity to any point in an electric field is stored as electrical potential energy of the test charge at that location.

The electrical potential energy of one coulomb of test charge at a certain location in an electric field is defined as the potential energy per coulomb, or voltage at that point. Points of equal voltage can be readily determined with a voltmeter. These points can be joined to produce equipotential lines. A test charge placed at any point along an equipotential line will have the same electric potential or voltage. Electric field lines can be determined by tracing the pathway of a hypothetical positive test charge as it is pushed by the repulsive electrical force of the positive electrode, and pulled by the attractive electrical force of the negative electrode. The electric field at any point along an electric field line is the vector sum of electrical forces exerted on a positive test charge. Electric field lines are perpendicular to equipotential lines. In this experiment the electric field between three different sets of electrodes on resistive paper will be examined.

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APPARATUS:

Banana wires (4)Paper, carbonPaperMultimeter, Extech (voltmeter)Alligator clips (2)Battery, 1.5-voltRulerFrench curvePencils, colorConducting leads, thin (2)Metal push pins (2)Plotting boardResistive paper

PROCEDURE:

1. Set up the apparatus as shown in Figure 1.

Voltmeter set to 1.5 volts a maximum of 1000 V

Alligator clips

White paper

Carbon paper

Resistive paper + -

Push pins

Fig. 1.

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2. Set the multimeter to function as a voltmeter by setting the dial to 1000 V _ _ _ (at the upper-left), plugging in a black wire into the COM port and a red wire into the extreme-right port. Press lightly across the electrode paper with the positive wire from the voltmeter to locate several points, one to two centimeters apart, at 0.30 volts. When you join all 0.30-volt points, you will have an equipotential line of 0.3 volts. Make a slight impression with the end of the positive wire about every centimeter. Avoid punching holes in the resistive paper.

3. Repeat step two for 0.60, 0.90 and 1.20 volts.

4. Trace the outline of the silver-painted shapes onto the white paper. Mark these electrodes as plus and minus to indicate their polarity. Use a colored pencil to connect equipotential points of the same voltage marked on the white paper with a smooth line, using a French curve.

5. Draw some electric field lines (about 12 of them) using a French curve and using a different colored pencil. Note that these lines have to be at right angles to the equipotential lines. Indicate the direction of the electric field lines with arrows. Shown below are representative electric field lines emerging from the positive electrode and terminating on the surface of the negative electrode in the dipole configuration:

6. Repeat the above procedure for two other electrode configurations: Parallel lines (plates) and a sharp point facing a long line.

7. In the presence of a strong electric field, the voltage change ΔV is large when you move a short distance s perpendicularly from one equipotential line to another. Calculate the electric

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field strength, E = V/s for the regions shown in Table 2 and Table 3.

8. Calculate the charge in units of coulombs on the positive electrode of a dipole system, calculated approximately as q = Vr/(9.0 × 109).

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2. THE OSCILLOSCOPEPURPOSE: To learn how to measure the frequency and voltage of a rapidly-changing voltage source.

APPARATUS:

Tuning-fork strikerTuning fork, 480-Hz or 512-HzBanana wiresAudio generator, RCA WA-504BPower stripElectronic timerSingle-pole, single-throw switchBNC-to-banana adapterBattery, 1.5-voltMicrophone, green with double banana plugLoudspeaker, PASCO or THORNTONOscilloscope

INTRODUCTION:

A 1.5-volt battery produces a voltage between its terminals that is constant over time, making it a source of DC (direct- current) power. While DC power is convenient for a few devices such as flashlights, much of our modern electronics uses AC (alternating-current) power. A graph of the voltage between the terminals of a household power plug would look like this:

Voltage(volts)

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P time (s) 1 2 3 60 60 60

The shape of this curve is called a sine wave and shows that the voltage changes very rapidly, reaching a maximum voltage ∆Vmax = +156 volts. The wave repeats itself every 1/60 seconds, and this length of time is called the period, P. In one second, the curve repeats itself 60 times, and this is called the frequency, f = 60 cycles per second. The unit cycles per second (abbreviated cps) is also called hertz (abbreviated Hz).

These voltage oscillations are much too rapid to follow on a voltmeter. Instead, an oscilloscope can be used to picture repetitive voltage changes on a screen.

PROCEDURE:

1. Place the oscilloscope on your table where it can be conveniently viewed. The oscilloscope can be propped up on its handle, by pressing the sides of the handle into the oscilloscope and rotating the handle downwards. Make sure that the handle clicks securely into a locked position, and keep the oscilloscope away from the edge of the table.

2. Examine the small BNC-to-banana adapter, which has a hollow shiny cylinder at one end and black and red plugs for wires at the other end. The black plug has a tab next to it labeled GND for ‘ground’, indicating that this is the negative terminal. Fit the two notches at the end of the aluminum cylinder through the two tabs on the INPUT on the lower-left of the oscilloscope, and rotate the aluminum cylinder 90° clockwise to lock it into position. Set the switch above the INPUT to GND. Turn the POWER ON dial at the upper-right of the screen until its indicator line is vertical. A horizontal green line, called the trace, should appear on the screen within fifteen seconds.

OSCILLOSCOPE

Screen 4 6 5 8 divisions

10 divisions

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1 3 2 + -

3. Connect the oscilloscope in series to the battery and the switch, as shown in the diagram. Be sure that the negative terminal of the battery is connected to the ground (black) terminal on the oscilloscope, and that the switch is open. The INPUT that the circuit is connected to is also labeled CH1. Notice that there are two other inputs, labeled CH2 and CH3, but we will not be using these channels.

Rotate the POWER ON dial, and notice that it changes the intensity of the trace. Set the trace so that it is easily visible, but not so bright that it becomes thick. Rotate the FOCUS dial until the trace is sharp and narrow. Rotate the SCALE ILLUM dial all the way clockwise. This illuminates the divisions in the screen, to permit measurements of the trace in a darkened room. Rotate this dial fully counter-clockwise to turn this illumination off.

4. Notice that the oscilloscope controls are grouped according to function, shown on the previous page:

- Section 1 controls channel 1, where the CH1 INPUT is.- Section 2 controls channel 2, which we will not be using.- Section 3 controls which channels are displayed on the screen,

and will be kept at CH1 throughout the lab.- Section 4 controls the appearance of the screen. You have

already adjusted these three dials.- Section 5 controls the trigger which flips the trace back to the

left-hand side of the screen. You may need to rotate the A TRIG LEVEL dial later in the lab, to stop the trace from flickering.

- Section 6 controls how rapidly the trace sweeps across the screen.

5. The vertical MODE switch in section 3 should be all the way up, on CH1, which indicates that the screen will display the voltages that are across the CH1 input. The five horizontal switches in sections 5 and 6 should be set all the way to the left.

The VOLTS/DIV dial in section 1 contains a small inner dial surrounded by a large outer dial. Rotate the inner dial fully clockwise, and push it in if it has been pulled out. Do the same for

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the POSITION and VAR HOLD OFF inner dials in section 6. This calibrates each outer dial.

6. Examine section 6. The TIME/DIV control has an inner dial (which controls channel 2) and an outer dial (which controls channel 1). Set the outer dial fully counter-clockwise to .5 seconds. You may need to reduce the trace’s intensity. Time how long it takes for the trace to sweep across the 10 divisions in the screen, divide by 10, and record this result to one hundredth of a second on page 5. This time should be close to 0.50 seconds. Rotate the dial clockwise, and notice that the sweep rate speeds up. The trace becomes difficult to follow when the dial is set to sweep at 20 ms (milliseconds, or thousandths of a second) per division. The sweep rate can be set to µs (microseconds, or millionths of a second) per division and ns (nanoseconds, or billionths of a second) per division. Set this dial back to .5 s.

7. In section 1, set the VARIABLE VOLTS/DIV to 1 (on the left side), and set the switch above the INPUT to DC. Push the POSITION dial in if it is out, and rotate it until the trace is exactly on the center horizontal axis. Close the switch connected to the 1.5-volt battery. Measure how many divisions (to the nearest tenth of a division) the trace has risen vertically, multiply this number by the number of volts per division (which equals 1, in this case) and write the result on page 5. This should be close to 1.5 volts, the voltage of the battery.

