Magnetism+(Ampere's+Law)

**Ampere's Law and the Relationship Between Magnetic Field, Current, and Distance** Yelim Youm
 * Magnetism**


 * Purpose **

The purpose of this project was the demonstration and exploration of Ampere’s Law, one of Maxwell’s equations. Ampere's Law is an equation that expresses the magnitude of the magnetic field generated by an electric current in a closed-loop path. In my project, it is more narrowly applied to a long, straight wire. The prime goal was to obtain field data that might express the relationship between an induced magnetic field and the magnitudes of current and distance as defined by this application of Ampere's Law. The two questions at stake were these: 1) does such a relationship exist? 2) how significant or strong is it?


 * Procedure **

The first step in the experiment was the gathering of required materials. Any object that fulfilled its role adequately would have been acceptable, but the materials used in this project were: a large resistor (maximum power dissipation: 75 W, resistance: 4.99 Ω), two long wires, a meter stick, a power supplier (maximum current: 4 A, voltage: 13.8 V), a magnetic field sensor, and a Vernier LabQuest 2.



// All of the materials assembled together. From left to right: LabQuest 2, power supplier, magnetic field sensor, resistor, wires, and meter stick. //

Before beginning the experiment in earnest and collecting data, I made several calculations to ensure the safety of using the resistor and the power supplier. The equations are:





The first and second (Ohm's Law) were used in order to determine that the resistor was safe to use in the experiment. If the power dissipated in the resistor had overstepped the resistor’s maximum tolerated amount, safety for all involved could have been risked. This resistor’s power rating was 75 W, while its resistance value was 4.99 Ω. The power supplier’s maximum current was 4 A, and its voltage was 13.8 V. The calculations were as follows:













Since the power dissipated in the resistor would be 38.16 W, much below the maximum value of 75 W, I continued the experiment as is.

In the process of setting up, I first connected the resistor and the power supplier using the two long wires. Then I stretched flat one of the wires on a surface, ignoring the other as it would not serve any purpose. I placed the meter stick on the same surface, so that the side reading 0 cm would be directly beneath the wire. Having set up the main materials, I proceeded to connect the magnetic field sensor with the LabQuest.

//The set-up of the resistor and power supplier //

A diagram of the simple circuit composed of the power supplier V and resistor R

I turned on the power supplier, and then began data collection, for time intervals of 10 seconds. At a point 0.5 m away from the wire, I measured the strength of the magnetic field using the sensor, pointing it straight down at the ground. Next, I repeated the measurement 0.4 m away from the wire. At each point 0.3, 0.25, 0.2, 0.19, 0.18, 0.17, 0.16, 0.15, 0.14, 0.13, 0.12, 0.11, 0.1, 0.09, 0.08, 0.07, 0.06, 0.05, 0.04, 0.03, 0.02, 0.01, and finally 0 m away from the wire, I measured the strength of the magnetic field. The sensor took 201 individual measurements in each 10 second interval, each measurement with 4 significant figures.

In my analysis of the raw data, I transferred the data files from LabQuest to the computer. Using the accompanying data analysis software Logger Pro 3, I generated statistics such as the mean, minimum and maximum values, and the standard deviation for each data set (one data set per distance value), and some visuals (such as graphs and plots) as well.


 * Analysis **

For each distance value, the sensor took 201 samples of the strength of the magnetic field. A summary of the results obtained are as follows:


 * (m) || Mean value (mT) || ∆B (mT) ||
 * 0.5 || 0.0512 || 0.0068 ||
 * 0.4 || 0.0516 || 0.0072 ||
 * 0.3 || 0.0518 || 0.0074 ||
 * 0.25 || 0.0581 || 0.0080 ||
 * 0.2 || 0.0698 || 0.0070 ||
 * 0.15 || 0.0773 || 0.0062 ||
 * 0.14 || 0.0732 || 0.0086 ||
 * 0.13 || 0.0783 || 0.0072 ||
 * 0.12 || 0.0885 || 0.0082 ||
 * 0.11 || 0.0880 || 0.0062 ||
 * 0.1 || 0.0885 || 0.0118 ||
 * 0.09 || 0.0900 || 0.0066 ||
 * 0.08 || 0.0965 || 0.0082 ||
 * 0.07 || 0.1072 || 0.0070 ||
 * 0.06 || 0.1022 || 0.0068 ||
 * 0.05 || 0.1092 || 0.0146 ||
 * 0.04 || 0.1017 || 0.0070 ||
 * 0.03 || 0.1015 || 0.0072 ||
 * 0.02 || 0.1173 || 0.0090 ||
 * 0.01 || 0.1318 || 0.0154 ||
 * 0 || 0.1712 || 0.0204 ||

