Group2_8_ch21

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=toc= =Lab: Multiple Capacitors=

Period 8 By: Chris Hallowell, Ryan Listro, Eric Solomon


 * OBJECTIVE:** Determine the relationship between voltage on a capacitor(s) and the format in which it/they are set up**.**


 * HYPOTHESIS:** According to conventions regarding capacitance in relation to voltage, there should be more voltage on capacitors in parallel versus capacitors in series. Also, capacitors in series should not deviate in accumulation of charge, only in capacitance and voltage. This should be because the charge off of one capacitor goes onto the adjacent capacitor.

1. Set up the following circuit (schematic diagrams shown below). 2. Calculate the voltage across each capacitor and resistor (when available). 3. Repeat step 2 if multiple trials are desired. Make sure you discharge the capacitors prior to performing additional trials. 4. Record the voltage values in an excel spreadsheet and compare with experimental values to determine the quality of your results.
 * PROCEDURE:**




 * DATA:**


 * SAMPLE CALCULATIONS:**


 * DISCUSSION QUESTIONS:**

1. How does the voltage on the individual capacitors in series compare to the voltage when they are in parallel? In series, just like with resistors, the potential difference from across a capacitor will split up. In parallel, the same voltage is received by each branch. This causes the voltage to be equivalent to the power source. In series, this results from the fact that charge can not pass through a capacitor. Therefore, whatever charge received by the first capacitor is received by all of them. And through the law of capacitance as defined by C=Q/V, with constant capacitance and charge, the voltage //must// deviate. This is not true, however, in parallel, where each branch can receive different amounts of charge, therefore substituting variables and allowing voltage to remain constant.

2. What is the effect of the resistor on the voltage of the capacitors? When accounting for equivalent resistance or equivalent capacitance, a circuit must be broken up into segments. There are two scenarios where there would be an addition of a resistor, in series or in parallel. In parallel voltage is not affect, as it never is in a parallel circuit. In series, however, the resistor is another component to the circuit which must be taken into account. This is where the circuit must be segmented. Take a look at circuit three, for example. The branched portion acts as one area where voltage is received, as it will be the same in both branches. The lone resistor is another segment. The voltage must be split between the two segments. Therefore, a resistor will take some of the voltage from the capacitor as it would from another resistor in a series circuit.

3. How does the potential difference of the capacitors in series compare to the voltage of the source? What about when they are in parallel? This is a simply answered question, exemplified perfectly by the discrepancies between circuit one and circuit two. In circuit one, which is an example of a series circuit, the voltage is calculated as such: Vtotal = V1+V2... This trend would continue for an infinite amount of capacitors, depending on the amount implanted into the circuit. In parallel, voltage is calculated as follows: Vtotal = V1 = V2... This trend also continues in a similar manner.

4. How is the amount of voltage on the individual capacitors related to the known capacitance? This question must be broken up, again, into scenarios of series circuits and parallel circuits (with a complex circuit behaving like a series circuit with the parallel portion acting as a single part of a series). As previously stated, the voltage on a capacitor in parallel does not deviate because charge received is variable. This follows the general parallel circuit trend of the voltage of the power source equaling that of each branch. In series, the more charge received by a capacitor, the higher the voltage will be. This is shown by a derivative equation of the normal capacitance equation, V=Q/C. Therefore, the higher the capacitance, because charge is constant, the lower the voltage.

5. Discuss the effect of switching out your bigger capacitor for one that is 10 times as big. Speaking quantitatively, having a larger capacitor will either lower its own voltage or have no effect. This depends on the circuit in question. In circuit one, the capacitance is larger, and therefore the same charge on it will not have as large of an effect (V=Q/C). The smaller capacitor will receive a larger portion of the voltage. In circuit two, which is parallel, more charge will be allocated to the large capacitor in comparison to the smaller capacitor, but the voltages will both remain equal to the power source's voltage. In circuit three, in which the larger capacitor was used, the voltage will be split up like it normally would in a parallel circuit, with one caveat. The effect of the branched portion, because of the increased capacitance, will increase. Because of this there will be a larger voltage amongst this branched portion. In circuit four, the same thing will happen yet again. The equivalent capacitance will be increased in the branched portion. Therefore, it will receive more voltage.