Set the VARIABLE VOLTS/DIV dial to 0.5 volts/division, and again calculate the voltage by multiplying the number of divisions the trace rises by the volts/division. Repeat with the dial set to 2 volts/division. All three numbers should be close to 1.5 volts and should include the appropriate units.

8. Remove the two wires from the battery, and connect them to the audio generator, making sure that the negative oscilloscope input is connected to the negative audio generator output. Connect the loudspeaker to the audio generator as well.

Set the audio generator LEVEL and RANGE output dials fully clockwise (for maximum power), press the gray switch in on the left to get a sine wave, set FREQUENCY to span between 200 Hz and 2K (2 kilohertz), and set the large dial to 20. Turn on the audio generator. The generator creates a sine wave of 200 hertz, which is audible as a low hum. Set the TIME/DIV on the oscilloscope to 1 ms,

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and rotate the VARIABLE VOLTS/DIV dial until the waves fill most of the screen. Slowly rotate the large dial on the audio generator. You should hear the audio tone rise in pitch, and see the waves in the oscilloscope screen crowd closer together. Set the audio generator to 2000 hertz, by keeping the FREQUENCY dial set to span between 200 hertz and 2 kilohertz, and setting the large dial to 200.

9. Adjust the TIME/DIV control on the oscilloscope so that one wave (from 0 volts to a maximum, back to zero, then down to a minimum and back up to 0 volts) is fully seen on the screen. You may have to rotate the A TRIG LEVEL dial to create a single, stable wave. Rotate the outer POSITION dial at the left of the TIME/DIV dial to place the wave in a convenient location.

Write down the number of divisions that this one wave crosses horizontally, to a tenth of a division accuracy. Write down the amount of time the trace takes to sweep across one division (the TIME/DIV dial points to this number) in seconds. Remember that if the unit of the dial setting is ‘ms’ you must multiply the number by 0.001, and if the unit of the dial setting is ‘µs’ you must multiply the number by 0.000001. Multiply the two numbers you have written down to get the period P in seconds. Calculate the frequency f = 1/P, and find the percent difference compared to the 2000 hertz setting on the audio generator dial.

10. Press the switch on the audio generator to get a square wave. Change the frequency, and notice that the sound has a high-pitched screech (called the overtones) compared to the pure sound of the sine wave.

11. Turn off the audio generator and unplug all devices from the oscilloscope. Plug the green microphone into the oscilloscope, and flip the microphone’s switch up to turn it on. Strike the top of the tuning fork with its striker (don’t hit the tuning fork against anything else) and hold it close to the microphone. Repeat activity 9, using the tuning fork instead of the audio generator. The nominal (theoretical) frequency of the tuning fork is printed on its base.

Try humming into the microphone with the same note as the tuning fork. Notice that your voice gives a differently-shaped curve, which is what gives your voice its unique qualities. Talk into the microphone, to see what your voice looks like.

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3. OHM'S LAWPURPOSE:

a) To learn about simple DC circuit elements.b) To verify Ohm's Law (V = IR).c) To calculate the power consumption by a resistor in a series circuit.

INTRODUCTION:

Ohm's law states the relation between the electrical potential V (in volts), the current I (in amperes) and the resistance R (in ohms, symbolized by the Greek letter Ω) in a circuit, as

VV = IR, so R = .

I

Circuit Elements: R

I

V

V = electrical potential, provided by a DC (direct current) power supply.

I = current flow of charges. The direction of current flow is denoted by an arrow and is the direction a positive test charge would follow.

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R = resistance or load. A resistor resists current flow, and consumes energy or dissipates power. A rheostat is a resistor with a resistance that can be varied.

P = electrical power measured in watts. The power P = IV.

APPARATUS:V

Rheostat, ~2800-ohmGraph paper Resistance Multimeter, BK (ammeter) BoxMultimeter, Extech (voltmeter)Spade lugs (3)Battery, 1.5-volt AStraightedge RheostatBanana wires (7)Decade resistance box

1.5 volts

PROCEDURE: V

(volts)

Slope = R

I (amperes)

1. Use wires and spade lugs to connect the negative (black) battery terminal to the lower-left port of the rheostat (the device with the long black wire coiled around a cylinder) and the positive (red) battery terminal to the lower-right port.

2. Connect a red wire to the upper-right port of the rheostat. Slide the tap to the left, and the voltage on this wire becomes small; slide the tap to the right, and the voltage becomes large. This device now functions as a variable voltage source.

3. Set the resistance decade box to 80 Ω. Connect the red wire

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from the variable voltage source to a port on the bottom of the resistance box. Set the BK multimeter dial to 200m A_ _ _ on the right-hand side of the dial, turn it on, run a wire from the other port on the bottom of the resistance box to the mA port of the BK multimeter, and run another wire from the COM port of the BK multimeter to the negative battery terminal. This multimeter now functions as an ammeter, measuring the current I through the resistance box, in milliamperes.

4. Set the Extech multimeter's dial to 2V _ _ _ . Run two wires from the COM and V ports (on the right-hand side) to the two ports on the bottom of the resistance box. This multimeter now functions as a voltmeter, measuring the voltage across the resistance box, in volts.

5. Slide the tap of the variable voltage source all the way to the right, to create the largest voltage available. Record the electric potential V (voltmeter reading) and the current I (ammeter reading). Calculate the power P being dissipated by the resistance box.

6. Decrease V by approximately 0.2 volts. Record V and I. Repeat this step for six more sets of readings.

7. Plot V vs. I as shown on page 2, and compute R as the slope of the line.

8. Compare this experimental value to the known value of R set on the resistance box, by calculating the percent difference.

9. Repeat steps 5 - 8 for R set at 160 Ω on the resistance box.

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4. JOULE HEATPURPOSE:

a) To determine the electrical power consumption of a heating coil.b) To determine the ratio between joules and calories, which are two units of energy.

INTRODUCTION:

When an electrical potential difference exists between the two ends of a resistance coil, current will flow through the coil. The electrical energy provided by the current flowing through the coil can be used to add thermal energy Q to water and to the calorimeter holding the water.

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I Heating coilof calorimeter

Electrical energy provided = Q added to water + calorimeter Power × time = Q added to water + cup + lid

(IV)t = mwcw∆T + mccc∆T+ mlcl∆TElectrical energy in joules = Thermal energy in calories

APPARATUS:

ThermometerCalorimeterHeating coilBalance, electronic

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Multimeter, BK (ammeter)Multimeter, Extech (voltmeter)Power stripTimer, electronicSpade lugsSwitch, single-pole, single-throwGlycerinBanana wires (6)Battery chargerIcePROCEDURE:

1. Determine the mass of the inner calorimeter cup mc. Record the mass. Fill the cup 3/4 full of water and add ice to cool it to 10 degrees Celsius below room temperature. Wait for all the ice to melt, then weigh the cup and water. Determine the mass of water mw and record the results in Table 1.

V

A

2. Set up the apparatus as shown. Use the BK multimeter as the ammeter, with the dial set at 20A _ _ _ , the left-side wire to the 20A port and the right-side wire to the COM port. Use the Extech multimeter as the voltmeter, with the dial set at 200 V _ _ _ , and wires from the COM and V ports to the two terminals of the calorimeter.

3. Lubricate the stopper hole with glycerin. Insert the thermometer about half-way into the calorimeter. Mix the water with the stirrer built into the calorimeter lid assembly. Record the water temperature, Ti, to a tenth of a degree.

4. Set the battery charger to 6V. Close the switch and start the timer simultaneously.

5. Record the voltage V and the current I, initially and for every five-degree increase in temperature.

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6. Allow the water temperature to increase by ∆T = 15.0 Celsius degrees, while stirring occasionally to ensure a uniform water temperature.

