Magnetic Susceptibility Experiment

Kelley Graham McNeil

Collaborators: Steven Maudrie

Abstract:

This experiment aimed to determine the magnetic susceptibility of solid graphite and titanium samples at room temperature while using the Teach Spin Gouy-Balance. These samples were subjected to a known magnetic field at varied distances, and the resulting mass of the system was measured to calculate their magnetic susceptibilities. The magnetic susceptibility of graphite was found to be and that of titanium was determined to be , indicating their diamagnetic and paramagnetic behaviors, respectively.

Introduction:
When an object is placed in a magnetic field, it is expected that the object either reacts in one of two ways or has no reaction at all. The magnetic susceptibility of a material, denoted with the symbol , is a measure of this response to the applied magnetic field. These reactions are defined as attraction and repulsion. A material that is repulsed is called diamagnetic and designated by a negative magnetic susceptibility. The opposite, an attracted material, is described as paramagnetic and has a positive value for magnetic susceptibility. This magnetic response of an object can be influenced by chemical or electrical composition, atomic structure, and density among other characteristics. For our purposes, we will focus on the changes in measured mass of a two magnet structure by how the sample responds to its magnetic field.

In this experiment, we use the Teach Spin Gouy-Balance to measure the change in mass of a permanent magnet structure set on the scale as the piquettes of titanium and graphite are inserted into this magnetic field at room temperature. An image of the Gouy-balance is shown in Figure 1, below, to provide context to the functioning of our experiment along with the magnet apparatus on the scale.

Figure 1: Image of the Tech Spin Gouy-Balance apparatus, including the labeled piquettes of samples as well as the graphite piquette passing through the magnetic field.

As can be seen, the sample piquette is introduced to the magnetic field through the top of the Gouy-balance with the lifter structure. We expect to see two different reactions from the paramagnetic and diamagnetic objects. The paramagnetic titanium will result in an initial decrease in mass as it is inserted into the permanent magnet structure, the original mass when it is directly in the middle, and finally an increase as it is lowered below the structure. The opposite will be true for the diamagnetic graphite, as it will initially result in an increase, the same original mass in the middle, and then a decrease. We will be able to explicitly demonstrate these responses in our results along with the magnetic susceptibility of both titanium and graphite.

Methods & Analysis:

Out of the sixteen different samples provided for the Magnetic Susceptibility Experiment, we chose to use only two: one paramagnetic (titanium) and one diamagnetic (graphite). We chose only two samples due to the limited working time we had for this experiment. The specific samples of titanium and graphite were chosen due to their most explicit demonstration of their magnetic responses out of all the options. Prior to measurement taking, the Gouy-balance will need to be calibrated following manual instructions. To do this, we needed to zero out the balance and then ensure that we continue to measure the same result for the permanent magnet structure.

Once we have determined that the Gouy-balance will provide accurate readings, we can begin to take our measurements. We will begin by inserting the piquette sample of titanium through the top of the Gouy-balance apparatus. With the piquette starting at the ceiling, we will begin lowering the sample using the lifter structure. Turning it from its initial position to about a quarter of a turn results in a lowered distance of 3.33 cm. We will perform a quarter turn of the lifter structure and then take a measurement of the mass reading, until the top of the sample passes below the middle point of the permanent magnet structure. An image demonstrating this method is shown below in Figure 2.

Figure 2: Close up image describing how the sample moves through the magnetic field between the two magnets.

Once we have reached this lowest height, we can remove the titanium sample. This method is repeated with the graphite sample.

Now, we have two data sets of the measured masses at each distance lowered through the permanent magnetic structure. With given and known values, we can use Equation 1, below, to determine the magnetic susceptibility of each sample.

(1)

Where is the permeability of free space (), is the gravitational constant (), is the height of the sample ( for titanium and for graphite), and is the magnetic field (). We calculate for different two values of magnetic susceptibility, and which will be positive and negative, respectfully. From these data sets, we will also create two graphs to demonstrate the opposite behavior of the mass changes as the samples move similarly through the permanent magnet structure.

Results:

We were successful in finding the magnetic susceptibility for both of our samples and in demonstrating their behaviors in graphs. We can explicitly see the behavior of a paramagnetic sample in the graph produced by the titanium sample, shown below in Figure 3.

Figure 3: Graph of measured mass for paramagnetic titanium, taken at room temperature with the Teach Spin Gouy-Balance.

This graph clearly shows the initial decrease, return to zero, and then the increase in mass that is to be expected of the paramagnetic object. The graph of the graphite’s response, in Figure 4, shows an exactly opposite behavior.

Figure 4: Graph of measured mass for diamagnetic graphite, taken at room temperature with the Teach Spin Gouy-Balance.

We can explicitly see the increase, return to zero, and then decrease in mass as the sample starts from the top of its path and down to below the permanent magnet structure. These graphs are significant as they accurately describe the opposite behaviors of the paramagnetic and diamagnetic objects. It is extremely powerful to observe that the maxima, minima, and transition points of these graphs occur at the same distance along the sample’s path. Another significant observance of these graphs is that the top of the path and the bottom of the path both return to a zero change in mass.

Additionally, let’s look at the calculated values of magnetic susceptibility. Using Equation 1 with the given values, we were able to take the average of these calculations at each change in mass. For titanium, we found a value of with an uncertainty of . Alternatively, the magnetic susceptibility for graphite was determined to be along with an uncertainty of . It is important to note that these values are dimensionless quantities that can have units of , but will remain without units for the purposes of this report. To determine the uncertainty of these values, we follow Equation 2 as a way to propagate error.

(2)

It will be noted that the general rule for uncertainty is followed for each part of the above equation, expressed with a general set of measurements in Equation 3.

(3)	; where n is the number of values for x.

Firstly, we do see the positive and negative behavior that we expected to compliment their graphs. We can compare these values to their accepted values: and . The resulting difference is found to be and , respectively.

Summary & Conclusion:

The purpose of this experiment was to determine the magnetic susceptibilities of titanium and graphite, as well as observe the behaviors of a paramagnetic and a diamagnetic object within a magnetic field. This was done using the Teach Spin Gouy-Balance, along with a technique to measure the change in mass at distance intervals rather than just at the top, the middle, and the bottom of the permanent magnetic structure. Though there are many potential variables that would influence the magnetic susceptibility, we decided to focus on how the mass of the permanent magnet structure changes. We were successful in our calculations when comparing how our values differ from the accepted values. The difference in titanium values of tells us that we are extremely successful, whereas the difference for graphite tells us that we have some tension in our result. This could be due to our ignorance of other factors contributing to magnetic susceptibility or human error. It would be advised to complete the method for more trials, as we would likely have a better calculation and better uncertainty results. Ultimately, we were successful in our demonstration of paramagnetic and diamagnetic material responses with the resulting graphs. This experiment provided a demonstration of the utility of a Gouy-balance in measuring the magnetic susceptibility and the different magnetic behaviors of objects.