Overall, we can conclude that our hypothesis was, in fact, correct. We saw that in circuit 1, which was a series circuit, the voltage over the two capacitors added up to equal the total experimental voltage. As a result, the voltages over the capacitors in circuit 1 were much less than the voltages over the capacitors in circuit 2, which was parallel. The voltages over the capacitors in circuit 2 were both very close to being equal to the experimental voltage of the power supply. The findings in circuits 1 and 2 also proved that in parallel circuits, both capacitors fill up to the maximum voltage supplied by the power supply. The findings also proved that in series circuits, the capacitor split the voltage. However, the amount of voltage over each of the capacitors depends on the capacitance of the individual capacitor. Because the amount of charge that each capacitor receives is equal, the voltage is found using the equation V= Q/C.
 * CONCLUSION:**

One source of error in this lab came from the fact that the true voltage of the power source was a little over 5 V. When we connected the voltmeter to the power supply, it showed that the true voltage was either 5.1 V or 5.11 V, rather than the expected 5 V. This definitely affected the percent error on several of our measurements. Another source of error came from the fluctuating readings on the voltmeter. When we connected the voltmeter to the locations throughout the circuits, the numbers on the screen would often change slightly every few seconds. Rather than waiting an extended period of time to get the perfect reading, we tried to get a number that it seemed to show the most and we used that as our measurement. However, the numbers that flashed up were very close to one another so this probably only cause a small amount of error.

In order to fix these sources of error, it would have been very useful to have a power source that we knew would supply 5 V. This would have greatly lowered our percent error for many of the capacitors and resistors. In addition, if we had waited a little more time for the voltmeter to stop fluctuating, we most likely would have gotten measurements that were a little more concrete and accurate. Overall, this lab is very important to understand because it is proving many of the formulas that we have discussed and worked with over the past few weeks. It is also backing up the ideas we learned in regards to how capacitors split up and share the voltage through series, parallel, and other complex circuits.

=Lab: Magnetic Field Strength= Period 8 By: Chris Hallowell, Ryan Listro, Eric Solomon


 * OBJECTIVE:** To determine the relationship between magnetic field strength and distance from the source of magnetism.


 * HYPOTHESIS:** There will be an inverse relationship between magnetic field strength and the distance from the source. For example, when distance increases, the field strength will decrease. Conceptually this makes sense because as the distance increases, the magnetism should be felt less and less. A similar analogy for magnetic field strength can be electric field strength.

1. Tape the measuring tape or meter stick to the table, and tape the Magnetic Field Sensor to a convenient location. The sensor should be perpendicular to the stick, with the white spot inside the rod facing along the meter stick in the direction of increasing distance. Carefully measure the location of the sensor on the meter stick. This will be your origin for all distance measurements.
 * PROCEDURE:**

2. As a convenient way to measure to the center of the magnet, and to ease handling of the small magnets, allow the two magnets to attract one another through an index card, about 0.5-cm from either edge near the corner. The magnets should stay in place on the card. The card itself will serve to mark the center of the magnet pair.

3. Connect the Magnetic Field Sensor to Channel A of the interface. Set the switch on the sensor to //1x//.

4. Open Data Studio and choose “Create Activity”. Click on “Setup” and add a Magnetic Force Sensor to the icon of the interface. For a display, click on “314 Digits”, which will show the magnetic field strength in Gauss.

5. Zero the sensor when the magnets are far away from the sensor in order to remove the effect of the Earth’s magnetic field and any local magnetism. The sensor will be zeroed only for this location, so instead of moving the sensor in later steps, you will move the magnets. > 6. Now you are ready to collect magnetic field data as a function of distance. > 7. Continue taking readings every 0.5 cm until you get no more change in Magnetic Field Strength.
 * 1) Move the magnets far away from the sensor.
 * 2) When the reading in the meter window is stable, click “Tare” on the sensor.
 * 1) Click “Start”to begin data collection.
 * 2) Place the card with the magnets against the meter stick, 2.0 cm from the Magnetic Field Sensor, so the card is perpendicular to the meter stick. Measure from the card to the center of the Magnetic Field Sensor.
 * 3) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">The current magnetic field measurement is shown in the meter window. If necessary, reverse the magnets so the reading is positive, and reposition the card 2.0 cm from the sensor.
 * 4) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">Carefully measure the distance of the card to the sensor.
 * 5) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">Record your data in a table.