7. Open the switch, and stop the timer simultaneously. Record Tf and the time t.

8. Discard the warm water in the calorimeter and repeat the procedure with the power supply set to 12 V. Remember to weigh the calorimeter cup with and without the new sample of water to ensure accuracy.

9. Calculate the instantaneous power P = IV at your four temperature readings, and average them. Calculate the electrical energy provided = average power × time, in units of joules.

10. The heat gained by the water is its mass × specific heat × temperature change, in units of calories. The specific heat of water is 1.00 cal/g·°C. The aluminum cup also gains heat, and the specific heat of aluminum is 0.22 cal/g·°C. The lid (including the heating coil and the stirrer) also gains heat, and the water equivalent (W.E.) in grams is written on the lid. These items absorb heat as if

they were water, with ml = W.E. and cl = 1.00 cal/g·°C. Calculate the heat gained.

11. The theoretical value of the mechanical equivalent of heat is 4.186 J = 1.000 calories. Calculate your values, and the percent difference when compared to the theoretical value.

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5. RESISTANCE AND RESISTIVITY

PURPOSE: a) To determine the resistance R and the resistivity ρ of a wire made of copper, and of wires made of the alloy called nickel silver (60% copper, 20% nickel and 20% zinc).

b) To observe the effects on R and ρ of the physical dimensions of the wires and the material characteristics.

LINTRODUCTION:

A I dR: Ohm's law states that V = IR,

where R is the resistance. The equation R = V/I permits R to be determined by measuring VV and I.

The resistance of a metal wire is proportional to its length L and inversely proportional to its cross-sectional area A = πd2/4, with d being the diameter of the wire. These factors may be combined to give R = ρ·L/A, where ρ is a proportionality constant called the resistivity.

ρ: Resistivity is a property which depends only on the material composition of a wire.

For copper, ρ = 1.72 × 10-8 Ω·m

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For nickel silver, ρ = 3.30 × 10-7 Ω·m

APPARATUS:

Multimeter, BK (ammeter)Multimeter, Extech (voltmeter)Alligator clips (2)Switch, single-pole, single-throwBattery, 1.5-voltRulerWire gaugeBanana wires (6)Resistance spool set

PROCEDURE:

1. Examine the circular wire gauge. One side lists the American Standard Wire gauge (or gage) number for each opening at the edge of the disk, from 0 to 30. For example, a bare, uninsulated wire of gauge 0 would just fit through the largest opening on the edge of the disk. The flip side of the disk indicates that this opening is 0.325 inches wide. Compare this to an inch on the wooden ruler, to convince yourself that these lengths are indeed in units of inches.

The wires used in this experiment are coated with insulation, so you can’t measure the wire diameters. However, the gauge numbers are given as either 28 or 30, so you may use this disk to look up the wire diameter. This can be converted to the metric system, as one inch equals exactly 2.54 centimeters.

V

A

2. Create this circuit, with the switch open and with spool #1 as the resistor. The BK multimeter will be the ammeter, with its dial set to 20A A_ _ _ , and the 20A and COM ports used by the wires. The Extech multimeter will be the voltmeter, with the dial set to 2V_ _ _ , and the COM and V ports used by the wires.

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3. For resistance spool #1, write down the length of the wire and the diameter of the wire on the data sheet. Close the switch, read the voltage in volts and the current in amperes, and open the switch.

4. Calculate the resistance of the spool, from Ohm’s law. Calculate the resistivity of the material making up the wire. Calculate the percent difference between your experimental value of the resistivity and the theoretical value.

5. Repeat steps 3 and 4 for the other six spools.

6. THE WHEATSTONE BRIDGEPURPOSE: To measure the resistance of a resistor by using a

null (zeroing) method.

bINTRODUCTION:

Rs I1 Rx

G

a d

I2 RL c RR

1.5 volts

Fig. 1 Wheatstone Bridge

Suppose that four resistors are connected together as shown in Figure 1. Usually, a different amount of current will flow through each resistor. However, suppose that a very sensitive ammeter called a galvanometer is placed between points b and c, and the resistance

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of the lower-left resistor RL and the lower-right resistor RR are varied until the galvanometer reads exactly zero. The resistors Rs and Rx must both experience current I1, and resistors RL and RR must both experience current I2, because no current passes from b to c.

Therefore, ∆Vab = ∆Vac and ∆Vbd = ∆Vcd.

From Ohm's Law, I1Rs = I2RL and I1Rx = I2RR.

Rearranging, I2/I1 = Rs/RL and I2/I1 = Rx/RR.

RREquating these two equations gives Rx = Rs × .

RL

a c d

LL LR

a c d

RL RR

Fig. 2Slide-Wire Apparatus

According to this equation, an unknown resistance Rx can be accurately measured if an accurately-known standard resistance Rs is available, and if the ratio of resistances RR/RL can be determined. A uniform wire 1.000 meters long can be tapped along its length to split the wire into two resistors, as shown in Figure 2.

Let ρ represent the resistivity of the metal in the wire, and let A represent the cross-sectional area of the wire.

ρ(LR/A)Since RR = ρ(LR/A) and RL = ρ(LL/A), Rx = Rs· , giving

ρ(LL/A)

LRRx = Rs· Equation 1.

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LL

Notice that neither ρ nor A need to be known.

APPARATUS:

Wheatstone bridge, slide-wire with tapSpade lugs (7)Tapping switchRheostat, 22-ohmAlligator clips (4)Battery, 1.5-voltBanana wires (7)Decade resistance boxWire spool #5, 30-gauge, 200 cmGalvanometer, studentResistor, standard, 50 ohmResistor, composition 47 ohm

PROCEDURE:

1. Set up the apparatus as shown in Figure 1. Use the 47Ω carbon resistor (the tiny cylinder with a wire sticking out of each end) as Rx. Both keys on the top of the galvanometer can be pushed down and rotated into a depressed position. Try this, then rotate both keys into their elevated positions.

2. Place the tapping switch over the wire, press its tap gently onto the wire and watch the motion of the galvanometer needle. Slide the tap to another position and press again, to see if the deflection is smaller. Continue changing the position until the

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galvanometer deflection is very small. Be careful not to drag the tap across the wire, as you will scrape some metal from the wire and it will no longer be uniform.

3. Push down and rotate the left-side key of the galvanometer to lock it into a depressed position, to increase the sensitivity. Locate the null position on the slide wire, which should be only a few millimeters from the position you have already located. Rotate and elevate the galvanometer key to its less-sensitive position.

4. Record LL and LR to the nearest millimeter. Calculate Rx from Equation 1, and record the nominal resistance. The nominal (named) resistance is the resistance assigned to it by the manufacturer. For the carbon resistor, the band colors yellow-violet-black-silver represent 4-7-0-10%, indicating a resistance of 47 × 100 = 47Ω with a typical scatter of 10% around this value. Calculate the percent difference, assuming the nominal value is the theoretical value.

5. Repeat steps 2 - 4 with the other resistors. Set the decade box at 70Ω. The nominal value of the spool can be calculated from R = ρ L/A, with ρ = 3.30 × 10-7 Ω·m, A = 5.067 × 10-8 m2 for gauge #30 wire, and L = 2.000 meters.

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7. THE RC TIME CONSTANTPURPOSE: To determine the time constant τ of an RC circuit.

INTRODUCTION:

The time constant τ = RC determines the rate at which a capacitor with capacitance C will be charged or discharged through a resistor of resistance R. The circuit diagram in Figure 1 shows a fully-discharged capacitor (Q = 0 on both plates) connected to an open switch. When the switch is closed, the voltage across the resistor will initially be large as current flows through the circuit, then gradually drops to zero as the capacitor plates become fully charged. The voltage across the resistor should decrease according to VR = V0e-t/RC = V0ce-t/τ

(Figure 2) as the charge Q on the capacitance plate builds up to a maximum value.

