PHY 132 Lab Manual



PHY 132 Lab Manual Revised December 2006

INTRODUCTION AND GENERAL INSTRUCTIONS:

Attached you will find instructions for this semester's labs, followed by answer sheets for you to use in your writeups. Please work in pencil so you can erase mistakes.

You are to write a report on each experiment, as in this example:

Report on Experiment #15: Volume of a Sphere

(Your Name)

The purpose of this experiment was to verify the equation for the volume of a sphere, V = 4/3πr3. We measured the volume of a thin-walled, hollow sphere by filling it with water and then pouring the water into a graduated cylinder. We measured the sphere's diameter with a ruler, and used this to calculate 4/3rπ3. The result agreed with the measured volume within our level of uncertainty.

(Attach to this the answer sheet showing the data and calculations, any graphs, etc.)

This discussion explains the other things you have attached and ties it all together. Your ability to explain what's going on also shows whether you actually understand it. It includes:

1) The Objective (What you were trying to do)

2) Apparatus and Procedure (How you did it)

3) Conclusion (Whether or not the experiment demonstrated what it was supposed to)

The apparatus and procedure section should describe, as clearly and simply as possible, the basic idea of how the experiment works. It might help to imagine yourself explaining the experiment to someone who is not in this class, but knows some physics. Even though it's what took most of your time and attention, you don't want to bore or confuse this person by going on at great length about trivial details like so:

Procedure: We had a thin-walled hollow sphere with a 3/8 inch circular opening in the top. Into this, we poured cold tap water, being careful not to drop the sphere because it is breakable. We then poured the water into a 100 ml graduated cylinder, shaking the sphere several times to be sure as much water dripped out as possible. The volume was read from the bottom of the meniscus, and .....

This not only takes longer to write than the first example, but it's also very easy, in all of that, to never actually get to the point. You also should not describe, step-by-step, the calculations on the answer sheet, which should be clear enough to speak for themselves. Here's another bad example:

The purpose was to check the equation v= 4/3πr3. We took the apparatus and made the measurements described in the instructions. The measured and calculated volumes did agree with other.

Your report should make sense on its own, and your hypothetical friend could not possibly picture the experiment based on just that.

Avoid simply copying from your partner or the instructions. Technically this is plagiarism. This applies not only to the discussion you write, but especially to the calculations: Everyone in a group usually has the same data, but you should do the calculations individually so that everyone understands how, and so that you can check each other for mistakes. Feel free to ask each other for help and to compare results, but some people should not be getting a free ride off of others. There's sort of a gray area between helping and copying; but, for example, four papers with the same arithmetic error would be a clear case of three of the people not giving the work any thought at all.

A maximum of four students is allowed in each group. (If the class is small enough, I may change this to three.) Except for a few experiments where it's necessary, if your paper is one of more than four with the same data, I'll send you back to do it over. (If yours isn't working, don't use your neighbor's data.)

Review of Uncertainties:

In general, the amount a calculation might be off can be related to the uncertainty of the numbers that went into it by taking the differntial and throwing away all the minus signs:

Example: If x = F cos(, then the uncertainty in x is (using the product rule and chain rule):

(F (-sin( d()( + (dF cos((. d( and dF are the uncertainties in ( and F.

From that general principle, the following rules for specific cases can be derived:

- When adding or subtracting numbers, add the uncertainties:

Example: (4.3 + .2 cm) - (2.7 + .2 cm) = 1.6 + .4 cm

- When multiplying or dividing, add the percent uncertainties:

Example: (4.3 + .2 cm)(2.7 + .2 cm)

= (4.3 + 4.65%)(2.7 + 7.41%)

= 11.6 + 12.06%

= 11.6 + 1.4 cm2

To find what percent .2 is of 4.3: .2/4.3 = .0465 = 4.65%

To find what 12.06 % of 11.6 is: (.1206)(11.6) = 1.4

Experiment 1: Electrostatics

Part I - Electric Field Mapping

BACKGROUND: A DC power supply charges two electrodes printed on a sheet. This sheet is coated with a poor electrical conductor; enough current flows between the electrodes to give a voltmeter something to measure. You will use the voltmeter to find several curves of constant potential. The direction of the field vector, [pic], is perpendicular to these equipotentials. (With gravity, water runs in the direction that the elevation changes fastest. Similarly, [pic] points in the direction that the potential changes fastest). So, you will sketch field lines by drawing them perpendicular to the equipotentials.

PROCEDURE:

1. Connect the posts on the mounting board to the DC terminals of a power supply. (Red and black.) Connect minus on the voltmeter to minus on the power supply and plus on the meter to the probe. (The thing with the black handle.)

2. With the probe touching the sheet’s positive electrode, set the power supply’s knob so that the meter reads 6.0 V.

3. Move the probe around the sheet to find points where the potential is 1.0 V. Plot enough on the answer sheet to accurately sketch a curve of constant potential all the way across the picture. (Maybe 2 or 3 cm apart; closer where the curve bends sharply.)

4 Repeat, obtaining equipotentials at 2, 3, 4 and 5 V.

5. Draw in some approximate lines of force by sketching smooth curves that cross all equipotentials at right angles. The edges of the printed electrodes are your equipotentials for 0 V and 6 V. Include arrowheads, and label which are the positive and negative electrodes. Draw enough to show the field in all areas of the picture.

(Since all you did was draw a picture, there is no conclusion to include when you write your discussion for this part.)

Part II - Coulomb's Law

You will verify that the force between two charges varies as the inverse square of their separation.

APPARATUS: Two balls are charged by touching them to a Van de Graff electrostatic generator. (The rubbing action of a belt inside it charges the metal globe on top.) They are then placed in a box to protect them from air currents. One ball is mounted on a movable base; the other hangs from a string. You can observe the force by how far the string is pushed to the side, x. The force is compared to the separation of the balls, r, to see if the inverse square law holds true.

PROCEDURE:

1. Run the Van de Graff for a second or two by plugging it in. Charge the balls by touching them to the globe, running the machine between balls. If they now touch you, the box, or anything else, the charge will escape.

2. Put the hanging ball in the box at the same height as the sliding ball. Cover the top with old transparencies. With the other ball outside, carefully position the hanging ball so that its left side is at 15.0 cm on the ruler behind it. The scratch on the window should line up with the ball's left edge each time you read the ball's position, as shown, so that your eye is in the same position.

3. Slide the movable ball into the box. Check that the balls are sufficiently charged: the hanging ball should be displaced over 2 cm when the other is close to it.

4. Record the positions of both balls, at 1 cm intervals as you slide the ball from the 26 cm mark to the 18 cm mark. For either ball, position your eye so that the scratch lines up with the left side of the ball as in the picture.

Before each reading, check that the balls are lined up parallel to the window.

Do not take data with the sliding ball's position, s, outside of the range 26> s > 18 cm. Further out, stray charge on the floor and walls of the box is nearly as close as the sliding ball, affecting the hanging ball's behavior. Farther in, the charge on the balls begins to polarize, superimposing an attractive force on top of the repulsion.

CALCULATIONS & ANALYSIS:

Assume the force obeys a power law of the form F = arn

where a and n are constants. You want to find n. (You want to see if the data obeys an inverse square law, as opposed to some other kind of function.)

1. Calculate x and r for each trial. Instead of measuring force in newtons or pounds, it's convenient to make up your own force unit, defined such that one unit of force causes one centimeter of displacement. This makes the force simply equal to x.