 * MATERIALS:** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">Magnetic Field Sensor, Data Studio and Science Workshop Interface, index card, Meterstick, neodymium magnet.

what a professional


 * DATA:**
 * Distance (m) || Magnetic Field (G) || Experimental Magnetic Field (T) || Theoretical Magnetic Field (T) || Percent Error (of exponent) ||
 * 0.03 || 59.1 || 0.00591 || 7.40741E-10 ||  || 29.56666667 ||   ||
 * 0.035 || 39.7 || 0.00397 || 4.66472E-10 ||  ||
 * 0.04 || 27.1 || 0.00271 || 3.125E-10 ||  ||
 * 0.045 || 20.4 || 0.00204 || 2.19479E-10 ||  ||
 * 0.05 || 15.6 || 0.00156 || 1.6E-10 ||  ||
 * 0.055 || 12.6 || 0.00126 || 1.2021E-10 ||  ||
 * 0.06 || 10.5 || 0.00105 || 9.25926E-11 ||  ||
 * 0.065 || 8.7 || 0.00087 || 7.28266E-11 ||  ||
 * 0.07 || 7.3 || 0.00073 || 5.8309E-11 ||  ||
 * 0.075 || 6.1 || 0.00061 || 4.74074E-11 ||  ||
 * 0.08 || 5.8 || 0.00058 || 3.90625E-11 ||  ||
 * 0.085 || 5.1 || 0.00051 || 3.25667E-11 ||  ||
 * 0.09 || 4.84 || 0.000484 || 2.74348E-11 ||  ||
 * 0.095 || 4.75 || 0.000475 || 2.3327E-11 ||  ||
 * 0.1 || 4.2 || 0.00042 || 2E-11 ||  ||
 * 0.105 || 3.9 || 0.00039 || 1.72768E-11 ||  ||
 * 0.11 || 3.85 || 0.000385 || 1.50263E-11 ||  ||


 * GRAPH:**




 * SAMPLE CALCULATIONS:**



<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt; line-height: 22px;">1. On Excel, create a graph of magnetic field //vs.// the distance from the magnet. Produce a best fit line using a “Power” function. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">Shown above under "Graph."
 * DISCUSSION QUESTIONS:**

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">2. Compare your data to the ideal inverse-cube model: <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;"> <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">a) What value do you get for the constant, //A//, or [(//m//0 2 //m// ) / (4p)]? How well does this agree with the value that the rest of the class measured? Our constant was 3x10^-6. When compared to other groups, it is generally larger. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">b) What exponent do you get for d? How well does this agree with the ideal expression? <span style="font-family: Arial,Helvetica,sans-serif;">For d, we got an exponent of -2.113. Theoretically,however, this should be -3. Because of the higher than normal error, there is not much agreement with the ideal expression. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">c) From the above comparison, does your magnet show the magnetic field pattern of a dipole? <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">The magnet does show the magnetic field pattern of a dipole because is somewhat follows the inverse cubic relationship.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt; line-height: 22px;">3. Use your value of //A// to determine the magnetic moment //m// of your magnet. m=15Am^2 See work under sample calculations.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt; line-height: 22px;"> 4. The units of //m// may suggest a relationship of a magnetic moment to an electrical current. In fact, a current flowing in a closed loop is a magnetic dipole. A current //I// flowing around a loop of area p//r//2 has a magnetic moment //m// = //I// p//r//2. If a single current loop had the same radius as your permanent magnet, what current would be required to create the same magnetic field?

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt; line-height: 22px;">r=?? <span style="font-family: Arial,Helvetica,sans-serif;">m = I( π)(r^2) 15= <span style="font-family: Arial,Helvetica,sans-serif;">I( π)(r^2) I=15/(πr^2) I=?

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt; line-height: 22px;">5. Discuss the precision of your data, referencing the correlation coefficient to support your conclusion. <span style="font-family: Arial,Helvetica,sans-serif;">According to our correlation coefficient, which was .984, our data was very precise.