Fig. 1The RC Circuit

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VR Q

V0 Qmax

0.632Qmax 0.368V0

0 τ t(s) 0 τ t(s)

VR = V0e-t/RC Q = Qmax(1 - e-t/RC)

Fig. 2 Charging a Capacitor

Q0

Fig. 3

VR Q

V0 Q0

0.368V0 0.368Q0

0 τ t(s) 0 τ t(s)

VR = V0e-t/RC Q = Q0e-t/RC

Fig. 4 Discharging a Capacitor

A circuit with a charged capacitor as shown in Fig. 3, will behave as shown in Fig. 4 when the switch is closed. The voltage across the resistor should decrease as the capacitor discharges, according to VR = V0e-t/RC = V0e-t/.

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APPARATUS:

Banana wiresGraph paper (3 sheets)Multimeter (voltmeter)Timer, electronicAlligator clipsSwitch, double-pole, double-throwBattery, 1.5-voltStraightedgeFrench curveCapacitor, 6- or 10- microfaradResistor, carbon, 20 mega-ohm

PROCEDURE:

A. Charging a Capacitor

1. Construct the circuit shown in the middle of Figure 5. Notice that when the switch is closed onto the lower terminals, the switch acts like a 1.5-volt battery that slowly charges the capacitor. When the switch is closed onto the upper terminals, the switch acts like a wire that slowly discharges the capacitor.

Charging: Discharging: C C C

VR R VR R VR R

1.5 V

1.5 VSwitch closed onto Switch closed ontolower terminals upper terminalsconnected to the connected togetherbattery by a single wire.

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Fig. 5

2. Zero the timer, and momentarily touch the ends of a wire to the two terminals of the capacitor to make sure it is completely discharged.

3. Close the switch onto the lower terminals. A couple of seconds later, when the voltmeter has adjusted to the larger voltages (~ 1.5 volts), start the timer and at the same moment press the HOLD button on the voltmeter to keep the reading visible. Press the HOLD button again as soon as the reading is recorded. Continue taking readings every 10.0 seconds by pressing the HOLD button, then pressing it again in order to release it for the next reading.

B. Discharging a Capacitor

1. Zero the timer. Unplug both voltmeter wires from the resistor, then plug each wire back in, but on the opposite side of the resistor. This reversal permits the voltage to remain positive, while the current reverses direction.

2. Close the switch onto the upper terminals. A couple of seconds later, once the voltmeter has adjusted to the larger voltages, start the timer and at the same time press the HOLD button on the voltmeter to keep the reading visible. Press the HOLD button again as soon as the reading is recorded. Continue taking readings every 10.0 seconds by pressing the HOLD button, then pressing it again.

C. Graphs and Calculations

1. Graph VR as a function of t for both the charging and discharging circuits. Use the French curve to connect the data points as smoothly as possible. When t equals the time constant τ, VR = V0e-1 = 0.368V0, where V0 = VR at t = 0.

Calculate VR = 0.368V0, find it on the graph, and obtain a value of τ, for each graph.

2. If VR = V0e-t/τ, then ln(VR) = ln(V0) - t/τ where ln is the natural logarithm function, to the base e = 2.7183 ... A graph of

ln(VR) vs. t will have a slope m = -1/τ. Construct this graph for

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the data obtained from the discharging capacitor. Draw the best-fit straight line through these data points, measure its slope m, and use this to calculate τ = -1/m. This will be your best estimate of τ, as it uses all the data obtained. This is a semi-logarithmic graph, as one axis is a logarithm of the data.

3. The theoretical value of the time constant is τ = RC. If the voltmeter had infinite resistance, the value of R would equal the resistance of the 20 MΩ resistor. However, the voltmeter has a resistance of 10 MΩ, so R is the equivalent resistance of these two resistors in parallel. Calculate R = 1/(1/20 + 1/10) in units of MΩ, and calculate τ = RC from this nominal value of R and from the nominal value of the capacitance.

8. SERIES AND PARALLEL CIRCUITSPURPOSE:

To learn to wire simple series and parallel circuits on a breadboard and to verify the rules pertaining to these circuits.

INTRODUCTION:

Breadboards allow an experimenter to build and modify circuits very easily, without soldering. The Protoboard.10 that you will be using contains sockets that are connected together in the following arrangement:

GLOBAL SPECIALTIES proto·board.10 V1 V2 GND

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33

Each vertical line represents a set of five sockets that are permanently wired together under the breadboard. Each of the eight horizontal lines represents 25 sockets wired together. Banana wires can be attached to the three terminals in the upper-right of the breadboard, and short wires can be run from the base of these terminals to the horizontal sockets to provide sources of voltage and a ground. Short wires can then connect different sets of sockets together, with circuit devices such as resistors and integrated circuits plugged in as well.

SERIES CIRCUIT GEOMETRY AND EQUATIONS:

R1 R2 R3 Io = I1 = I2 = I3

I1 I2 I3 Vo = V1 + V2 + V3

Req = R1 + R2 + R3

Io Io Vo = Io·ReqVo

Fig. 1a

PARALLEL CIRCUIT GEOMETRY AND EQUATIONS:

Io = I1 + I2 + I3 R1

I1 Vo = V1 = V2 = V3

1 R2 Req =

34

1/R1 + 1/R2 + 1/R3 I2

Vo = Io·Req

R3

I3

Io Io Vo

Fig. 1bAPPARATUS:

Multimeter, BK (ammeter)Multimeter, Extech (voltmeter)Spade lugsSwitch, single-pole, single-throwBreadboard wiresAlligator clips (4)Battery, 1.5-voltBanana wires (7)Breadboard (proto·board.10)Resistor, composition, 270, 470, 1000 ohm

PROCEDURE:

1. Determine the nominal Resistor Color Coderesistance of each carbonresistor by using the color Black = 0code bands. For example, Brown = 1yellow-violet-brown-silver Red = 2becomes 4-7-1-10%, so the Orange = 3resistance of the resistor is Yellow = 447 × 101 = 470 ± 47 Ω. Green = 5

Blue = 6Violet = 7

Tens digit Grey = 8 White = 9

Ones digit

35

Exponent Silver 10% tolerance Gold 5% tolerance

Tolerance

2. Set the Extech multimeter to 2000Ω to function as an ohmmeter, connected to two banana wires, two alligator clips and two short breadboard wires. By inserting these two short wires into various sockets of the breadboard, convince yourself that the sockets are connected to each other as described in the Introduction. Remove the two breadboard wires, measure the resistances of the three resistors, then turn off the multimeter.

3. Unscrew the plastic insulators around the V1 and GND ports of the breadboard, insert a short breadboard wire into each exposed hole so that metal touches metal, then tighten the plastic insulators. Connect the V1 wire to a socket in the nearest row, and connect the GND wire to a socket in the row below it. Connect the battery in series with the open switch to the V1 and GND ports on the breadboard. As shown on the first page, the adjacent 24 sockets are connected to each wire, and can serve as a power source for the breadboard.

4. Set up the circuit as in Figure 2a, with the switch open and with the BK multimeter as the ammeter set at 20m A_ _ _. The resistors can be plugged directly into the breadboard sockets. Close the switch and measure the current through and the voltage across the resistors with the multimeters. Calculate R = V/I, from Ohm's law.

5. Repeat step 4 for the other seven circuits.

6. Disconnect the breadboard from the power supply, and set one of the multimeters to act as an ohmmeter. Measure the resistance of the two arrangements directly with the ohmmeter, and calculate the percent difference of these readings from the values calculated from Ohm's law.