2. Graph the data on the log-log graph paper provided. Include error bars based on your own estimate; notice that the logarithmic scale makes these look larger in the lower part of the graph. Draw what appears to be a best fit line through your data.

The slope of this graph is n:

Take the log of both sides of F = arn :

log F = log a + log rn

or,

So, plotting log F as a function of log r should give a straight line whose slope is n. (If our assumption of a power law is false, the graph won’t come out straight.) Using the log-log paper allows you to just plot the data itself instead of calculating logs.

3. Find its slope, measuring the rise and run with a ruler, not from your data or the scale of the graph. Measuring the logarithmic graph with the linear ruler has the effect of taking the log of F and r. Indicate which points on the line you measured between.

4. Round n to the nearest integer - it's only good to one significant figure. In your conclusion, state whether your result agrees with Coulomb's.

5. After the instructor has approved your results, put the hanging ball back in its little container, so it doesn't get lost

Experiment 2: Dipoles & Superposition

(The calculations are a large part of this lab; if you’d like to get out early, you could work out how to do them ahead of time.)

A dipole is a pair of equal and opposite charges. You will verify that the direction of a dipole’s field matches what you get from superposing the fields of the two individual charges. Although we do not cover magnetism until later in the course, you will use a bar magnet because it is easier to work with than an electric dipole. The bar magnet is placed on a sheet of graph paper. For each of five points on this grid, the magnetic field vector from one of the magnet’s poles is added to the field vector from the other pole, and the direction of the resultant is calculated. You will then check your results experimentally by observing the direction of a magnetic compass at those points. The graph paper must be oriented such that the earth’s magnetic field does not turn the compass away from the predicted direction.

Background: We've seen that the field of a pointlike electric charge is given by E = 1 q

4πεo r2

The magnetic field, B, of a magnetic pole is similar to the electric field, E, of a charge (and to the gravitational field of a mass): B is proportional to the pole strength, qm, and inversely proportional to the square of the distance, r. The proportionality constant would be different from the Coulomb's law constant.

It is also convenient to not use SI units. Instead, make up your own unit for B such that the proportionality constant is the reciprocal of your magnet's qm. In this unit, B is simply

B = 1

r2

Calculations:

In the diagram below, the origin is at the center of the magnet, x and y are the coordinates of point P, and a is the distance from the origin to either magnetic pole.

Measure the length of your magnet, and divide by 2 to get a.

Use a spreadsheet (Excel) for the rest of this. Compared to a calculator, a spreadsheet doesn’t save any time if you do a calculation just once. But, if you do it repeatedly, each time takes as long as the first on a calculator, whereas the rest appears with just a couple of clicks on a spreadsheet.

Use + for addition, - for subtraction, * for multiplication, / for division, and ^ for an exponent. Sine of what's in cell c2, the box in column c and row 2, would be =sin(c2), arcsin would be =asin(c2). For the spreadsheet to recognize something as a formula, it must begin with a character from a certain list; use an equal sign. So, for example, to divide what's in c2 by what's in d2, and then take the square root, you would type =(c2/d2)^.5 in the cell. When you move elsewhere, the formula no longer appears in the cell, but if you come back, it will be in the box near the top of the screen.

Have the spreadsheet’s second row find the direction of [pic], the magnetic field vector at point P, as follows. (Use the first row for headings.)

- Put x and y in the first two columns. (2 and 8)

- Find r1 and r2 from x, y, and a.

- Find θ1 and θ2 from x, y, and a. (Two problems to avoid: 1. Use arctan not arccos, because arctan makes angles below the horizontal negative but arccos does not. 2. Do not convert to degrees. When Excel uses them in a later column, it will expect them to still be in radians, just the way it first calculated them.)

- Find B1 and B2 from r1 and r2.

- Find the components of [pic] and [pic] from B1, B2, θ1 and θ2. (Get the signs right: N is to magnetism what + is to electricity, and S is like -. So, [pic]points away from the N pole, but [pic]points toward the S pole.)

- Find the components of [pic] from the components of [pic] and [pic].

- Find θ, the direction of [pic] relative to the positive x axis, from the components of [pic].

- The spreadsheet uses radians; have it multiply by 57.3 to get degrees.

Once the formulas are set up in the second row, copy them down through row six: Highlight your row of formulas. Click Edit then Copy. Highlight the same columns in rows 3 - 6. Click Edit then Paste. The row numbers in your formulas will automatically be adjusted.

To experimentally check your results, a sheet has been printed with outlines of the magnet and compass. Use this to see if the magnet makes the compass point in the predicted direction: (The direction of the field is the same as the direction that a compass points.)

1. Place the compass on the sheet so that east on the compass is the x direction, and north on the compass is the y direction.

2. Orient the sheet of paper so that the earth's field does not affect the answer: With the magnet very far away, rotate the sheet until the compass is pointing in the expected direction. Then put the magnet in place.

3. Record the field's direction. (The marks on these little compasses are 15( apart. Don't try to substitute a larger compass; they are easier to read, but the field differs too much between one end of the needle and the other.)

Estimate an uncertainty, based on how closely you can read the compass and on how badly it seems to stick. Within the expected level of accuracy, does theory match experiment?

Experiment 3: The Charge of the Electron

You will measure the electron's charge using a version of Robert Millikan's oil drop experiment (1909). Instead of oil drops, you will use tiny latex spheres, which simplifies things since we already know their mass.

A solution containing the spheres is sprayed into the space between two charged metal plates, inside a chamber where they are viewed through a microscope. Rubbing against the sprayer gives the spheres a small electric charge. You pick one sphere to watch, then adjust the electric field from the plates until the electric force pushing the sphere up equals the its weight pulling down, so the sphere neither rises or falls. Since you know the weight of the sphere and the strength of the electric field, F = qE tells you the charge on the sphere.

You repeat this many times, determining the charge on many spheres. Each sphere gains or loses a whole number of electrons, so its charge is a multiple of e. Your results, therefore, should clump up around charges of e, 2e, 3e, 4e, etc. The spacing between these clumps of data gives you e.

Anticipate spending most of the period gathering data; a computer quickly does the calculations. Good results require care, patience, and a certain amount of luck. If you are easily frustrated, you may want to bring some inexpensive breakable objects to smash from time to time.

There is no answer sheet for this lab; your data sheet and calculations will be printed out by the computer. Anything not apparent from the printouts, such as how you calculated the weight or the formulas used in the spreadsheet, should be stated when you write your discussion.

Procedure:

Do not adjust the light or the focus of the microscope without checking with the instructor. They should already be adjusted; you will probably only mess them up.

Accuracy is critical in getting any result from this experiment. These bad results are bad because something was just a little off on just a few trials. To avoid this,

- Make sure the sphere has the proper weight: They can stick together, or you can get a foreign object. With the electric field off, a good sphere will take between 7.7 and 10.2 seconds to fall two of the small divisions in the microscope. (The variation is due to reaction time with the stopwatch, random movement of the air around the spheres, and possibly the shape of the "spheres".)

- Watch each sphere for enough time to be sure the forces are accurately balanced: Once you think you've got it right, watch it for at least another 20 seconds to be sure.

- Do not use spheres that balance at less than 40 V: The uncertainties get too large.

1. Set up the spreadsheet, as described below under "calculations". Some of you may want to start gathering data while others do this. (Gathering the data takes quite a bit of time.)