 * CONCLUSION:**

Hypothesis As represented by our data and the curvature of the graph, our hypothesis was decently correct. We did not achieve the exponent of negative three (indicating inverse //cube//), which we had at first predicted. However, we did achieve a fairly similar relationship. Originally we had indicated this hypothesis based on the fact that, logically, magnetic field should decrease with more distance. From my experiences, this seems to be an easily definable characteristic of fields of this nature. For example, gravity will have a greater affect on an object the closer it is to a source. The same applies for electricity. The specific inversely cubic relationship derives from the equation which has been found to be true. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;"> Error As we had originally thought (albeit hadn't written), there would be massive amounts of error. As indicated from the data, when the theoretical variables and other such numbers are plugged into the aforementioned equation, we had produced infinitely smaller values than were described by the actual experiment. This comes from a number of expected sources, but one in particular. The particular source happens to be the Earth itself, which has inherently flawed this procedure. The Earth has its own magnetic field that interferes with our own field produced by the neodymium magnets. This will drastically throw off the results of the experiment by radically increasing the field. Also in play and unaccounted for are the other magnetic objects. These range from laptop components to various metallic surfaces, and will in turn throw off the results. The last important source of error would be the lack of a "zero-ing" button on the sensor. A zero button would allow for all of the surrounding fields to be taken into account and set that as the default level of magnetic activity. This would effectively eliminate all outside sources of magnetic field and allow for the sole association of magnetic field to be focused on the magnets in question. Application This phenomenon can be seen in many places everyday. For example, in scrapyards magnets are used to transport equipment from one area to another. The strength of these magnets must be gauged prior to implementing them or else they might not be strong enough or unnecessarily strong. Magnets are also used in everything from medicine to security scanners. They can detect harmful magnetic fields within the human body, such as foreign objects that have been mistakenly placed inside of the body. Magnetic fields are also used to see if any harmful objects are passed through a security checkpoint. The scanner must allow for some activity, though, because of the body's natural field, the Earth's magnetic field, amongst others. Magnets in general are ever-present and play a crucial role to society, despite the fact that their importance is often underestimated.

=LAB: Magnetic Force on a Wire= By: Chris Hallowell, Ryan Listro, Eric Solomon 12/1/11

HYPOTHESIS: The magnetic force will be directly proportional to the strength of the magnetic field, the current, the length of the conductor, and the angle between the current and the magnetic field. When the magnetic force increases, the other variable will have increased as well. This happens because of the equation, F=B(I)(//l//)(sin ** θ). **

PROCEDURE/MATERIALS: Measuring Force vs. Current 1. Set up main circuit. 2. Using a manual power supply, change the current that goes through the circuit. 3. Observe and record the values of mass, convert to kilograms, and multiply by 9.8 to obtain the force for each trial. Measuring Force vs. Length 1. Set up main circuit. 2. Change the loops that complete the circuit in order to change the length of the conductor. 3. Observe and record the values of mass, convert to kilograms, and multiply by 9.8 to obtain the force for each trial. Measuring Force vs. Number of Magnets 1. Set up main circuit. 2. Vary the number of magnets that surround a part of the circuit to simulate varying the strength of the magnetic field. 3. Observe and record the values of mass, convert to kilograms, and multiply by 9.8 to obtain the force for each trial. Measuring Force vs. sin(theta) 1. Set up main circuit. 2. Connect the loop pieces to a rotating device to change the angle between the current and the magnetic field. 3. Observe and record the values of mass, convert to kilograms, and multiply by 9.8 to obtain the force for each trial.

DATA: Measuring Force vs. Current Measuring Force vs. Length Measuring Force vs. Number of Magnets


 * Theoretical Force (N) || Experimental Force (N) || Magnets || Magnetic Field (T) || Current (A) || Length of Wire (m) || Angle Between Field and Current (degrees) || Percent Error (%) ||
 * 0.000632386 || 0.000588 || 1 || 0.013766667 || 1.98 || 0.0232 || 90 || 7.548571429 ||
 * 0.001264771 || 0.001274 || 2 || 0.027533333 || 1.98 || 0.0232 || 90 || 0.724395604 ||
 * 0.001897157 || 0.00196 || 3 || 0.0413 || 1.98 || 0.0232 || 90 || 3.206285714 ||
 * 0.002529542 || 0.002548 || 4 || 0.055066667 || 1.98 || 0.0232 || 90 || 0.724395604 ||
 * 0.003161928 || 0.003136 || 5 || 0.068833333 || 1.98 || 0.0232 || 90 || 0.826785714 ||
 * 0.003794314 || 0.003626 || 6 || 0.0826 || 1.98 || 0.0232 || 90 || 4.641853282 ||

Measuring Force vs. sin(theta)

GRAPHS/ANALYSIS:

Measuring Force vs. Current This graph shows that current and magnetic force have a direct relationship, thus proving our hypothesis correct.