// The mean value designates the average of the 201 samples taken per distance r from the wire; // // ∆B designates the difference between the lowest and highest values. //



//The mean of each data set, plotted against distance and magnetic field //

The table and graph presented in the beginning of this Analysis section present a summary of the raw data obtained. In total, I was able to obtain 21 sets of 201 samples, with one set per distance value. Overall, the magnitude of the magnetic field increased if the distance to the wire was shortened. This trend produced a graph that somewhat resembles an exponential function. 0 m from the wire, the magnetic field was around 0.1712 mT. However, instead of decreasing steadily with the distance, the magnetic field dropped sharply in strength in the first 0.05 m or so, then decreased more and more gradually as time passed. In accordance with this, the magnetic field dropped about 0.0394 mT in the first 0.01 m away from the wire. However, in the next 0.05 m, it dropped 0.0296 mT. The magnetic field seemed to level around the 0.3 m mark. Any measurement beyond 0.3 m showed similar magnitudes of 0.0500 mT or so. Additionally, ∆B, the difference between the highest value and lowest value within a data set, never went above 0.0205, and most were around 0.0080, which is a very reasonably accurate range.

As the graph clearly shows, the magnetic field B decreases as the distance r from the wire increases. The current I is kept constant. As the magnitude of the magnetic field did not change significantly at a distance more than 0.15 m away from the wire, measurements were taken more sporadically in the distance interval from 0.5 m to 0.15 m.

The fact that the magnetic field B is directly proportional to the current I is easily explained by Ampere’s Law. However, Ampere’s Law makes no mention of distance or radius; this is because this law applies to any closed-loop path and therefore refers to a general dL instead of a more specific subtitute. The application of this law to a straight, current-carrying wire results in a new equation through the following mathematical process:



The above equation is the general form of Ampere's Law. In the case of a long, straight wire, since B is constant in a uniform circle in which all points are equidistant from the center:



Since dL simplifies to be the circumference of the circle with a radius of r, 2πr may be substituted for it. Then B may be isolated.



Based on the results, a different form of Ampere's Law was needed to better clarify the obtained data. In this form, it is easier to see the mathematical relationship between the magnetic field B and the variables I and r. Therefore, it is easier to compare this resulting equation to the obtained data.

Error may have arisen from a possible lack of precision. For example, the distance between the wire and the sensor may not have been measured totally accurately. Likewise, fluctuations in the surrounding magnetic field may have impacted readings. Possible errors, though probably present, are not likely to have had a devastating impact. Also, the chance of inaccuracy is lowered by the large number of samples taken. The experiment could be improved by fixing these errors, especially the meter stick issue.


 * Results **

//The graph from above, modified to better show the exponential shape of the graph and to display the best-fit line //

This data turned out to be in support of Ampere's Law, and this may be seen most easily in the application of Ampere's Law to a long, straight wire:



According to this version of Ampere's Law, as the current I increases, the magnetic field B increases. As the distance r increases, the magnetic field B decreases. I, the permeability of free space, and 2π remain constant. The results state something similar. According to the data, presented in the graph above, the magnetic field deteriorates more the farther away from the wire it is measured. It is strongest when the measurement is taken closest to the wire, and it is weakest when the measurement is farthest from the wire. Accordingly, it also takes the shape of an exponential function; Ampere's Law allows us to predict this, since r is in the denominator of the fraction. Also, the best-fit line of the graph (shown above) displays a downward slope, which means that the magnetic field will generally decrease as the distance increases. These observations support Ampere's Law. However, some of the data suggest something entirely different. For example, according to Ampere's Law, the magnetic field should keep decreasing as the distance increases. Ideally, there should be no spikes or confusions in the data. But the fact is that at some points, notably at 0.07 m and 0.05 m, the magnetic field was larger than at the points preceding them. Though this may deserve attention, the data in support of Ampere's Law is much more prevalent. In addition, these spikes in magnetic field may have been from unknown disturbances in the surrounding magnetic field and nothing to do with the current.

In conclusion, the data gathered in the experiment supports Ampere’s Law. It shows clearly the inversely proportional relationship between the magnetic field B and the distance from the wire r. It also shows the directly proportional relationship between the magnetic field B and the current I. Clearly and simply, the strong mathematical relationship between these variables can be well seen in the results of this experiment. Therefore the answers to the two questions states in the Purpose are: 1) yes, the relationship of Ampere's Law does exist according to the data; 2) this relationship is significant enough for the pattern to take on the form of an exponential function-- therefore, important.