2. If necessary, connect the wires: Black and red from the apparatus go to black at the lower right of the power supply and the red +250 V connector above it. (Don't touch the connections with the power on.) Connect the green wires from the light to the 6 V DC connectors at the bottom center of the power supply; which is which doesn't matter. Connect a digital meter, set for DC volts, to the red and black sockets on the back of the apparatus.

3. Turn on the power supply. Turn the knob on the Millikan apparatus all the way up, so the voltmeter reads about 250 V. To turn on the field in the chamber, put the 3-way switch on the Millikan apparatus either toward you or away from you. Toward versus away determines which plate is positive and which is negative; center is no field.

4. Darken the room. Look into the microscope and squeeze the atomizer two or three times. You should see little dots of light like faint stars. These are the tiny spheres.

5. Select a sphere:

a. Turn on the field, at 250 V. Pick one that is rising slowly. (Fast ones have too much charge.) If you don't see a good one, try reversing the field, or spray in more.

b. Turn off the field by moving the switch to the center. Check that the sphere you picked takes between 7.7 and 10.2 s to fall two small divisions. (Clunps or pieces fall faster or slower than that.) The numbered lines are five divisions apart, not one.

c. If it's ok, record this time in the first coulmn of the spreadsheet. If not, disregard that sphere, and go back to (a).

6. Turn the field on, then adjust the voltage knob near the switch (not the one on the power supply) until the sphere is neither rising or falling. It saves time if your partner watches the voltmeter while you're looking in the microscope, alerting you if you're about to invest a lot of time balancing something below the 40 V cutoff.

Move the right side of the knob in the direction the sphere is drifting. They sometimes move a little at random; watch a few seconds before deciding some gradual trend is real. Ignore horizontal motion.

7. Once you think you have the sphere balanced, watch for at least 20 seconds to make sure. If you need to adjust the voltage, start the 20 seconds over. If there is no long term vertical motion at all for 20 s after the last time you touched the knob, and you have 40 V or more, record the voltage in the second column of the spreadsheet.

Hopefully, it won't drift out of sight before you're done. This is where those inexpensive breakable objects might be useful.

8. Turn the voltage back up to 250, spray in more spheres, and repeat. It's easiest to just type data into the spreadsheet, instead of writing it down first. Save to the hard drive after every trial. Be sure everybody in the group gets at least a brief turn at the microscope, so they can see what's going on.

Make as many trials as necessary for the pattern to emerge; get at least three clumps of values in the charge column containing at least three trials each. (More clumps containing more trials would be even better.) You will probably end up doing around 15 or 20 trials. If most of your data seems to be falling into the same clump, try to select spheres that rise with a slightly different speed at 250 V. In particular, there should be some which rise very slowly; people often seem to overlook these.

9. When the instructor says you're done, remove the atomizer and pour the remaining fluid into the container he has. Then, rinse thoroughly with tap water. Squeeze the bulb a number of times to work water through the sprayer, and run some water through the tube that comes out of the lid.

Calculations:

1. The mass of each sphere is 6.01 x 10-16kg. (Based on a diameter of 1.03 μm and a density of 1.05 g/cm3, printed on the bottle.) Calculate a sphere's weight.

The rest of this is quite repetitive, so you will automate it with a spreadsheet. You will then export the results to our usual software package, Science Workshop, to plot them on a histogram. (A graph of number of occurrences versus the value of some variable.)

2. Open Excel, and type the following headings in the top row: t, V, E and qx10-19. The first two columns are for you to record the time to fall two divisions, and the equilibrium voltage.

3. Enter appropriate formulas in the last two columns. Review last week's lab, if necessary, for how the spreadsheet works. (Type the formulas in row 2, then copy them from row 2 into the cells below, through row 30. Once again, you highlight the cell to copy from, click edit, click copy, highlight the cells to copy to, click edit click paste.) The ideas behind the formulas are as follows:

a. Column 3, the electric field strength: Recall that in a uniform field, ΔV = [pic]. Have it calculate E from the voltage you measured and the plates’ separation, .0050 m.

b. Column 4, the charge on the sphere: Since the sphere is in equilibrium, the forces shown here are equal. Determine a formula for q from this. Multiply this formula by 1 x 1019. That makes 3.2 x 10-19, for example, appear as just 3.2, which is easier for the computer to deal with.

4. When you have enough data, export it to Science Workshop and have it do a histogram:

Highlight the last two columns. (You don't care about the second to last, but the software insists on two variables.) Click Edit, then Copy. Minimize the spreadsheet with third button from the upper right corner.

Open Science Workshop. Click Edit, then Paste. Type "charge" for name, and "10-19c" for units. Click ok. Drag the graph icon to channel A. Click on the circle full of dots with the A next to it, at left. Click on data cache, then on charge to its right.

Click . Click the box by Integral Histogram to make the check mark go away. Click ok. Click Σ. Click Σ on the menu which appears, hold down the mouse button as you go down to histogram then release it on 100 divisions. A sideways histogram should appear. Ignore the graph to the left of the histogram.

5. Your data should fall into equally spaced clumps, as discussed earlier. Some clumps might be missing, and you might also have some inaccurate points scattered around. It's ok as long as you have enough good results to spot the pattern. If not, ask the instructor, but you will probably have to go back for more data.

6. When you have enough data and a good histogram, print a copy of each. (Data from the spreadsheet, histogram from Science Workshop. One of each per person.) To read the charge of each clump, first click on [pic]

to turn the pointer into a set of cross hairs. Put the horizontal cross hair through what looks like the center of a clump, then look to the left of the graph. Write the results on your printed copy.

7. Each clump should be a multiple of the elementary charge. If no clumps are missing, the average charge of the first clump divided by 1 should be e, the second clump divided by 2 should be e, and so on. If clumps are missing, you can tell by the spacing on the histogram. Calculate an e for each clump, then average to obtain your result for the experiment. Don't bother calculating an uncertainty. If this workes at all, you should be within just a few percent of the accepted value.

There is usually someone with just numbers everywhere and no clumps. It only takes mistakes observing a few spheres to fill in the gaps between clumps. If the number of spheres observed was increased enough, it should make the clumps stand out from these mistakes. But, it's probably the end of the period by now. So, if this is you, just make it clear that you understand what should have happened, grovel a bit, let me grumble at you some, and you will get your 10.

Experiment 4: The Potentiometer/ Capacitor Combinations

How to use meters:

Analog meters (the kind with a needle): Red connectors are for + and black for -, or they are just labeled + or - . There is one of one sign, which you always connect to, and several of the other, which you use just one of. Which of these you use determines which scale you read. For example, if you put the wire on a connector labeled 30 V, read the scale which goes from 0 to 30 V. Also, check that the meter is correctly zeroed, and have the instructor adjust it if it isn't.

Digital meters: Turn the dial to the function you want. For example, 20 in the area marked DCV turns it into a DC voltmeter which reads from 0 to 20 V. DCA stands for DC amps. If the meter has more than two connectors, the negative wire always goes to COM. Where the other wire goes depends on whether you want to measure volts, amps, or milliamps; the sockets are labeled. On the yellow meters, the dial goes in the same place for the 10 A scale and for the 20 mA scale; which you are using depends on which socket the + wire is in.

PART ONE: The potentiometer

We wish to measure a battery’s emf, E; meaning the “voltage” created by the chemicals in it. (Don’t confuse E with E, the electric field strength. E is in volts, E in volts per meter.) It might seem that you could just measure E with a voltmeter. But, the materials the battery is made of have resistance, and Ohm’s law says there is a voltage drop across this “internal resistance” if a current flows. Since a voltmeter draws current from the battery, the potential difference between the battery’s terminals is E – Ir, which is less than E.