Measuring Force vs. Length This graph shows that length and magnetic force have a direct relationship, thus proving our hypothesis correct

Measuring Force vs. Number of Magnets This graph shows that magnetic field strength and magnetic force have a direct relationship, thus proving our hypothesis correct

Measuring Force vs. sin(theta) This graph shows that sin(theta) and magnetic force have a direct relationship, thus proving our hypothesis correct

CALCULATIONS:

Analysis: 1. Using the equation of the trendline from the graph of Force vs. Current, find the magnitude of the magnetic field. Show your work.

The Force vs. Current graph trendline was y=.0174x. In this scenario, y is the magnetic force, .174 is BLsin(theta), and x is the current. By using x=.5 as a point, and figuring out that .174/(LIFsin(theta))=B where F=6 and I=1 and L=.0232 we found that the magnetic field was .0125 T.

2. Discuss the relationship of the quantities shown in the graphs. How do they agree with the theoretical relationships?

Each relationship appears to be linear. In a tested scenario, only one variable was changed at a time. All of the other values were the components of the slope. Also, the lack of error further asserts the validity of a direct linear relationship between Force and either B, I, L, or sin(theta).

3. Do the experimental relationships shown in the 4 graphs validate the theoretical relationships? Explain your reasoning using specific evidence from the lab to support your answer.

Since all of the relationships on the graphs appeared linear, and the error was very little, the relationships can be validated. Also, the r^2 value was very close to 1. As previously stated, the slope is formed from the multiplication of the other three components of Force, while the x will represent the variable in any of the direct linear relationships.

4. Is it reasonable to assume that the strength of the magnetic field is directly proportional to the number of magnets? Why or why not? This is reasonable because of the evidence previously stated. Each magnet, also, carries a magnetic field, which is added to the previous magnetic fields in place when a new magnet is added, or subtracted when a magnet is taken away. The graph shows that the force increases with the number of magnets. This is also proven by the fact that the equation describes B as a component to be multiplied in with the other components of Force.

CONCLUSION: As we had originally stated, the magnetic force from a wire's magnitude will be directly proportional to all four of the following: magnetic field strength, current, length of the wire (or section of a wire), and the angle at which the field and current directions interact. This leads to the equation, F=BILsin(theta). As shown by the graphs, calculations, data, and miniscule percent error shown above, this hypothesis has been effectively proven correct. Specifically, our graphs all show a linear relationship with their respective variable.

Although there was very little error here, ranging from 1.59% to 12.55%, some was still present, otherwise our results would have been perfect. The major source which can't be overlooked is the natural magnetic fields surrounding us. They intrinsically effect any and all magnetic experimental data, thus throwing it off from the expected values. Also, the placement of the circuit between the magnets might have been a., not perfectly at the correct angle, or b, slightly more towards one side of the magnets. This would lead one to believe that the placement, if not directly between the magnets, could produce a nonlinear function. This can be clearly seen by analyzing the graphs and seeing that the lines would perhaps more correctly correlate with another type of function.

Magnets and induced magnetic fields make motors function and help provide vital functions to medical equipment as well. Another function is the potential usage in a generator or, as described in various textbook problems, rail-guns. They are also a vital part of certain circuits.

=Lab: Magnetic Field in a Solenoid= Ryan Listro, Chris Hallowell, Eric Solomon Date: 12/8/11

