Ch20.lab_SolomonE

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= = =Kirchoff's Rules Lab=


 * __ Objective(s) __**** : ** How do currents split in multi-loop circuits?


 * __ Equipment: __** Resistors, Wire Leads, D-cell Batteries, Several Digital Multimeters (DMM), (2) power supplies, 3 – 4 resistors, connecting wires


 * __Procedure:__ **

1. Go to a circuit 2. Use ammeter/voltmeter to collect data regarding experimental voltage/current for given resistors 3. Draw a diagram of the complex circuit 4. Calculate theoretical current 5. Compare this data to the experimental data and analyze 6. Continue to do this for the remaining three circuits

The current through a complex circuit should split at points of a circuit becoming parallel. This follows the rule that I1=I2=I3... for a series circuit and I1=I2+I3... for a parallel circuit.
 * __Hypothesis:__**

Circuit A
 * __Circuits:__**
 * The top 100 ohm resistor is marked with a subtext two. The lower 100 ohm resistor is marked with a subtext one. The right 200 ohm resistor is marked with a subtext two. The left 300 ohm resistor is marked with a subtext one.

Circuit B

Circuit C

Circuit D


 * __Data:__ **


 * Current A ||  ||   ||   ||   ||   ||
 * || Resistance (Ω) || Voltage (V) || Current (experimental) (A) || Current (theoretical) (A) || Percent Error (%) ||
 * 1 || 300 || 3.96 || 0.0129 || 0.0125 || 3.100775194 ||
 * 2 || 300 || 2.39 || -0.0078 || -0.01 || -28.20512821 ||
 * 1 || 100 || 1.54 || 0.0157 || 0.007 || 55.41401274 ||
 * 2 || 100 || 2.05 || 0.0206 || 0.0226 || 9.708737864 ||
 * || 560 || 4.46 || 0.008 || 0.009 || 12.5 ||
 * Current B || 1000 || 6.14 || -0.0062 || -0.0062 || 0 ||
 * || 750 || 1.17 || -0.0015 || -0.0015 || 0 ||
 * || 500 || 3.84 || -0.0077 || -0.0077 || 0 ||
 * Current C ||  ||   ||   ||   ||   ||
 * || 1000 || 9.2 || 0.0096 || 0.0095 || 1.041666667 ||
 * || 820 || 0.48 || 0.0006 || 0.0006 || 0 ||
 * || 680 || 5.5 || 0.0079 || 0.0081 || 2.53164557 ||
 * || 560 || 0.54 || 0.0009 || 0.00086 || 4.444444444 ||
 * Current D ||  ||   ||   ||   ||   ||
 * || 100 || 4.87 || 0.045 || 0.0435 || 3.333333333 ||
 * || 200 || 1.97 || 0.0094 || 0.0083 || 11.70212766 ||
 * || 47 || 1.67 || 0.0394 || 0.0352 || 10.65989848 ||
 * || 100 || 4.87 || 0.045 || 0.0435 || 3.333333333 ||
 * || 200 || 1.97 || 0.0094 || 0.0083 || 11.70212766 ||
 * || 47 || 1.67 || 0.0394 || 0.0352 || 10.65989848 ||

__A__ Matrix 0 0 1000 300 6 -1 1 1 0 0 0 0 0 -1 0 1 1 1 0 0 0 1 1 0 -1 0 0 560 0 0 100 0 6 0 560 0 300 0 300 6
 * __Calculations:__ ** Some values may be slightly off because of sig figs.

__B__

Matrix 750 1000 0 5 -1 1 1 0 750 0 500 -5

__C__ Matrix -1 1 - 1 1 0 1000 820 0 0 0 10 1000 0 0 680 0 15 0 0 -1 1 1 0 1000 0 0 0 560 10

__D__

Matrix -1 1 1 0 100 200 0 6 100 0 47 6

Sample Matrix Equation (Circuit B) I1=I2+I3 -.0077=-.0062-.0015

Sample Percent Error (Circuit B) (|Theoretical-Experimental|/Experimental)*100 ((|-.0077+.0077|)/-.0077)*100 0%

__** Discussion Questions **__

1. Are the experimental values of the currents for the entire laboratory generally larger or smaller than the theoretical values expected for the currents?