A potentiometer measures the voltage without drawing any current, so E – Ir just equals E , and you get the true emf. The voltage from a power supply is put across a long, bare, high resistance wire, CE, as shown. The connection at point D is moved until VCD is equal to the emf of the battery. You can tell the right spot for D because when the voltages balance, the galvonometer (a sensitive ammeter) reads zero. You can then get VCD (which equals E) from a proportion: Voltage CD is to the wire’s total voltage as length CD is to the wire’s total length (100 cm):

[pic] so, [pic]

PROCEDURE:

Record the value for its emf printed on the battery.

Connect the battery directly to an analog voltmeter (the kind with a needle) and record the reading. Ask for a different battery if it isn’t between .1 V and 1.0 V. (The effects we are studying are more pronounced for an old, dead battery because of its large internal resistance.)

Set up the potentiometer circuit. Don't put the battery backwards: The wire from + on the battery should lead to + on the power supply. Set the power supply at between 2 and 4 volts. Keep the power off when not taking readings so the wire doesn't get hot.

Tap the key at point D on different parts of the wire to find where the galvanometer reads zero. Do not hold the key down with the galvanometer at full scale, or it may suffer. Read the length l, from the positive end of the wire. (The same section of wire the battery is connected across.)

Find the electromotive force of the battery.

In your conclusion, comment on which of your three values represents the true emf of the battery and why.

Part Two: Capacitor Combinations

A group of capacitors is assembled, as shown. You will predict what the voltage across each capacitor in the group should be, then measure them with a digital meter to see if you were right.

Some capacitors are labeled without units: "2-25DC" means 2 μF, and a maximum voltage of 25 V DC. Select four different capacitors of at least 1 μF each (the higher the better). The largest capacitance should be no more than ten times the smallest.

Don’t hook capacitors up backward: Some have arrows pointing to the negative end; with this kind of capacitor, that's important.

CALCULATIONS :

1. Choose some potential difference, V, to put across the group. (How much doesn't matter.)

2. Calculate the equivalent capacitance of the group of four capacitors.

3. Calculate V1: Since C1 and the rest of the circuit are in series, QTot = Q1. So, Ceq V = C1 V1.

4. Find V2 from V and V1.

5. Calculate V3: Since C3 and C4 are in series, Q3 = Q4, so C3V3 = C4V4. Putting that together with V4 = V2 - V3 gives C3V3 = C4(V2 - V3). You know everything in that except V3.

6. Calculate V4.

MEASUREMENTS. Use a digital voltmeter: (The capacitors would almost immediately discharge through an analog meter.)

1. Put the voltmeter across the power supply and set it to the voltage you chose.

2. Connect the meter to a capacitor with the power supply off. If it doesn't read zero, short out the capacitor to discharge it. (Temporarily run a wire from one side of the capacitor to the other.)

3. Turn on the power supply. Record what the meter said before it started decreasing. (The capacitors will slowly discharge even through a digital meter.) Promptly turn the power supply back off.

4. Move the meter to a different capacitor and repeat 2 and 3.

5. If you need to double check anything, turn off the power supply and short out each capacitor first. Otherwise, some charge which flowed through the meter might affect your results.

Compare the measured voltages to what you calculated. The main source of experimental error is the accuracy of the capacitances. Based on this, the difference between your calculated and observed values should be no more than 10%.

Experiment 5: Ohm’s Law; Electric Power.

Review of how to use meters:

Analog meters: Red connectors are for + and black for -, or they are just labeled + or - . There is one of one sign, which you always connect to, and several of the other; which of these you use determines which scale you read. Also, check that the meter is correctly zeroed.

Digital meters: Turn the dial to the function you want. For example, 20 in the area marked DCV turns it into a DC voltmeter which reads from 0 to 20 V. DCA stands for DC amps. If the meter has more than two connectors, the negative wire always goes to COM. Where the other wire goes depends on whether you want to measure volts, amps, or milliamps. On the yellow meters, the dial goes in the same place for the 10 A scale and for the 20 mA scale; which you are using depends on which socket the + wire is in.

We usually don't get through this lab without someone blowing the fuse in a meter: Voltmeters go in parallel, ammeters in series. Start with the least sensitive scale (such as 0 to 10 A), then change scales (such as to milliamps) once you know the approximate reading.

Also, don't take circuits apart until the instructor approves your results. You may need to double-check something.

Part 1: Ohm’s Law.

In part a, a resistor is connected to a DC power supply and some meters. (This resistor has a coiled up piece of fine wire inside; a more common type contains graphite.) Different voltages are applied to it by adjusting the power supply, and the resulting current is observed. You then see if the results are consistent with Ohm’s law. In part b, this is repeated using a light bulb as the resistor.

Part 1a, Resistor:

Select a resistor of 100 to 1000 Ω. Connect a circuit as shown. (The power supply includes both a source of emf and a key-operated on-off switch. R is the resistor, V is the voltmeter, and A is the ammeter.) Start with the ammeter reading amps, not milliamps; switch to milliamps if it turns out that the current is small enough.

Each line on the diagram is a wire. Follow the wires like roads on a map. The current leaves the + (red) terminal of the power supply, and travels to the + connector on the ammeter. So run a wire between those points. After flowing through the ammeter, the current goes through another wire to the resistor. And so on. Finally, run a wire from each end of the resistor to the voltmeter. (It measures the difference in potential between the two points you connect it to.)

Check that the knob on the power supply is turned all the way down (counterclockwise). Keep an eye on the meters as you turn the power on. IF ANY METER GOES OFF SCALE, TURN THE POWER OFF IMMEDIATELY. (When a digital meter goes off scale, the display goes blank.) As you slowly turn up the voltage, continue to be careful of this. Don't use a scale which barely moves the needle either; that isn't very accurate. If the ammeter shows that the current is small enough, use a milliammeter instead. However, do not use the 1.5 V or 3 V scales on an analog voltmeter; they will not read accurately in a circuit with this much resistance.

Get several pairs of values for potential difference and current, covering as wide a range of values as possible (0 to 23 V.). Graph the potential difference as a function of current. For error bars, estimate how closely you can read an analog meter; with a digital meter, if you used the proper scale, the error bars aren’t any larger than the dots you drew on the graph. Neatly draw the best average line or smooth curve through the data.

Pick a point that lies right on your line, and use it to compute the resistance. The data points that lie somewhat off the line presumably contain random errors due to your experimental uncertainty.

In your conclusion,

1. Compare the value you obtain to the value printed on the resistor. Do they agree within the uncertainty printed on the resistor?

2. Comment on the shape of your graph. (The graph of V = IR, where R is a constant, is a straight line. A linear voltage-current relationship is called ohmic behavior because it's what Ohm's Law leads you to expect. Say whether the resistor is ohmic or nonohmic.)

Part 1b, Light Bulb:

Replace the resistor with a light bulb, leaving the circuit the same otherwise. Start again with an ammeter, not a milliammeter, since the current might be larger now.

Get several pairs of values for V and I covering as wide a range as possible (all the way up to 23 V). Include a few readings under 2 V. Include (0,0) as one of your data points, too. Nothing is wrong if the bulb doesn't glow; it's just because of the low voltages you are using.

Graph the potential difference against the current as before. Calculate the resistance twice, using points from near opposite ends of the graph.