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">1. The objective is stated in the title. What is your hypothesis? (Attempt to answer the question, to the best of your knowledge.) > The magnetic field in a solenoid will equal the number of coils per length times the vacuum permeability constant times the current (B=NµI). Generally, the magnetic field will be largest in the center of the solenoid. > Measure the magnetic strength at different locations inside the solenoid. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">2. Read the entire procedure through. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;"> 3. Design __data table(s)__ in order to record your observations __and__ calculations. You can do this by hand (acceptable) or in Excel (preferable). <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;"> 4. Answer the following questions: > The magnetic field strength will increase as the position approaches the center of the solenoid. > No, the location inside or out of the solenoid will be an influencing factor of the magnetic field. > B=µ(N/L)I > There is no direct relationship to radius. However, there is an indirect relationship between field strength and magnetic flux, where radius would affect the area and field strength.
 * PRE-LAB**:
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">Include the rationale for your hypothesis (Provide detailed reasoning here. This may take the form of a list of what you already know about the topics, with a summary at the end.)
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">How do you think you might test this hypothesis? (What might you measure and how?)
 * Length into Solenoid (m) || Experimental Magnetic Field (G) || Experimental Magnetic Field (T) || Current (A) || N (coil #/length) || Theoretical Max Magnetic Field (T) || Percent Error (%) ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">a) How does the strength of the magnetic field inside a solenoid relate to the position inside?
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">b) Is the magnetic field the same strength at every location within the solenoid?
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">c) What is the magnitude of the magnetic field inside a very long solenoid?
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt; line-height: 24px;">d) What is the relationship of the magnetic field strength and radius of the coil?


 * OBJECTIVE**: The objective is to determine the magnetic field strength in a solenoid.


 * HYPOTHESIS**: The magnetic field in a solenoid will equal the number of coils per length times the vacuum permeability constant times the current (B=NµI). Generally, the magnetic field will be largest in the center of the solenoid both by way of side length as well as radius.


 * MATERIALS**: The materials used in this were a magnetic field sensor, power supply, meter stick, patch cord, solenoid and a ruler.

1. Connect the solenoid to the power supply. 2. Plug the magnetic field sensor into a computer and open the application, EZScreen. 3. Test the magnetic field strength at various points in the solenoid by referencing a ruler next to the solenoid. 4. Record the values for distance and field strength.
 * PROCEDURE**:


 * DATA**:

Determining the Magnetic Field in a Solenoid
 * Length into Solenoid (m) || Experimental Magnetic Field (G) || Experimental Magnetic Field (T) || Current (A) || N (coil #/length) || Theoretical Max Magnetic Field (T) || Percent Error (%) ||
 * 0 || 62.01 || 0.006201 || 3.3 || 3289.473684 || 0.0136 || 9.227941176 ||
 * 0.005 || 78.14 || 0.007814 || 3.3 || 3289.473684 ||  ||   ||
 * 0.01 || 90.1 || 0.00901 || 3.3 || 3289.473684 ||  ||   ||
 * 0.015 || 98.8 || 0.00988 || 3.3 || 3289.473684 ||  ||   ||
 * 0.02 || 106.4 || 0.01064 || 3.3 || 3289.473684 ||  ||   ||
 * 0.025 || 111.4 || 0.01114 || 3.3 || 3289.473684 ||  ||   ||
 * 0.03 || 113.22 || 0.011322 || 3.3 || 3289.473684 ||  ||   ||
 * 0.035 || 116.2 || 0.01162 || 3.3 || 3289.473684 ||  ||   ||
 * 0.04 || 117.24 || 0.011724 || 3.3 || 3289.473684 ||  ||   ||
 * 0.045 || 118.9 || 0.01189 || 3.3 || 3289.473684 ||  ||   ||
 * 0.05 || 120.11 || 0.012011 || 3.3 || 3289.473684 ||  ||   ||
 * 0.055 || 120.9 || 0.01209 || 3.3 || 3289.473684 ||  ||   ||
 * 0.06 || 121.64 || 0.012164 || 3.3 || 3289.473684 ||  ||   ||
 * 0.065 || 122.08 || 0.012208 || 3.3 || 3289.473684 ||  ||   ||
 * 0.07 || 122.09 || 0.012209 || 3.3 || 3289.473684 ||  ||   ||
 * || Maximum --> || 0.012345 ||  ||   ||   ||   ||
 * The data table above shows the experimental magnetic fields that were measured at different lengths of the solenoid. After finding the values, we then used the length and magnetic field data to create a trend line that would give us our maximum value for magnetic field. After finding this value, we then found the theoretical maximum magnetic field using the constant values for current, number of coils and vacuum permeability. Finally, we found the percent error between the experimental and theoretical maximum values of magnetic field.