It would seem that most of the experimental current values are larger than the theoretical values.

2. It was pointed out in the laboratory that some error might be caused by neglect of the internal resistance of the //emf//. Would the internal resistance cause an error in the direction shown in your answer to question 1? State your reasoning for the direction of any error caused by the internal resistance.

When taking emf into account, the total voltage of the battery/batteries would be smaller. Therefore, the theoretical current, using Ohm's Law, would be smaller. In my calculations, the theoretical value was smaller. This confirms the theory about the error analysis presented here. With the experimental values, we measured actual voltage from the battery, thus taking into account emf (in the end there will be less voltage). And yet again, the theory presented is confirmed.

3. An ideal ammeter has zero resistance. Real ammeters have small but finite resistance. Would ammeter resistance cause an error in the proper direction to account for the direction of your error indicated in question 1? State your reasoning.

This results in more resistance through the circuit. More resistance equals less current. Current is smaller theoretically, however, which makes no sense according to the resistance of an ammeter. The added resistance would not effect theoretical values, because they do not take this resistance into account. According to this, experimental values should be smaller.

4. The connecting wires in the experiment are assumed to have no resistance, but in fact have a finite resistance. Would this error be in the proper direction to account for the direction of the error stated in your answer to question 1? State your reasoning.

This is the same scenario as in the previous question. The current should be less in experimental scenarios because resistance is larger. This is again proven wrong through the results.

5. What is the meaning of any current values obtained in your solutions that are negative?

The current values, to begin any set of equations, are represented with a direction in a given diagram. A negative current solution simply means that the current flows in the opposite direction than is represented in the equations.

My hypothesis was proven correct. It was stated that at points of a circuit becoming parallel, current will split. This can be summed as current splits at junction points. The values of current make sense also according to Ohm's Law and the idea that charge takes the path of least resistance. In terms of Ohm's Law, the current paths in parallel circuits will all receive equal voltage, but not necessarily equal resistance. Therefore, the current will need to be present, but not equal. In terms of the idea of charge wanting least resistance, there was a greater magnitude of current flowing through paths of less aggregate resistance.
 * __Conclusion:__**

Error can be seen as present in this lab at a number of places. The percent error in all circuits except for B asserts that there was indeed error. This may have resulted from, as mentioned in the discussion questions, emf, ammeter resistance, and wire resistance. These three values are all assumed to be negligible. However, this is clearly not true because of the presence of error through experimental/theoretical current. The emf existed because the used batteries were, well, used. They were not completely new and through their use were worn down even further. This decreased voltage in the circuit and therefore decreased current because of the stable resistance values. Also was the resistance from the ammeter/wire (they served the same function). The added resistance of these two values would prove to decrease current experimentally. The emf, though, would possibly (probably) affect the current more than that of the ammeter/wire.

This lab, theoretically, required that there be only one variable. However, all three are in question. To address this, there should be a constant power source in all of the currents and not just some of them. This would eliminate emf. Also, the resistance in the wire should be accounted for, or the wires should be thicker/shorter to reduce resistance as much as possible. In real-life scenarios, currents need to be split all the time, especially when certain appliances can only handle a certain flow rate. And since the resistance is not negotiable, and the voltage is rarely ever so, the manipulation lies in the current. The knowledge of how to split current through junction points could prove crucial in operating a household circuit.

=Ammeter/Voltmeter Lab=

Hypothesis: Shown below in picture in form of color coded circuits on left side. Also shown in terms of arrow-tails. Keep in mind that more arrow heads equals higher flow rate. This equates to the ordering from highest to lowest of E, F, H, A, G, C/D, B. This was estimated using the idea that the round bulbs have an approximate resistance of 10 ohms and the long bulbs have an approximate resistance of 60 ohms.