In your conclusion,

1. State whether the bulb is ohmic or nonohmic. (The graph of V=IR, where R is a constant, is a straight line.)

2. Ohm’s law implies that R is related to the slope of this graph. Comment on what the resistance does as the bulb gets hotter, and how this relates to the graph’s shape.

Part Two: Power.

An electric heating element is immersed in water, and runs for a time t, producing a temperature change ΔT. You will calculate its power from the formula covered in class, and see if it matches the rate you observe energy being delivered as heat.

Procedure:

1. In the same circuit from part 1, replace the light bulb with the heating coil. The meters should be able to handle around 3 A and 15 V.

2. Remove the inner aluminum cup, and measure its mass empty. Measure the mass again with at least enough water to cover the heating coil. Subtract to obtain the water's mass.

3. Hang the inner cup through the ring in the top of the outer cup, and put on the lid which includes the heating coil. (The stagnant air between the cups makes good insulation.)

4. Insert the thermometer through the hole in the lid, and record the initial temperature, Ti.

5. Start the stopwatch at the same time that you start about 3 amps through the heater. Run it until the temperature changes 10°C or so. The water needs to be kept well stirred; the best way is to pick the apparatus up and swirl it more or less the whole time. (Not so hard you spill it.)

6. Record the voltage and current.

7. Record the time, t, at which you turn off the power.

8. The temperature will probably continue to rise, for maybe half a minute, because the heat hasn't finished flowing from the heater to the where the thermometer is. It's especially important to stir the water well now. Record the highest temperature reached, Tf.

9. Determine the amount of temperature lost to the surroundings: With the water at some temperature T1, restart the stopwatch from zero. Allow the water to cool off for the same amount of time, t. Record T2, the temperature at the end of this time.

Calculations:

1. Correct Tf to compensate for heat leaks:

a. How much temperature was lost between T1 and T2?

b. Add this to your measured Tf to get (approximately) what it would have been if no heat escaped from the apparatus. (For example, if T1 was 35.0( and T2 was 34.0(, 1.0( was lost. If Tf was 35.2(, the corrected Tf would be 36.2(.)

2. Compute the heater’s power from the voltage and current.

3. Compute the rate heat energy was delivered to the cup:

a. Calculate how much heat was added to the water. Calculate how much heat was added to the cup. Use the corrected final temperature. Add to obtain the total thermal energy produced. (If necessary, review the material on heat and specific heat from PHY 131. The specific heat of water is 4186 J/kg·°C; the specific heat of aluminum is 900 J/kg·°C.)

b. Power is energy delivered per unit time. (A watt is a joule per second.) So, divide the energy by the time to deliver it.

Conclusion: Does the power calculated from I and V match the rate you observed energy being delivered? (Expect a difference of 10% or so.)

After you're sure nothing needs to be rechecked, please dump out the water and leave the lid off so everything will dry.

Experiment 6: Kirchhoff's Laws

To check whether Kirchhoff’s Laws actually work, you will set up the circuit shown, measure all of the voltages and currents, and see if the equations are true with the measured numbers in them.

Set up this circuit, using two power supplies and three resistors between 100 Ω and 1000 Ω. Add two voltmeters to measure the potential differences across the power supplies. Get it approved by the instructor. Make E1 and E2 different from each other and in the range 1 V to 5 V. Record all voltages and resistances.

Measure I1 with a digital multimeter by disconnecting a wire at R1 and inserting the meter in series, so the current to be measured flows through it. (Never connect an ammeter in parallel.) A current flowing in the direction opposite that shown in the picture should be recorded as negative. Similarly measure I2 and I3. Don't take the circuit apart before your calculations are done, so you can double-check data.

There are three independent equations that can be written for this circuit, based on Kirchhoff's two laws. Write these down, and verify that they are being obeyed by plugging in your values for the E 's, I's, and R's. (You are not being asked to solve for anything. Fill in numbers for everything.)

To find the uncertainty: The main source of error is that the resistors were made to a 10% tolerance. (If they say something else, use that percentage.) So, assume each IR term might be off by 10%, and each term from a voltmeter is accurate. (The meters do contribute small errors, but let’s not complicate things.) Once you know how many volts each term might be off, add to get the possible error in the entire summation around the loop.

Example: ΣV = 3.00 V + (.0056 A)(200 Ω) – (.0081 A)(500 Ω)

= 3.00 + 1.12 – 4.05

Uncertainty: 0 + .112 +.405 = .517, which rounds to .52

So, the sum of the voltages around the loop is .07 + .52 V.

The point rule should work within a few hundredths of a milliamp.

In your conclusion, comment on whether Kirchhoff's Laws worked: Do the left sides of all three equations, in fact, agree with their right sides?

Experiment 7: Charge to Mass Ratio For the Electron

Shock hazard: There will be uninsulated exposed connections at high voltage. The possible current is limited to 3 mA, so it won't kill you, but try to avoid getting an unpleasant zap.

Part One: Sign of the electron’s charge.

You will observe whether electrons are positive or negative. A beam of electrons in an evacuated glass bulb is shot across a screen coated with a material that glows where electrons hit it. There is a magnetic field from a set of Helmholtz coils just outside the tube, whose direction is observed with a compass. From the electron’s direction of motion and the direction of the field, the direction of the force on a positive charge is determined. This is compared to the observed direction of the force on the electron beam.

1. Connect the tube to the 5000 V power supply as shown. (Yours may not look exactly like the picture.) Bear in mind that the tube is fragile. The 6 V AC heats the filament which is the source of the electrons. (Do not connect to 12 V.) The high voltage applied between the filament and the anode accelerates the electrons.

2. Connect a

second circuit from the 0 - 16 V DC power supply to the coils as shown. Use a digital ammeter, on a scale suitable for around an amp. Obtain the instructor's approval.

3. Turn on the tube's power supply, and set it somewhere between 2000 and 5000 V. The blue line you see is the beam's path. Send some current through the coils. If there is little effect on the beam, their fields are canceling not adding; switch the wires on one of them.

4. Observe whether the beam is positive or negative:

a. To see the direction of the field, [pic], hold a compass just outside the tube, in front of the center part of the screen.

b. Using arrows, dots and x’s, draw a diagram showing, as you look at the screen:

- the direction of [pic] when the electrons first enter the screen.

- the direction of [pic]

- the direction of the force on a positive charge, from the right hand rule.

- the direction of the observed force.

c. In your conclusion, state whether the direction of the force you actually saw indicates a positive or negative charge for the rays.

Part Two: e/m

The main part of today’s lab is to do a modified version of J. J. Thomson’s 1897 experiment in which he discovered the electron. (The experiment showed “cathode rays” to be a stream of particles, each with the same charge to mass ratio, which he measured. e and m were not known individually until Robert Millikan measured e in 1909.)

Notation: v = velocity, V = voltage

The apparatus is the same as in part one. As the electrons move through the magnetic field, the magnetic forces make them follow a circular path. The magnetic force q v B is providing a centripetal force, m v2/r. Setting these equal and doing a little algebra gives you a formula for q/m in terms of v, B and r; show the details when you write your discussion.

Finding q/m, then, amounts to finding, v, B, and r: The speed, v, depends on the voltage used to accelerate the electrons; the higher the voltage, the faster they go. You find B from the current in the coils and their dimensions. The radius of the beam's circular path, r, comes from observing the beam's path as it curves across a luminescent screen in the tube.