 * GRAPH**:

Field Strength vs. Position into Sensor The shape of this graph is parabolic because it corresponds to the magnetic field strength in a solenoid. As the sensor just enters the solenoid further, the field strength increases. Once the sensor is in the middle, the strength is at a maximum. When the sensor continues past the center, the field strength decreases. In other words, the graph should start with increasing slope, reach a maximum, and then end with a decreasing slope. This is exactly what is shown in the above graph. The coefficient represents the relationship between distance into a solenoid and magnetic field. It mainly shows that a solenoid is at its maximum in the very center of the solenoid. Also, it shows that this relationship, while being direct (distance in vs. magnetic field strength), is not linear, but is quadratic.

Theoretical Maximum of Magnetic Field
 * CALCULATIONS**:

Percent Error

Percent Difference between our Experimental vs. Class Average


 * DISCUSSION QUESTIONS**:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">1. Did the axial reading change when the sensor was moved radially outward from the center toward the windings on the coil?

The axial reading did change as the sensor was moved radially outwards. As theorized by the hypothesis, the reading would be greatest not only at the center of length, but also at the center of the circle forming the solenoid. As the sensor was moved towards the sides, the reading would lessen. This proved that the reading was greatest when in the very center, or equidistant from the side of the circle at all angles.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">2. Was the axial reading different from the reading in the middle of the coil when the sensor was inside but near the ends of the coil? Why?

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">Yes, the reading was different. Similar to the first question, the strength of the magnetic field read by the sensor depends on the proximity to the center. And while the center for the first question was in terms of the circle, the center in this question corresponds to the side length. As the sensor was moved outwards from the middle, it would become smaller. This reflects the fact that the magnetic field, at some point, experiences a maximum. We have shown that this maximum happens to be in the middle.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">3.By comparing the axial and radial readings, what can you conclude about the direction of the magnetic field lines inside of a solenoid?

To reiterate upon the aforementioned points, the maximum magnetic field is greatest at the center. This would show that the magnetic field lines are the most densely packed at the exact center and middle of a solenoid because of the fact that density indicates strength. We can ascertain from this that the field lines flow into and out of the center (depending on the orientation of the solenoid in the field). We can not, though, figure out the specific direction without discovering which side will be positive and which side will be negative.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">4. At what position in the solenoid should you get the greatest magnetic field strength?

Again, to reiterate here, the greatest magnetic field strength should hypothetically be at the exact center and middle (radially and longitudinally). There was error, however, because the field was basically read by the sensor as the same about a certain specified region.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">5. How does the theoretical value compare to the value at this position?

We got a maximum value of .012345T, while the theoretical value was .0136. In terms of relation, the theoretical value was the assumed value of the magnetic field of the solenoid. In other words, the theoretical value was equal to the maximum reading's value.

Overall, after completing the lab, we can conclude that our hypothesis was basically correct. We stated that the magnitude of the magnetic field would be greatest at the exact center of the solenoid. Our results showed that this was, generally, correct. The experimental maximum value of magnetic field was found to be .012345 T after using the trend line to find the maximum value. According to the trend line, this maximum "y" value would be at a distance of just about .06 m from the end of the solenoid. Due to the fact that the solenoid was measured to be .152 m, this experimental value was very close to the center but not perfect.
 * CONCLUSION**:

After calculating our theoretical maximum magnitude of magnetic field, we then used our two values to find the percent error. The percent error turned out to be 9.2279%. Although this number was not too high, there were still two sources of error that could have caused our lab to be slightly off. First, the sensor that we used to measure the magnetic field fluctuated greatly as we were trying to get our values. When we would put the sensor in the solenoid, the values would fluctuate to the point that we were just trying to be a good guess as to what the true value of the magnetic field was. Another source of error came from the fact that because our changes in length were very small from one trial to another, it caused the changes in the magnetic field to be very small. This did not make our task of collecting the magnetic field data any easier. It was also possible that the distances that we were changing were slightly off.

In the future, if we could someone get a device that would measure the magnetic field strength of a solenoid without fluctuating greatly, it would help to improve the results. In addition, if we had used a longer solenoid, we could have made the changes in distance greater, which as a result, would have helped to make better guesses for the values of the magnetic field. This lab is important to understand because in addition to being seen in the physics classroom, solenoids have many real-life applications that can be seen in our everyday lives. Solenoids help to operate many different machines and products.