I1 I2 A A

36

R1 R2 R3 R1 R2 R3

V V

Fig. 2a Fig. 2b

V

I3 A

R1 R2 R3 R1 R2 R3

V I0

A

Fig. 2c Fig. 2d

I1 R1

A

V V I2

A R2

Fig. 3a Fig 3b

V

37

I3 V

AR3

I0

A

Fig. 3c Fig. 3d

38

9. CIRCUIT ANALYSISPURPOSE: To analyze the current and voltage distribution in an electrical circuit.

INTRODUCTION:

The rules governing the relationship between current, voltage and resistance for series and parallel circuits are outlined below:

Kirchhoff's Kirchhoff'sLoop N Junction NRule: ∑Vi = 0, with Rule: ∑Ii = 0, with

i i

Vi +ve for a voltage rise Ii +ve for current into junctionVi -ve for a voltage drop Ii -ve for current out of junction

R1

I1

39

R1 R2 R3 R2 b

I2

R3

I3

I a I0

V0 V0

Loop Equation: Junction Equation:

Clockwise from a, At junction b,

V0 + (-V1) + (-V2) + (-V3) = 0 -I0 + I1 + I2 + I3 = 0

so V0 = IR1 + IR2 + IR3 so I0 = I1 + I2 + I3

APPARATUS:

Banana wiresMultimeterBreadboard wiresAlligator clipsSwitch, single-pole, single-throwBattery, 1.5-voltBreadboardResistors, composition 100, 330, 470, 1000 ohm

PROCEDURE:

Part A:

1. A breadboard allows an experimenter to build and modify circuits very easily, without soldering. The protoboard.10 that you will be using contains sockets that are connected together in the following arrangement:

40

GLOBAL SPECIALTIES proto·board.10 V1 V2 GND

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Each vertical line represents a set of five sockets that are permanently wired together under the breadboard. Each of the eight horizontal lines represents 25 sockets wired together. Banana wires can be attached to the three terminals in the upper-right of the breadboard, and short wires can be run from the base of these terminals to the horizontal sockets to provide a ground and two sources of voltage. Short wires can then connect different sets of sockets together, with circuit devices such as resistors and integrated circuits plugged in as well.

2. Set the multimeter to function as an ohmmeter, plug it into the V1 and GND terminals, insert the exposed end of each of two short wires through the hole at the base of the two terminals and screw them tightly in place. By inserting these two short wires into various sockets, convince yourself that the sockets are connected to each other as described in the previous paragraph.

Part B:

1. Measure R1, R2 and R3 by using the multimeter as an ohmmeter, and measure V0 by using the multimeter as a voltmeter. Record these values on the data sheet, then set up the apparatus as in the figure below.

41

V0 ~ 1.5 V R1 R2 R3 ~ 100 Ω ~ 330 Ω ~ 470 Ω

I1 I2 I3

2. Calculate the theoretical current and the voltage for each resistor by using Kirchhoff’s Rules and Ohm’s Law. Use your measured values of resistance and voltage from step 1.

3. Measure the voltage across each resistor, and the current through each resistor. Record your experimental measurements in the data table.

4. Compare the experimental and the calculated I and V values by computing the percent differences.

Part C:

1. Measure R1, R2, R3 and R4 by using the multimeter as an ohmmeter, and measure V0 by using the multimeter as a voltmeter. Then, set up the apparatus as in the figure below.

R2 ~ 330 Ω

I1 I2 I4

R1 ~ 100 Ω R2 ~ 1000 ΩI3

R3 ~ 470 Ω

42

V0 ~ 1.5 V

2. Repeat steps 2 to 4 of part B.

10. MAPPING THE MAGNETIC FIELDPURPOSE: a) To determine the shape of magnetic fields around

magnetic poles.b) To map the magnetic field around a bar magnet.c) To determine the strength of the north pole of a bar magnet.

INTRODUCTION:

1. Each pole of a bar magnet placed in an external uniform magnetic field B experiences a force F proportional to the magnitude of the field.

N F B

43

The magnetic pole strength m of a magnetic pole is the ratio between the force and the external magnetic field;

Fm = or F = mB . B

2. At moderate distances from a magnet’s north magnetic pole, the pole’s magnetic field can be approximated as pointed radially outward with a magnitude of

km·m BN = , with the constant km = 1.0 × 10-7 T·m/A. r2

3. If a pole of one magnet of magnetic pole strength m1 is placed at a moderate distance r from a pole of another magnet of magnetic pole strength m2, each pole will lie in the other pole's magnetic field, and will experience a force

km·m1·m2 F = , analogous to Coulomb's Law for electricity. r2APPARATUS:

Meter stickMasking tapePaper, 22 x 34 in.PaperPlywood board, largeMagnetic field boardProtractorMagnet, horseshoeMagnetizerMagnetic compassMagnets, bar, 2Plexiglas sheetIron filings

PROCEDURE:

44

PART A: THE SHAPE OF THE MAGNETIC FIELD

1. Use the magnetizer to refresh the magnetic fields of your horseshoe magnet and two bar magnets. Be sure to place them in the correct orientation, with the south pole on the bottom left.

2. Move your table closer to another table, and suspend the magnetic field board between them. This will minimize the magnetic deviations due to the steel beam running underneath each desk. Place two bar magnets on your magnetic field board, lined up horizontally North pole to North pole and 5 centimeters apart, as shown on the top of Figure 1.

3. Place the 8-1/2" × 11" paper on top of the Plexiglas sheet placed above the magnets, then gently sprinkle a very thin layer of iron filings over the paper. Be careful not to spray iron filings onto the board or the floor.

4. Tap the edges of the paper lightly until the field lines gradually show up, and sketch the results on the front or back of the data sheet. Try not to spill the filings, as they are difficult to clean up.

5. Repeat steps 2 to 4 with the other three arrangements shown in Figure 1. When finished, carefully return the iron filings to the container they came from. Throw the paper into the garbage when finished, without scattering stray iron filings on the floor.

S N N S

5 cm----------------------------------------------------------

S N S N

----------------------------------------------------------

N N S

45

S S N

----------------------------------------------------------

N S N

S N S

Fig. 1

PART B: MAGNETIC FIELD LINES

1. Place your magnetic field board at a location between two tables where the compass points approximately north (away from the blackboard), as shown in Figure 2. Tape a magnet to the middle of the edge of the paper with the North pole of the magnet pointing toward magnetic North, and draw the magnet’s outline on the paper.

2. Draw the line AA’, pointed towards the center of the magnet, as shown in Figure 2. Draw line BB’ about 5 cm from the top edge of the paper, parallel to line AA’. Draw line CC’ about 5 cm from the bottom edge of the paper, parallel to line AA’.

46

Magnetic North N

A A’ S

Fig. 2

3. Starting on line AA’ 10 cm away from the bar magnet, mark the position of the tip and tail of the compass needle.

4. Move the tail of the needle to coincide with the tip of the previous compass position. Continue the process in both directions until the field line terminates at a pole of the magnet or at the edge of the paper.

5. Join the dots which mark the changing positions of the compass needle tip with a smooth line. Repeat this process for positions of 20, 30, 40 and 50 cm from the magnet.

6. Repeat steps 3 to 5 along line BB’, and along line CC’.

PART C: MAGNETIC POLE STRENGTH (m)

1. At point A’ the magnetic field of the Earth dominates, and the compass points towards magnetic North. At point A the magnetic field of the bar magnet dominates, and the compass points in the opposite direction. At some point along the line AA’ the two fields cancel, as shown in Figure 3, and the compass points aimlessly when it is shaken. Use the compass to find this neutral point, and label it.

BEarthN

r

47

A θ A’

BS BN

S BS Bmagnet

Fig. 3

2. For Harbor College, the horizontal component of the Earth's magnetic field is BE = 2.46 × 10-5 teslas. The neutral point is a moderate distance from the bar magnet, so the bar magnet may be considered as a magnetic dipole, with each pole contributing a magnetic field BN and BS at the neutral point. Assuming the poles have equal magnetic pole strengths, BN·sinθ = (1/2)·BE. Calculate BN, and the magnetic pole strength m of the bar magnet.

3. When finished, move your table back into its original position, and sweep up all remaining iron filings.

48

11. THE TANGENT GALVANOMETERN

PURPOSE: To determine the horizontal component of the Earth's magnetic field.