PROCEDURE:

[pic]

1

1. Measure D and s for the coils, the average diameter and separation. Side to side, as shown, gives the same thing as center to center, and is easier to measure. Also, the coils are not exactly parallel, so measure s on both sides and average.

2. Turn on the tube, leave the field off. Around x = 8 or 9, where you will take your data, notice whether the top, bottom, or center of the beam runs along the x axis. Use the same part when observing y, below. If the beam isn’t on the axis at all, notice where it is and correct y by the difference.

3. Turn on the magnetic field. From the centimeter grid, record the x and y coordinates of a point on the beam, somewhere around x = 8 or x = 9 cm. (The equation for B is actually only valid near the center of the screen. If you use a point too far to the right, the beam is just entering this part of the field when you stop looking at it.) Be accurate to the nearest millimeter. y should be 1 cm at the very least to minimize the effect of measurement errors. More is better.

4. The screens were not positioned accurately in many of these tubes. Correct x by the amount shown on the tag on your tube.

5. Repeat for a total of three trials, using different combinations of V and I.

CALCULATIONS:

1. From your formula sheet, the magnetic field of a flat coil is B = N μoI R2

2 (R2 + x2)3/2

Multiply by 2 because there are two coils. Substitute x = s/2 and R = D/2. The result is:

B = 2 NμoID2

(D2 + s2)3/2

N is the number of turns (printed on each coil) and I is current. (A large source of error is that this equation is actually only valid on the axis of the coils; B varies somewhat between different parts of the screen. A better setup would use larger coils for a more uniform field.)

Put in numbers for everything but I, and simplify into the form B = (constant)I. Show the details in the space provided. Then, use this result to find B for each trial.

[pic]

2

2. For each trial, calculate the radius of the beam’s path: The + plate of the electron gun, where the electrons enter the magnetic field, is at the point x = 0, y = 0. If the electron beam passes through the later point (x,y), we have from the Pythagorean theorem:

r2 = (r-y)2 + x2

r2 = r2 - 2ry + y2 + x2

2ry = x2 + y2

r = x2 + y2

2y

3. Get a formula for the electrons’ speed: Consider an electron being accelerated through the gun. The potential energy it has on one side of the gun equals the kinetic energy it has on the other. In the space provided, solve this conservation of energy equation for the electron's speed WITHOUT FILLING IN NUMBERS YET. (The usual mistake here is to look up q and m and calculate v from them. q/m is what you've trying to measure; assuming values for q and m instead of treating them as unknowns is circular reasoning.)

4. Substitute this into the equation you derived earlier (from q v B = m v2/r) to eliminate v. Square both sides and simplify to obtain a formula for q/m. Use the result to find q/m for each trial.

5.Average your three values. Assume the result has a 20% uncertainty, primarily due to the nonuniformity of the magnetic field.

CONCLUSION:

Divide the accepted e by the accepted m and compare to your results.

The write-ups for this lab seem particularly prone to people telling all about how to connect the wires, which meters to read, and which equations to use, without ever mentioning how the experiment works. As usual, I want to know what you did more than the details of how you did it. Completely describe the line of reasoning that leads from your measurements to e/m.

Experiment 8: The Current Balance

Object: To see if the measured magnetic force between current carrying wires matches that predicted by theory.

Apparatus: A current carrying bundle of wires, B2 in the diagram, creates a magnetic field around itself. Another bundle of wires hanging from a balance, B1, feels a force from this field, pushing the balance upward. The device measures this force the same way a triple beam balance works: A weight (a small wire) pushing it back down is slid along the arm until the balance balances.

To balance, the torque from the magnetic force must equal the torque due to the weight:

(6 cm)(Fmag) = (d cm)(W)

So, the magnetic force is d/6 times the hanging weight.

Procedure: Connect the circuit as shown. Use a low-voltage, high current power supply with fins on the sides, not our usual gray ones. Starting at (+) on the power supply, current goes through the ammeter, then through the bundle on the balance, B1, then to the double-pole switch. With this in one position, the current then goes straight back to (-); in the other position, it goes through B2 and then back to (-). So, current passes through the B1 either way, but in one position the magnetic field from B2 is off, and in the other position it is on.

Check that the needle points which the balance pivots on don’t bind on the rims of their cups, and that the pointer does not rub on the scale. The counterweight, m, should be adjusted so that the balance almost, but not quite, tips over backward. (The pointer does not have to be at 0.) If necessary, adjust the wires on the base so that the balance does not rest against them. A good way to check if the balance moves freely is the blow on it: It should bob up and down a few times. Also, check that the wires under the base run directly under the balance.

You have an approximately .2 gram wire. Go to the chemistry lab and measure its mass on a balance which can be read to the nearest hundredth of a gram. Calculate its weight.

With the little wire off the arm, send 5.0 amps through B1 only, and note the position of the pointer. (Check that the wires on the balance are not resting against those on the base.)

CAUTION: The needle points or other loose connections could get hot.

Set the switch for current in both sets of wires. If the pointer goes down not up, switch the current’s direction in one set of wires. Readjust the power supply as necessary to keep the current at 5.0 amps (it tends to drop as connections heat up, increasing their resistance.) If you can’t maintain 5 amps, turn it off for a few minutes to let things cool.

Hang the wire over the arm as shown above, and move it until it has the right amount of leverage, d, to bring the pointer back to its position with current in B1 only. To find the uncertainty in d, slide the wire closer to the pivot until you find the minimum distance where the pointer sometimes returns to the right mark when gently disturbed. Then, slide the wire the other way, and similarly find the maximum d.

Find the maximum and minimum values for the "measured force" by using (6 cm)F = d W with both the maximum and minimum d.

Theoretical Force: With the pointer still in the same position (you can turn off the current and just hold it in place), measure r, the distance between the centers of the two bundles of wires, as shown above. A good way to do this is to measure left to left, then right to right, then average. Estimate an uncertainty. (Tryinig to measure r between these wiggly bunches of wires is the largest source of error in this experiment.)

Also measure the length of B1, the wires along the bottom of the balance, and count the number of strands in both bundles. Record which setup you have from the tag on top of it. When you're done, stick your piece of wire to the base with some scotch tape so it won't get lost.

CALCULATIONS:

1. Find the magnetic field which one wire on the base creates at the bottom of the balance. (With its percent uncertainty.)

2. Multiply this by the number of strands on the base to obtain the total field.

3. From the field, find the force on a single wire on the balance. (Keep going with the percents.)

4. From the total number of wires, find the total force, with its uncertainty. (Ignore the other three sides of the coil on the balance.) This is your theoretical value for the magnetic force.

In your conclusion, state whether the measured and theoretical forces agree.

Experiment 9: Magnetism/The Oscilloscope

(This lab’s "write up" is integrated into the answer sheet. You don't need to attach a separate one.)

Part I: Magnetism and Coils

A) Obtain a magnet from the instructor. Connect a coil to a galvanometer (a sensitive ammeter). Do the things described on the answer sheet, record your observations and answer the questions.

(B) You have a device consisting of a coil between the poles of a magnet. Connect a galvanometer, then spin the coil. (Try not to send the meter off scale.) Unlike part A, the strength of the field at the coil's position is not changing. In view of that, how do you explain what you saw?

(C) The same machine you used as a generator in part B, can also be used as a motor. Get rid of the galvanometer and connect it to a DC power supply as shown. Analog meters work best in this case. Use scales suitable for around 1 A and 3 V. Adjust the power supply so that the motor runs at a moderate speed. (Avoid running it too fast.) You'll probably have to push it to get it started. Record the meter readings, and answer the questions.