Data: Shown below in forms of voltages and corresponding flow rates. Sample Calculations:

__Parallel__ Voltage: (Circuit H) 4.27V=4.14V=4.09V=4.15=4.07V Current: (Circuit H) .38A=.09A+.09A+.1A+.1A
 * These values should be all the same, but this is not an ideal scenario. They are therefore only approximately the same.
 * This one worked quite well

__Series__ Voltage: (Circuit D) 4.48V=3.88V+.58V Current: (Circuit D) 85mA=85mA=85.5mA
 * These values should be all the same, but this is not an ideal scenario. They are therefore only approximately the same.
 * These values should be all the same, but this is not an ideal scenario. They are therefore only approximately the same.

Discussion Questions:



Analysis:

Conclusion: This lab proves the rules of parallel circuits as well as series circuits in terms of the correlation between flow rate and voltage differences throughout the circuit. We have found that these two types of circuits react inherently differently. The series circuit, because of the single continuous conducting loop, has equivalent flow rates throughout. Each part of the flowing charge has to pass through each resistor and each battery cell. However, because of this same principle of a single conducting loop, there are multiple different voltages. On the other hand, the parallel circuit reacts in the opposite manner. The flow rates, because of the multiple conducting paths, are split up (also related to Kirchoff's Rules, but that's a different lab). The voltages in a parallel circuit, however, are all equivalent. In terms of error analysis, this may have stemmed from the resistors used. For one, the bulbs (resistors) were approximated to all be the same depending on the type of bulb. In reality, though, the bulbs will deviate, thus resulting in differences between theoretical and actual values of resistance, flow rate, and voltage. The batteries, too, have an estimated voltage output, which is most likely not the exact voltage output.

=Ohm's Law Lab=

Hypothesis: The relationship between potential difference and current, when resistance is at a constant, is direct. One will increase as the other does. We suspect this because when we had previously increased voltage by adding batteries to a circuit, the bulbs became brighter, thus indicating an increase in current.

Data:


 * Volts (V) || Current (A) || Predicted Resistance (Ω) || Actual Resistance (Ω) || Percent Error (%) ||
 * 0 || 0 || 68 || #DIV/0! || #DIV/0! ||
 * 1 || 0.0142 || 68 || 70.423 || 3.563 ||
 * 2 || 0.0274 || 68 || 72.993 || 7.342 ||
 * 3 || 0.041 || 68 || 73.171 || 7.604 ||
 * 4 || 0.0538 || 68 || 74.349 || 9.337 ||
 * 5 || 0.0689 || 68 || 72.569 || 6.719 ||
 * Avg. Numbers ||  || 68 || 72.701 || 6.913 ||
 * 0 || 0 || 22 || #DIV/0! || #DIV/0! ||
 * 0.5 || 0.0218 || 22 || 22.936 || 4.254 ||
 * 1 || 0.0366 || 22 || 27.322 || 24.193 ||
 * 1.5 || 0.0508 || 22 || 29.528 || 34.216 ||
 * 2 || 0.0755 || 22 || 26.490 || 20.409 ||
 * 2.5 || 0.0958 || 22 || 26.096 || 18.618 ||
 * Avg. Numbers ||  || 22 || 26.474 || 20.338 ||
 * 0 || 0 || 10 || #DIV/0! || #DIV/0! ||
 * 0.5 || 0.071 || 10 || 7.042 || 29.577 ||
 * 1 || 0.1138 || 10 || 8.787 || 12.127 ||
 * 1.5 || 0.133 || 10 || 11.278 || 12.782 ||
 * 2 || 0.159 || 10 || 12.579 || 25.786 ||
 * 2.5 || 0.1869 || 10 || 13.376 || 33.761 ||
 * Avg. Numbers ||  || 10 || 10.613 || 6.125 ||
 * 0 || 0 || 60 || #DIV/0! || #DIV/0! ||
 * 1 || 0.0364 || 60 || 27.473 || 54.212 ||
 * 2 || 0.0485 || 60 || 41.237 || 31.271 ||
 * 3 || 0.0613 || 60 || 48.940 || 18.434 ||
 * 4 || 0.075 || 60 || 53.333 || 11.111 ||
 * 5 || 0.084 || 60 || 59.524 || 0.794 ||
 * Avg. Numbers ||  || 60 || 46.101 || 23.165 ||
 * 3 || 0.0613 || 60 || 48.940 || 18.434 ||
 * 4 || 0.075 || 60 || 53.333 || 11.111 ||
 * 5 || 0.084 || 60 || 59.524 || 0.794 ||
 * Avg. Numbers ||  || 60 || 46.101 || 23.165 ||