W E

INTRODUCTION:S

BH is the horizontal component of theEarth's magnetic field. The tangent galvanometer's compass needle points in this direction when no electricityis supplied to the galvanometer, as BH B

49

shown in Figure 1. θ

BI is the magnetic field created bythe current I through the galvanometer. BIB is the vector sum of BH and BI. Fig. 1The compass needle points in thisdirection when the current I flows through the tangent galvanometer. 5 10

Turns Turns

I

BI I BI 15 Turns

Tangent Magnetic Field Binding PostsGalvanometer and Current Configuration

BI may be found from BI = NµoI/2R, where N = number of turnsµI = 4π × 10-7 T·m/AI = currentR = radius of coil

Once θ and BI are known, BH may be calculated from BH = BI/tanθ.

APPARATUS:

Banana wires (7)Bubble levelTransfer caliperRheostat, 22 ohmsMultimeterSpadelugs (6)Switch, reversingPlywood board, largeRulerBattery charger, 6-volt G ATangent galvanometer

Reversing

50

Switch

Rheostat- +

6 volts

Fig. 2

PROCEDURE:

1. Use the transfer calipers and the metric ruler to measure the diameter of the tangent galvanometer's coil to the nearest millimeter. Divide by two to get the coil radius R, and enter this value on the data sheet.

2. Set the multimeter dial to 20A on the right-hand side of the dial, and use the 20A and COM terminals. The multimeter is now an ammeter. Set the power supply to 6V.

The tangent galvanometer should be set on a plywood board between two tables, to minimize the magnetic deflection due to the metal support rod that runs along the underside of each table. Set up the apparatus as in Figure 2, using the 5-turn posts with the switch open (raised), so the current will run through N = 5 loops of wire.

3. Set the rheostat to minimum voltage (and minimum current) by sliding the contact to the negative terminal.

With the switch still open, rotate the galvanometer so the short, blue-and-silver compass needle is parallel to the loops of wire (coil), as seen from above. Place the circular bubble level on the base of the tangent galvanometer, and adjust the height of the feet until the galvanometer is level. Rotate the compass dial until the long needle (perpendicular to the compass needle) reads 0°.

4. Close the switch and deflect the compass needle clockwise 45° by adjusting the rheostat, and record the absolute value of the

51

current Icw. Close the switch on the other side and deflect the compass needle counter-clockwise 45° by adjusting the rheostat, and record the absolute value of the current Iccw. Open the switch.

5. Calculate Iaverage. Calculate BI and BH, and convert them to microteslas (1 µT = 10-6 T).

6. Repeat steps 4 and 5 for a 60° deflection.

7. Repeat steps 4 to 6 using the 10-turn posts on the tangent galvanometer, so the current will run through N = 10 loops of wire.

8. Repeat steps 4 to 6 using the left and right terminals, so N = 15. Leave the switch open when finished, and unplug the battery charger.

9. Calculate an average value of BH from these six measurements, and calculate its standard deviation and standard error, with n = 6.

10. Calculate the percent difference between your value of BH and the accepted value at the location of Los Angeles Harbor College, BH = 24.62 µT.

52

12. THE BALLISTIC GALVANOMETERINTRODUCTION:

‘Ballistics’ is the study of projectiles, which receive a sudden impulse and then move freely. A galvanometer is a device that accurately measures small amounts of electric current, by passing the current through a coil of wire close to a magnet, creating a torque (a twisting force) on the coil that is proportional to the

53

current.In the ballistic galvanometer, a small pulse of electric charge

Q passes through the circuit so quickly that the coil barely begins to move before the current returns to zero. Over the next few seconds, the coil (and the mirror attached to it) rotates because of this sudden impulse, stopping at a maximum deflection ‘x’ before returning to its original reading.

Because the discharge is so rapid, Q = mx + b, where

Q is the charge that passes through the galvanometer,m is the ballistic constant of the galvanometer, x is the maximum deflection, andb is the zero-point correction if the mirror was not centered exactly.

For a capacitor, Q = C·∆V, where

Q is the charge (in coulombs) on the positive side of the capacitor, the other side having a charge -Q,C is the capacitance (in farads) of the capacitor, and∆V is the potential difference (in volts) across the capacitor.

In this lab, you will measure a known ∆V across capacitors of known capacitance. Each resulting charge Q will be sent through the ballistic galvanometer and readings of ‘x’ taken, permitting you to calculate ‘m’ and ‘b’. The capacitance of unrated capacitors can then be determined.

APPARATUS:

Circular bubble levelGraph paperMultimeter, BK (voltmeter)Switch, Single-pole, double-throwSpade lug (2)Alligator clip (4)Battery, 1.5 voltStraightedge

54

Banana wires (10)Decade capacitor box, brown (CDE)Capacitor labeled ‘A’, ~0.5 mFCapacitor labeled ‘B’, ~1.0 mFBallistic galvanometer

PROCEDURE:

1. Carefully carry the ballistic galvanometer to the cabinet-top around the side of the room. Place the bubble level on top of the galvanometer, and rotate the vertical screw on the galvanometer’s feet to level it.

2. On the pipe that sticks up above the black casing on top of the galvanometer, loosen the lower screw, then slowly and carefully lift the crossbar up until the brown rectangular coil is centered top-to-bottom around the shiny cylindrical magnet, rotate the crossbar so the mirror points outwards, then tighten the lower screw. Caution! Don’t raise the coil too high, or you will snap the thin wire that holds the coil up.

3. Rotate the vertical screw on the left or right foot of the galvanometer until the coil is centered left-to-right around the magnet. The mirror should swing freely.

4. Loosen the upper screw, lower the crossbar to touch the thicker tube, then tighten the upper screw. Attach the long black arm to the middle of the galvanometer, slide the scale (with the numbers upside-down) through the upper half, and center it. Look through the hole below the scale, and rotate the screw at the bottom of the arm near the galvanometer until the scale seen in the mirror is centered.

5. Insert the telescope in the hole, and pull out the black rim of the eyepiece until the crosshair is in focus. Pull out the knurled ring near the eyepiece until the scale is in focus. Rotate the eyepiece to align the crosshair with the scale.

CIRCUIT

Ballistic Galvanometer

55

Switch

1.5-volt battery

- +

Voltmeter

- +

- +

Capacitor

6. Loosen the lower screw at the top of the galvanometer, rotate the crossbar to zero the scale on the crosshair (±1.0 cm), and tighten the lower screw.

7. Turn the voltmeter off if it is on, rotate the dial to 2 V_ _ _, then turn the voltmeter on. Construct the circuit on the next page. The capacitor in the diagram is the decade capacitor box, assumed to be exactly calibrated, set at

56

0.20 µF.

8. Flip the switch to the lower terminals. The capacitor now contains the charge Q = C·∆V, with ∆V being the potential difference across the battery, as read by the voltmeter. Flip the switch to the upper terminals, discharging Q in the capacitor almost instantly through the galvanometer. Measure the maximum deflection ‘x’. Calculate the value of Q. Notice how small it is!

9. Repeat step 8 for the other capacitor settings listed on the data sheet.

10. Draw a graph of Q (on the vertical axis) vs. x (on the horizontal axis), draw a single best-fit straight line through the data points, and determine the slope ‘m’ and the y-intercept ‘b’.

11. Replace the capacitor box with the unknown capacitor ‘A’, and determine its capacitance, from Q = mx + b and C = Q/∆V. Repeat with the unknown capacitor ‘B’. Careful! Orient the capacitors correctly:

- +

12. Place the two capacitors in parallel, and determine x, Q and the equivalent capacitance of this arrangement. Calculate the theoretical value Ceq = CA + CB.

13. Repeat step 12 with the two capacitors in series. The theoretical equivalent capacitance is Ceq = 1/(1/CA + 1/CB).

14. Loosen the upper screw of the crossbar, raise the crossbar to the top, and tighten the screw. Loosen the lower screw while holding the crossbar, and gently lower the coil so that it fully rests on the magnet. Tighten the lower screw and dismantle the circuit.

13. RL AND RC CIRCUITSPURPOSE:

To study the role of an inductance coil (inductor) in anAC series circuit as a low-pass filter, and to study the role

57

of a capacitor in an AC series circuit as a high-pass filter.