Part II: The oscilloscope:

BACKGROUND:

The basic purpose of an oscilloscope is to draw graphs on its fluorescent screen. The graph is drawn by a beam of electrons projected from behind. The electron beam passes between two pairs of plates; the electric fields between the plates deflect the beam, moving it around to draw the picture on the screen.

Procedure:

To turn the scope on, turn brilliance. (Red knobs are labeled with red writing, black with black. So, brilliance is the outer, red part.) It should take about half a minute to warm up; if the screen remains blank, try the following:

1. Turn brilliance up higher.

2. Turn stability clockwise until a trace (green line) appears or until it stops turning.

3. Turn Y shift from one extreme to the other: The trace may be beyond the top or bottom of the screen.

Once you've got a trace, turn brilliance to the minimum convenient level. (Too bright can be bad for the screen.) For a better idea of how the scope works, turn time per centimeter fully counterclockwise. This turns off the sweep circuits; there should now be just a stationary spot where the electron beam hits the screen. As you slowly turn time/cm clockwise, you should see the spot move faster and faster, the rate it moves being labeled next to the knob. After a certain point, it's going so fast that it just smears out into a line.

Adjust focus, x shift, and y shift if necessary for a sharp, centered picture.

Now, put some test signals into the scope:

You have an audio oscillator (marked "Function Generator"), which produces electronic signals. Run wires from its output to the scope's input, red to red and black to black. Also connect a loudspeaker to its output.

To read the function generator’s frequency, multiply the setting of its frequency dial by the factor on its frequency multiplier knob. Notice that, depending on the setting of the function knob, it can put out a square wave, the familiar sine wave, or a triangle wave. Attenuation and variable attenuation are coarse and fine adjustments for the amplitude.

Set the knobs on the function generator oscillator for a sinusoidal vibration, volume at some convenient level, and frequency somewhere in the hundreds of hertz. (Frequency multiplier on 100.) On the oscilloscope, adjust volts per centimeter and time per centimeter so that the picture is not too stretched out or squashed up, horizontally or vertically.

If the picture is running along horizontally instead of standing still, check the following. (These control the scope's triggering circuits, which are what tell the spot when to begin its trip across the screen. If it starts at the wrong time, the next wave it draws will not be on top of the previous one.)

a. Stability should be as far counterclockwise as you can get it without the trace disappearing.

b. Trig level should be at auto. (It clicks in when turned fully counterclockwise.)

c. The bottom button of the trig selector should be in. (set for internal triggering.)

Draw two pictures from the scope to show the difference between loud and soft sounds, with the pitch kept the same. Draw two more to show the difference between high and low pitches, with the loudness kept the same. Don't change any knobs on the scope between pictures.

Replace the wires from the audio oscillator with a microphone as shown. Turn volts per centimeter all the way clockwise. You should now see a graph of the sound if you sing or hum. To get the strongest signal, hold the microphone right under your nose or against your throat. Measure this sound’s period and frequency:

a. Check that variable time/cm (a red knob) and x gain (black) are all the way to calibrate. ALWAYS check this just before taking a reading – it often causes grief.

b. Determine f and T as in this example. (You do not have to have the knobs set the same as in the example. Rather, for the best possible look at the sound, set time/cm to stretch the wave horizontally as much as you can, and still see at least one complete period.):

[pic]

3

EXAMPLE. From the display pictured,

T = (6.2 cm)(.5 ms/cm) = 3.1 ms

The 6.2 cm comes from the screen, as shown. .5 ms/cm is the setting of the time/cm knob in this example. (You can use something different.) ms stands for millisecond.

f = 1/T = 1/.0031 s = 323 Hz

Convert 3.1 ms into .0031 s so f comes out in cycles per second, not cycles per ms.

Repeat this for the sound of the plastic flute, with all of the holes open. Hold the mike near the

hole in the mouthpiece; blow moderately. Rinse off the mouthpiece before and after use.

As a comparison to what you just measured, calculate the flute's frequency from its dimensions: The frequency produced by the flute is determined by a standing wave inside it, whose wavelength is about twice the distance to the first open hole. So,

a. Measure from the tip of the mouthpiece to the (center of) the first round hole.

b. Double this to get the wavelength.

c. Use this and the speed of sound to calculate what the frequency should be. (See Phy 131.)

Comment on how well the measured and calculated frequencies match. (The value you calculated from the distance to the hole may be a little higher than what you measured with the scope, because the calculation is somewhat oversimplified. But, the difference between them should be no more than 10% of the smaller frequency.)

Oscilloscopes can pick up undesired signals, called electrical noise, broadcast by our industrial civilization. To look at this, connect a single wire to the red input terminal, and hold the other end in your hand by the bare metal. You and the wire are now acting as an antenna. Determine the signal's frequency. From the frequency, decide on the probable source of this electrical noise - if necessary, use the electromagnetic spectrum chart on the wall in the back room.

Sometimes you want your signal to move the spot horizontally rather than vertically. To do this, turn time/cm all the way counterclockwise to EXT X, disconnecting the scope's sweep generator. Have brilliance no higher than necessary - a stationary spot is especially prone to damaging the screen. Connect the signal from the function generator (red terminal) to EXT X on the oscilloscope, and ground (black terminal) to[pic]on the scope. Set the function generator so that its output has a frequency of about 1 Hz. Put attenuation at 0 and variable attenuation around the middle of its range. Notice how the spot now moves horizontally rather than vertically.

Oscilloscopes can be used to compare frequencies by putting one signal into EXT X and another signal into the regular y input. A good standard for comparison is commercial alternating current, since its frequency is a very accurate 60 Hz. From the AC (yellow) terminals of a gray power supply, run wires to both red and black on input. Start with the knob all the way counterclockwise. Leave the function generator on the EXT X, and set it for a 60 Hz sine wave. Adjust the amplitudes of both signals and volts/cm on the scope so that the vertical size of the display roughly equals the horizontal size. If the two frequencies were exactly equal, you would see a perfectly stationary circle or ellipse. If they were nearly equal, the circle would seem to rotate. With unrelated frequencies, you just get a mess. Adjust the oscillator's frequency until you get as stationary a circle as you can. How far off is your oscillator, both in hertz, and as a percent of the true frequency?

The circle you've been looking at is just the simplest of a whole family of curves called Lissajous figures. One of these figures is formed whenever the two frequencies are exactly in the ratio of two small integers. For example, you got the circle when they were 1 to 1. Adjust the oscillators so that the vertical frequency is exactly twice the horizontal frequency. Sketch the 2:1 Lissajous figure. Also, find and sketch the 3:2 figure. These sketches should show all parts of the figure: As they appear to rotate on the screen, there are positions where some parts are lined up on top of other parts. Do not sketch them in these positions.

Might want some of thei for lab 10 (or 9):

To read voltages from the oscilloscope:

1. Be sure variable volts/cm (a red knob at the lower left) is all the way in the direction of the red arrow by it.

2. Read the peak to valley distance from the screen: 4.4 cm in the example shown.

3. Multiply this distance by the setting of the volts/cm knob. (The black knob in the same place as the red one you just checked.) In the example, (4.4 cm)(2 V/cm) = 8.8 V.

Experiment 11: RC and LC Circuits

Part I: RC Circuits

You will check on whether the time constant and voltage observed for an RC series circuit match those predicted by theory. These things are observed on a graph of capacitor voltage versus time created by a computer interfaced to a voltage sensor.