Graph:



Calculations:

__Resistance Calculation__



__Percent Error Calculation__



__Percent Difference Calculation__



Discussion Questions:


 * 1) In terms of experimental data, how is resistance defined and what are its units?

Resistance is defined as the slope of the progression lines. This means that it equals the V/A.


 * 1) Imagine that you had a third resistor that has a much smaller resistance than the ones used in the lab activity.
 * 2) Sketch a graph of pressure difference vs. flow rate that shows your 2 original resistors and this new resistor (sketch them on the same axes). Clearly label the lines.




 * 1) Explain why you drew it this way.

Because the resistance is equal to the slope, the new slope will be much smaller.

In order to keep the same resistance, the flow rate would need to change in direct
 * 1) How would the flow rate through this resistor change as the pressure difference decreases?


 * 1) Assume that resistor A has 10 times the resistance of resistor B. What would a graph of resistance vs. current look like for these two resistors (sketch them on the same axes)? What about a graph of resistance vs. voltage? Justify your answers.



Examine the graph of electric pressure difference vs. flow rate below. The resistor is Ohmic This is shown through the linear fit according to the equation of Ohm’s Law.
 * 1) Is this resistor Ohmic or non-Ohmic?

R= V/I R= 5V/1A R= 5 Ohms
 * 1) What is the resistance of the object from which this data was collected? (Show your work.)

Conclusion:

The hypothesis that I suspected asserted a direct relationship between electric potential difference and current. If this were true, then the graph of the two would look like a line increasing on the x-axis as it increased on the y-axis. This was correct, as shown in the graph of voltage vs. flow rate. When testing this, we were able to insert a certain voltage as the power source and then test the circuit to see flow rate, which was equal on both sides of the resistor in the circuit. When, for example we tested the resistor that was supposed to be 68 Ohms according to the label, we inserted a voltage of 1 V and ascertained a current of .0142 I. When we tested 2 V, we got a current of .0274. This trend continued for the remainder of this resistor as well as the other resistors. For sources of error, we certainly had some. This was able to be dissected and shown through the deviation of the resistor’s predicted and actual resistivity. Even though the resistors are allowed to deviate, some of the percent errors were disproportionate to what the deviation allowed, which was 5%. For the first resistor, for example the actual measured value of the resistor was an average of 6.913% off, at an average of 72.701 Ohms. This may have resulted from a few variables, namely the possible additional resistance from the voltmeter or the bulbs’ inconsistency in their resistivity. With regards to the first possible error source, the voltmeter is advertised as without resistance when placed on the setting measuring flow rate. However, we as a class have never proven this to be truthful. Therefore, the (if minimal) resistance could possibly cause slight deviations in the measured vs. predicted resistances. With regards to the second possible source of error, the bulb has undergone a lot of previous stress, as well as the fact that these bulbs are mass-produced. It would come as no surprise if the manufacturers skimped a bit on the production and/or materials. This, in conjunction with previous stress, could cause a different resistance than anticipated. The fixes for these sources of error would not be complicated to address. To get rid of the first possible source of error we could measure the resistance from the voltmeter and take it into account. Possibly it is at no resistance, which in and of itself would quell that source of error. For the next source of error we could use fresh bulbs for each set of tests. This would be tedious and a pain to say the least, but at least it may preserve the integrity of this experiment. In a real life scenario this lab also proves to be one of decent importance. Resistors are used every day to make circuit boards or house circuits. It would certainly help, if a circuit happened to malfunction, to be able to address the problem on your own.