INTRODUCTION:

The RL series circuit:

Input Function OutputVoltage Generator Inductor Resistor Voltage

L= 10 mH Vs ~ R VR Vs VL

VR

Fig. 1

Experimentally, XL = VL/I, with VL calculated as VL = √Vs2 - VR2.

Theoretically, XL = 2πfL.

The RC series circuit:

Input Function OutputVoltage Generator Capacitor Resistor Voltage

VR

58

C = 0.5 µFVs ~ R VR Vs VC

Fig. 2

Experimentally, XC = VC/I, with VC calculated as VC = √Vs2 - VR2.

1Theoretically, XC = .

2πfC

APPARATUS:

Graph paperMultimeter, BK (voltmeter)Multimeter, Extech (voltmeter)BNC-to-banana adapterAlligator clips (2)StraightedgeFrench curveBanana wires (7)Inductor, L = 10 mHDecade capacitor box, C = 0.5 microfaradFunction generatorResistor, composition 470 ohm

PROCEDURE:

PART A: INDUCTIVE REACTANCE

1. Set up the apparatus as in Figure 1. Set the multimeters to 2V~ to function as alternating-current voltmeters, measuring Vrms.

59

The BNC-to-banana adapter is fitted to the Output port of the function generator.

2. Set the frequency of the function generator to 4000 Hz by turning the Frequency dial to 4.0, the Frequency Multiplier to 1k, and the Function dial to the sine wave. Set the root-mean-square voltage level to 1.00 volts by turning the Level to 1V-10V, and rotating the Fine dial until the Input Voltage voltmeter reads 1.00 volts.

3. Record Vs and VR in the data table.

4. Determine VL from VL = √Vs2 - VR2 and I from I = VR/R .

5. Repeat steps 2 through 4 for f = 6000 Hz to 18000 Hzin steps of 2000 Hz. Be sure to rotate the FINE dial to maintainVs = 1.00 volts every time you adjust the frequency.

6. Calculate the experimental value of XL.

7. Calculate the theoretical value of XL, assuming that L = 0.01 henrys.

8. Plot the experimentalvalue of XL versus f. XLFrom the slope of this (Ω)straight line, calculate L.

1 XL 1 L = · = · slope f (Hz) 2π f 2π

9. Plot VR versus f.Notice that the output voltage VRis large only for low (small) (volts)frequencies. The circuit actsas a low-pass filter.

f (Hz)PART B: CAPACITIVE REACTANCE

1. Set up the apparatus as in Figure 2.

2. Set the function generator at 1.0 Vrms and 100 Hz.

60

3. Record Vs and VR in the data table.

4. Determine VC from VC = √Vs2 - VR2 and I from I = VR/R.

5. Repeat steps 2 through 4 for f = 200 Hz to 800 Hz in steps of 100 Hz. Maintain Vs at 1.00 volts.

6. Calculate the experimental value of XC, and its inverse.

7. Calculate the theoretical value of XC, assuming that C = 0.5 microfarads.

8. Plot the inverse of the experimental value of XC 1/XCversus f. From the slope of this (Ω-1)straight line, calculate C.

1 1/XC 1C = · = · slope f (Hz) 2π f 2π

9. Plot VR versus f.Notice that the output voltage VRis large only for high (large) (volts)frequencies. The circuit actsas a high-pass filter.

f (Hz)

14. THE RLC CIRCUITPURPOSE: a) To determine the resonance frequency fo

b) To determine the bandwidth |f2 - f1|

61

c) To determine the quality factor Q = fo/|f2 - f1| of this RLC series circuit.

INTRODUCTION: In an RLC series circuit:

Input AC power OutputVoltage supply Resistor Voltage

L C Vs ~ R VR

The impedence of this circuit is Z = √R2 + (XL - XC)2 .

Also, I = VR/R.

At the resonant frequency, XL = XC, so

1 12πfoL = . Solving, fo = . 2πfoC 2π√LC Vs VRAlso at the resonant frequency, Irf = = . Zmin R

I Power Irf

Small R Po 0.707·Irf

Larger R

Po

f1 f0 f2 f f1 f0 f2 f

APPARATUS:

BNC-to-banana adapter (3)Alligator clips (2)

62

RulerFrench curveGraph paperInductor, L = 10 mHDecade capacitor boxFunction generatorOscilloscopeResistor, composition, 10 and 100 ohm

L C Frequency ~ R Counter

Oscilloscope

CH1 CH2 INPUT

Fig. 1

PROCEDURE:

PART A:

63

1. Set up the apparatus as in Figure 1. Be sure to connect the ground wires of CH1, CH2 and the AC function generator together, as shown. Use R = 10Ω.

2. Calculate fo from the nominal values of capacitance and inductance.

3. Maintain Vs at a constant amplitude of 0.40 V by monitoring Channel 1 of the oscilloscope.

4. Starting with fo - 2000 Hz, take voltage readings at +500-Hz intervals up to f = fo + 2000 Hz, recording them in Data Table A.

5. Calculate the corresponding amplitude of the current from the recorded voltages, and the maximum power.

6. Plot I vs. f and P vs. f.

7. Determine the bandwidth and quality factor of the circuit from the values of fo, f1 and f2 from the graphs.

foBandwidth = |f2 - f1| and Quality Factor = Q = .

|f2 - f1|

8. Determine fo from the maximum of the graph and compare with the value calculated from step 2.

PART B:

1. Repeat steps 1 through 8 for R = 100Ω.

64

65

15. THE VISIBLE SPECTRUMPURPOSE:

The wavelengths of electromagnetic waves in the visible range will be determined with a diffraction grating.

INTRODUCTION:

A diffraction grating consists of a number of closely-spaced parallel grooves ruled on a transparent surface. It is a useful device for dispersing the waves of different wavelength in a source of light. The effect of a grating is similar to a prism but exhibits greater resolving power.

The two dots in Figure 1 represent adjacent grooves on a diffraction grating, a distance d apart. When the two rays of light entering from the left strike the two grooves, each ray is scattered in all directions. A bright spot will appear on a distant screen if the two scattered rays heading toward the lower right are in phase, so that constructive interference occurs. As may be seen in Figure 1, δ is the difference in the path lengths of the two scattered rays of wavelength λ. The two rays will be in phase if δ = 0λ or 1λ or 2λ, for example. Call n = 0, 1, 2 ... the order of the diffraction. Then constructive interference occurs if δ = nλ. From the geometry of Figure 1, δ = d·sinθ. Equating these two, nλ = d·sinθ, so

d·sinθλ = .

n

Grating

LLaser δ Laserlight light

θ Card-

d Grating board xLaser θlight

66

Fig. 1 Fig. 2APPARATUS:

Blinds for doorsBuret clamp (2)GratingCardboard, largePaper, 11 x 17 in.Laboratory jackLaser, helium-neonMasking tapeGrating stand and holderRingstand (2)Meter stick (2)Light source aperturesLight source, incandescentPower supply for lamp, 12V

PROCEDURE:

PART A: DETERMINATION OF THE GROOVE SEPARATION d

1. To determine the diffraction grating spacing d, set up the grating and helium-neon laser as shown in Figure 3, with L = 1.000 meters. Orient the grating perpendicular to the beam so the reflected beam shines back into the laser. Measure the distance x for the orders n = 1 and n = 2. Calculate θ from tanθ = x/L, as shown in Figure 2. Determine an average value for the grating groove spacing d from d = nλ/sinθ. The wavelength of the laser light is 633 nm.

xleft

He-Ne Laser L xcenter ~ 0.000

θ meters

Lab Jack xright

67

Fig. 3

PART B: DETERMINATION OF THE WAVELENGTH RANGES FOR VISIBLE LIGHT

1. Set up the apparatus as shown in Figure 4, replacing the laser with a white light source, without moving the diffraction grating or the cardboard. Adjust the lens or the filament position to create a sharp image of the filament. Place the light-source aperture cap in front of the light source lens, to create a narrow vertical beam. Place the ledger paper at the location of the n = 1 visible spectrum.

2. Record L. Record x for the boundary between each color. For example, the horizontal line between violet and blue in Figure 4 is the location of xlower(Violet) = xupper(Blue).