Select a capacitor somewhere around 1 μF. Calculate the resistance needed with it to make the time constant at least 20 ms, and obtain such a resistor. (Larger is fine as long as you don’t have more than a few hundred thousand ohms. Do not trust the label on the drawer the resistor was in; read the resistor itself.) Record R and C.

Connect things as shown. The grounds (black terminals) of the voltage sensor and power supply must be together.

Get a graph showing how voltage builds up after you close the switch:

In the computer's Science Workshop window, double click on science workshop. On the picture of the interface which appears, drag the analog plug from near the right to channel A. (Using channels B or C seems to cause problems.) Double click on voltage sensor. Drag the graph icon to channel A. Then, click on the interface box to put it back on top. Click on sampling options near the left, then run "periodic samples" up to 500 Hz. Click ok.

Put the knob on the power supply one notch clockwise from its off position. (Off is fully counter clockwise.) With the switch off, turn on the power supply. Short out the capacitor for a moment (connect a cord from one wire to the other) to be sure it is discharged.

Click on REC at the upper left of the screen. When REC gets fainter, close the switch. After a second or two click STOP. Open the switch back up.

Click on the graph to bring it back on top. Click the button near the upper right to make it bigger. Your data points might originally be beyond the edges of the picture; use the + and - buttons at lower right repeatedly to change scales. Also, if you click on the magnifying glass:

then outline part of the graph with the mouse, the outlined part will be magnified to fill the graph when you release the mouse's button. Using these functions, get a good look at the part of the graph which shows the voltage building up after the switch was closed, like the picture below.

If you need to do another run, don't bother setting up another graph. Just downsize the one you've got, take the data, and then bring the graph back up. The new data will automatically be there.

You don't need a hard copy, but have the instructor approve your data before deleting the graph.

On this graph, observe the following:

The "smart curser" will be helpful: After clicking on , click a point on the graph then hold down the button and move the mouse. The vertical and horizontal difference between the point you clicked and the mouse's current position is shown next to the axes of the graph.

1. Emf: The capacitor might have started out with some charge on it. Observe the difference between its voltage just before the switch was closed and the final voltage it builds up to. Record this as ε, the full-charge voltage on the capacitor.

2. The time constant. (Remember what the time constant means: Read how much time passes from closing the switch until the voltage is 63% of its final value.)

3. The voltage (above the initial) when t = half the measured time constant.

Calculate these last two things, and compare to the measured values. Taking the calculated values as exact, and the measured values as 10% uncertain, do they agree?

Part II: LC Circuit:

Use the same circuit, except replace the resistor with a 5 mH inductor. Change the sampling rate to 10,000 Hz; you have to exit Science Workshop, then come back in and set things up again. When you click on record, close the switch immediately after the writing gets faint. Then, wait about 20 or 30 seconds before you click stop. When you look at the graph, there will be only a second or two of data, but at this high sampling rate, the computer needs that long to absorb it from the interface.

Look at how the voltage across the capacitor varies with time just after the switch is closed: When you close the switch, the pulse of voltage hits the circuit like striking a bell; it makes the circuit "ring" for a little while. So, the curve should suddenly spike up when you closed the switch, then briefly oscillate around the final voltage with a decreasing amplitude. Read the period from the graph. (Use the smart cursor again. For best accuracy, measure across several periods and divide.) If the oscillations damp out before a couple of periods are completed, try turning up the power supply somewhat.

Calculate what the circuit's period ought to be from its capacitance and inductance. Does this agree with what you saw within 10% ?

Experiment 12: An AC Circuit

PART ONE: AC Voltages

In an AC circuit, we wish to see if calculated and observed voltages match, and if the phases of the voltages are as predicted. A resistor, inductor and capacitor are connected in series across a function generator. The peak to valley voltage and the phase of the voltage across each of these is observed using an oscilloscope.

CALCULATIONS: Unless you are informed otherwise, R = 500 Ω, L = 5.0 mH, C = .01 μF. These are to be placed in series with a 2 volt, 15,000 Hz emf. Find the impedance of the circuit, the current, and the potential difference across each of the three circuit elements: the capacitor, resistor and inductor. (Solve this problem mathematically.)

EXPERIMENTAL PROCEDURE: Now, check your results for VL, VC, and VR experimentally. Normally, V and I stand for RMS values. However, peak to valley values are easier to read from an oscilloscope and behave the same way. So, if you set the source for 2.0 V peak to valley, the peak to valley values for VL, VC, and VR should match what you calculated.

Obtain the resistor, inductor, and capacitor, set up the circuit at left, and adjust attenuation on the function generator to a 2.0 V output as measured by the scope. The three boxes are the resistor, inductor, and capacitor. Which one is which is not important. The extra wire from external trigger, together with setting the scope's triggering selector on external (bottom button out) will synchronize the scope with the function generator's voltage, enabling you to see the phase differences between the different components of the circuit. All of the trig selector's other buttons should also be out, and trig level should be on auto.

To set function generator to 2V: To measure VL, VC, and VR:

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The frequency dials on some of the function generators are a little off. Check it: At 15,000 Hz, a period should be 6 2/3 cm long at 10 μs/cm. If not, ask the instructor if it's the function generator or scope which is miscalibrated.

When the function generator has been set to 2 V, move the scope's input wire so that it now displays VL, VC, or VR, as shown at right. If the start of the graph is beyond the left edge of the screen, adjust x-shift so you can see it. If it doesn’t start right on a peak, a valley or the centerline, ask the instructor to adjust trig level so that it does. (Then, don't touch trig level again.) Then,

- Record the peak to valley voltage, being sure variable volts/cm is on calibrate.

- Sketch the trace, paying attention to where in the cycle it begins at its left.

Next, switch the circuit element you just measured with one of the others by unplugging them then reconnecting them in each other's place. Repeat the same measurements. (Do not change how the wires are arranged; just switch components under the wires. You must keep the scope's ground connected to the function generator's ground.)

Repeat, to look at the last of the three.

Get the instructor’s approval of your sketches before going on; there are often problems.

In your conclusion,

1. Comment on whether your measured and computed values for the three voltages agree with each other if the measured values have a 10% uncertainty. (A typical tolerance when components like these are manufactured.)

2. Comment on whether the voltages had the expected phase relationship:

a. Should VL lead or lag VR? Is this what you see?

b. Should VC lead or lag VR? Is this what you see?

PART TWO: Resonance

You will see if a circuit’s calculated resonant frequency matches the observed resonant frequency.

Use the same circuit from part I, except switch to a 47 Ω resistor. (Less resistance means more current around the resonant frequency, making the peak stand out better on your graph.) Have the scope measuring the voltage across the resistor. By I = V/R, the resistor's voltage is proportional to the current, which is what we're really interested in.

Take data for a graph of resistor voltage as a function of frequency:

- Turn the frequency and frequency multiplier knobs on the function generator through their whole range, and notice how the amplitude of the current changes. Locate the approximate value of the resonant frequency. Record the frequency and resistor voltage.

- Go just a couple of kHz below this, and record them again. Repeat a couple of kHz above your first frequency. (This is the part of the graph we are the most interested in, so we want fairly closely spaced points here.)

- Take more widely spaced points from 100 Hz to 100,000 Hz, to fill out the rest of the graph.

Read the resonant frequency from your graph. Also, calculate it from L and C. If your observed (graph) value has a 10% uncertainty, do these agree?

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