White-light 0th order source

_Violet Blue Green Yellow

Fig. 4 Orange Red

3. Calculate θ from tan θ = x/L, as before. Calculate λ = d·sin θin units of nanometers, with d being determined from Part A. Calculate the percent difference between your results and the (rather arbitrary) values given below. Indigo is missing, as it is too difficult to distinguish from violet and blue in this experiment.

Color λupper λlower

Violet 400 nm 424 nmBlue 424 nm 491 nmGreen 491 nm 575 nm

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Yellow 575 nm 585 nmOrange 585 nm 647 nmRed 647 nm 700 nm

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16. REFLECTION AND REFRACTIONPURPOSE: To show by means of ray tracing:

a) The angle of incidence θ1 equals the angle of reflection θ1‘.

b) The virtual image is the perceived location of the object behind the mirror.

c) The angle of deflection φ is twice the angle through which a mirror is rotated.

d) The index of refraction can be determined usingSnell's Law.

normal normal

θ1 n1 n2

θ1 θ1‘ θ2

Law of reflection: Law of refraction (Snell’s Law):

θ1 = θ1‘ n1sinθ1 = n2sinθ2

APPARATUS:

CardboardLedger paper (11" X 17")RulerProtractor, largeCompass, pencilStraightedgePencil, colorMirror, planePins, long

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Refraction cubeRefraction plate

PROCEDURE:

PART A: VIRTUAL IMAGE IN PLANE MIRROR

A’

Mirror Line

L2 R2

A L1 R1

Fig. 1

1. Support the mirror as shown in Figure 1, on the ledger paper above the cardboard. Draw a line on the paper, directly underneath the back of the mirror (at the plane of the reflecting surface), and draw a triangle in front of the mirror.

2. Place a pin at A as shown in Figure 1.

3. Place a pin at R1, about 10 to 15 cm to the right of the triangle.

4. View R1 from behind and place R2 so that R1, R2, and A’ (the reflection of A in the mirror) appear in a straight line. See Figure 2.

A’

Mirror Line

L2 R2

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A L1 B R1

C

Fig. 25. Repeat steps 3 and 4 from the left side with pins L1 and L2.

6. Draw R1R2 and L1L2 to the mirror with a colored pencil. Remove the mirror. Extend this as a dashed line behind the mirror until R1R2 and L1L2 meet at A’. Place a dot at A’ and label it.

7. Move the pin from vertex A to vertex B of the original triangle and repeat steps 3 to 6 with a differently-colored pencil. Move the pin from vertex B to vertex C of the original triangle and repeat steps 3 to 6 with a differently-colored pencil.

8. Join points A’, B’ and C’ to construct the virtual image of the original triangle. Fold the paper on the mirror line. What can you say about the mirror image of the triangle?

PART B: ANGLE OF INCIDENCE AND ANGLE OF REFLECTION

1. Join points A and P on the ledger paper, as shown in Figure 3.

A’

Mirror Line

θ1 θ1’ R1 A

R2normal

Fig. 3

P

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2. Draw the normal at P, perpendicular to the mirror surface.

3. Measure the angle of incidence θ1, and the angle of reflection θ1’.

PART C: ROTATION OF MIRROR

1. On a new sheet of ledger paper, set up the mirror with pins at A and B, as shown on the left side of Figure 4. Draw a line on the paper, directly underneath the back of the mirror. Draw the line AB to the mirror surface at P, and label it.

A’new

A’ B’new

B’ Mirror P Line α

B C B E

A D A F

Fig. 4

2. Draw line CD by aligning pins C and D so C, D, and the reflection of A and B (A’ and B’) appear on a single line.

3. Leave A and B in place. Rotate the mirror clockwise through an angle α = 20°, with P as the fixed pivot point. Then, looking into the mirror, line up pins E and F so that E, F, and the reflection of A and B (A’new and B’new) appear on the same line.

4. Measure the angle between the reflected rays CD and EF. This angle φ is the angle of deflection of the reflected ray.

5. How well does your value of φ compare to the theoretical value? From Part B, how well does your value of θ1’ compare to the theoretical value?

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PART D: INDEX OF REFRACTION

1. Place a glass cube or a glass plate on the ledger paper and trace its outline.

2. Draw a normal to the surface at R close to the top-right corner, and draw the lines AR, BR and CR at 20, 40 and 60 degrees to the normal respectively.

3. Place pins at R, A, B, and C as shown in Figure 5.

A B C normal

Medium 1air R

Medium 2glass

n2n1 C’ B’ A’

Fig. 5

4. View A and R through the glass and place pin A’ on the edge of the cube so that A, R and A’ all appear on a straight line through the glass.

5. Repeat for pins B’ and C’.

6. Remove the glass and join points R and A’, R and B’, and R and C’.

7. Measure θ1A, θ1B, θ1C, θ2A, θ2B and θ2C at point R, from the normal.

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8. Calculate n, the index of refraction of the glass (n = sinθ1/sinθ2) for each of the three rays.

9. Calculate the velocity of light through glass v = c/n, from an average value of n.

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17. THE THIN LENSPURPOSE: a) To measure the focal length of thin lenses.

b) To observe characteristics of images formed.c) To verify the lens equation.d) To practice drawing ray diagrams.

INTRODUCTION:

Object

RealObject F Image Virtual F

Image

Fig. 1 Fig. 2

A biconvex lens makes A biconcave lens makes parallel rays converge. parallel rays diverge.Its focal length f Its focal length fis a positive number. is a negative number.

1 1 1Thin Lens Equation: + = , with p q f

p = distance from the lens to the objectq = distance from the lens to the imagef = distance from the lens to the focal point

APPARATUS:

PaperRulerLamp

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Optical benchMeter stickLight source, optical bench object lampOptical bench accessoriesScreen, frostedDiopter gaugeLens, +15-cm convexLens, -10-cm concaveRod for virtual-image placementPROCEDURE:

Part A: FOCAL LENGTH OF BICONVEX LENS

1. Set up the optical bench, biconvex lens and screen as in Figure 3.

To a distant object

f

Fig. 3

2. Slide the screen back and forth to focus the image of a distant object onto the screen. The distant object may be the reading lamp provided, held temporarily at the other side of the lab room.

3. Measure the image distance q. This is equal to the focal length f of the lens since the first term in the lens equation goes to zero. Write down this result on the top of Table 1.

PART B: VERIFYING THE LENS EQUATION EXPERIMENTALLYAND WITH RAY DIAGRAMS

1. Set up the apparatus as in Figure 4.

2. Place the object at the following position:

(a) p > 2f.

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p q

Fig. 4

3. Adjust the position of the screen to create a focused image at a new image distance q. Record p and q in Table 1. Calculate f using the thin lens equation. Determine the magnification M = -q/p and describe the image.

4. Place the object at the following positions, and repeat step 3:

(b) p = 2f(c) f < p < 2f(d) p = f(e) p < f

For positions (d) and (e), q cannot be measured, but can be calculated by using the value of f from part A.

5. Draw ray diagrams to scale for cases (c) and (e) on white paper.

PART C: FOCAL LENGTH OF BICONCAVE LENS

1. Remove the metal cover from the top of the diopter gauge. Place the central prong against the middle of one side of the lens, and record the reading on the outer (red) scale. Repeat with the other side of the lens. Add these two numbers. The inverse of the sum, multiplied by -1, is the approximate focal length in meters. Write down this result on the top of Table 2, and recap the diopter gauge.

PART D: VIRTUAL IMAGES PRODUCED BY A BICONCAVE LENS

1. Set up the optical bench, light source and biconcave lens as in Figure 4.

2. Repeat step 2 of part B, recording the object distance p for each position. Calculate the image distance q by using the value of f (a negative number) from part C. Determine the magnification M = -q/p and describe the image.

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3. For each of the five positions, place the support rod vertically at the calculated image distance, and note that the image (seen through the lens) appears to be at the same distance as the support rod (not viewed through the lens) as you move your eye left and right.

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