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Measurement of the Magnetic Susceptibility of Liquids

Hyelee Seo

Methods of Experimental Physics Spring 2013

University of Minnesota

Abstract

The magnetic susceptibilities of the several liquid samples have been measured by applying only the order of 0.25 T magnetic fields with simple apparatus and technique. The energy balance between the magnetic energy and the gravitational potential of the liquid interaction is used to determine the susceptibility. The measurement proceeded by neglecting the surface tension effect. Water was used for the calibration for this experiment. Three CuSO₄5H₂O solutions with different densities, and NaCl have been selected in a range of diamagnetic to paramagnetic and their magnetic susceptibilities have been determined(-2.78 ± 0.76)x10^-6, (5.81± 0.71)x10^-6, (2.11 ± 0.22)x10^-5 and (-1.67 ± 0.16)x10-5 respectively.

Introduction

Due to the well-established fact that magnetic effects on nonmagnetic materials are smaller than the effect on ferromagnetic materials, most of studies on magnetic properties dealt with ferromagnetic materials. However, since repulsion of the surface of diamagnetic fluids has been observed when strong magnetic fields are applied, exploration of magnetism has been extended to nonmagnetic materials [1]. Nonmagnetic materials are mainly classified by diamagnetic and paramagnetic materials where their physical properties are in opposite to each other.

The class of diamagnetic fluids which includes the most practical fluids, such as water, is much larger than the class of paramagnetic fluids. While most inorganic compounds are paramagnetic, most organic compounds are diamagnetic. Diamagnetic fluids contain atoms or molecules that have no intrinsic magnetic moment. Therefore, under an applied static magnetic field, a small current inside each atom or molecule is induced by the change of the field and a magnetic moment is also induced and directed opposite to the applied field. This results in a repulsive magnetic force in diamagnetic fluids. Paramagnetic fluids act in an opposite way to diamagnetic fluids under the magnetic field; paramagnetic fluids have an attractive force to the non-uniform field. [2]

The produced magnetic force by the applied magnetic field causes deformation of the surface of the fluid as meeting equilibrium of the fluid by balancing the gravitational, surface tension, and magnetic force. Therefore, the deformation of fluid depends on the magnetic force which means that under the applied magnetic field, diamagnetic fluid will have a dip on its surface since it has the repulsive magnetic force while paramagnetic fluid will have a rise on its surface because of the attractive magnetic force. This physical fact gives a big clue to determine the magnetic susceptibility of the fluids.

Observing this phenomenon requires expensive equipment to apply a large magnetic field and requires a complicated data collection technique. However, Chen and Dahlberg [3] have introduced a method that requires a simple technique and minimal equipment by applying only about 0.3 Tesla of the magnetic field. From a careful study of the reflection of a laser beam from the surface of fluid, they have quantitatively analyzed the deformation of the surface by the magnetic field and measured the magnetic susceptibility of fluid. This project has adapted their method [3] in order to determine the magnetic susceptibility of fluid. The five different aqueous solutions have been selected in a range of diamagnetic to paramagnetic fluids; water, three different copper sulfate pentahydrate solutions, and sodium chloride solution (see Table1).

Table1. Solutions used in this experiment and their physical properties [5]

Theory

When the magnetic field is applied perpendicular to the surface of the sample liquids, the deformation of the liquid surface occurs and causes the energy densities on the surface to change. Since the deformation is numerically affected by the physical property of each solution, balancing the energy densities and arranging their relationship will allow determining the magnetic susceptibilities of the sample solutions. For the sake of the feasibility condition, this project ignores the effect of surface tension since Chen and Dahlberg [3] have determined the gravitational potential and magnetic energy are dominant and the surface tension energy is only about 10% of the gravitational energy. The change in the gravitational and magnetic energy densities can be determined by [3,4]:

By balancing the equations of (1) and (2) for minimum surface energy, the displacement of the surface by deformation can be given by:

Experimental Setup

Fig 1: Schematic depiction of the experimental setup:

As shown in the figure, a magnetic field on the order of 0.3T was produced by a PASCO EM-8641 Variable gap magnet which had a U-shaped yoke with two neodymium magnets each 2.5cm diameter attached at about 4cm apart from each end. The sample solutions were placed in a plastic petri dish with 9cm diameter and was filled up to the top of the dish. To avoid any contact between the dish with the liquid and the magnets, the dish was placed on non-ferromagnetic material, a wood board, which was elevated off of the magnets. A 5mW 670nm laser source was attached to a ring stand with a test tube holder and placed behind and about 15cm above the petri dish. At this point, the laser beam was adjusted to point to the surface of the liquid of center of the dish for all the measurement. The distance between the center of the petri dish and the wall, x in the figure, was fixed to 330.9cm. The height y was measured from the liquid surface level to the center of the reflection of the laser beam without applying any magnetic field.The shifted height by applying the magnetic field was measured as changing the position of the PASCO magnets. The values of y and were numerically dependent on the physical properties of the solutions. The applied magnetic field on the center of the petri dish was measured with a F.W.Bell model 4048 Gaussmeter in the perpendicular direction to the surface of the sample solution.

Data Collection Strategy and Analysis

Measuring the magnetic field with the gauss meter brought about a large random error since the measurement was very sensitive with orientation of the gauss meter probe. Moreover, it was not able to figure out how the orientation quantitatively affected on the measurement. Therefore, we decided to take the measurement with water for our calibration since water is the most common diamagnetic fluid and it is often used for calibration or correction in many researches about the magnetic susceptibilities of materials. By using the measurement data with water and the published magnetic susceptibility in Table 1, the magnetic flux density was determined for every displacement of the magnets.

As seen in the experimental setup, when the liquid surface was deformed by the magnetic field, there was a change in the incident angle of the laser beam by that caused a shift in the position of the reflected beam by 2. By simple trigonometry, the shift angle was calculated with the measurement value of the shifted total height of the reflected laser beam, the distance x, and the initial reflection angle with no magnetic field applied [3,4]:

The calculated value was used to obtain the slope of the liquid surface at every point wherever the position of the magnet is changed. The obtained slope was the derivative of the magnet position dependent on height, thus it could be integrated as a function of the magnet position, d. This integration provided the description of the deformation shape across the whole liquid surface. With that,, the vertical displacement of the liquid surface under the applied magnetic field, could be determined by a Riemann sum [3,4]:

where was the measured slope with an initial height was assumed as zero. Finally, the magnetic susceptibility of the liquids could be determined by using the equation (3) in the theory section.

<a name="Results and Discussion"></a>Results and Discussion

The measured reflection angle as a function of the magnetic displacement along the translation axis is shown in Figure 2. Since all of the selected solution has similar shape of plot as Figure 2, only the measurement data of the distilled water has been plotted. Since this figure was plotted from the calculated result of Eq. (4) mentioned above, we could tell a large error in the y-axis for each data points mostly came from measurement of the vertical displacement of the reflected beam, and x in Figure 1. Moreover, the determination of the center on the reflected beam to the wall was one of the dominant uncertainty sources since it was considered as ±0.3 cm on each measurement of the shifed height.

Figure2. The slope of the liquid surface as a function of the displacement of the magnet where the initial magnet’s position arbitrarily set to zero

Figure 3 (left) shows the corresponding magnetic field which was determined from the calibration using the water. As it shows, the maximum magnetic field was 0.27T when the magnet was placed at 0.03m in our configuratio. From these data, Figure 3 (right) was plotted for the magnetic field squared as a function of the magnet’ displacement and used for the calculation of Eq. (3).

With those data and Eq. (5), the results for the five different liquid samples are shown in Figure 4. As it shows, we could see that the two of , (II) and (III), had positive vertical displacements under the applied magnetic field thus they are paramagnetic solutions. On the other hand, the other solutions, water, NaCl solution, and CuSO4 5H2O solution (I), had negative vertical displacements when the magnetic field was applied to the solution. They are diamagnetic solution.

Figure 4. The change of the liquids’ surface level as a function of position of the magnet are shown.

<a name="Conclusion"></a> Conclusion

The measured magnetic susceptibilities for the selected solutions are listed in Table 2. According to Chen et al. [4], ignoring the surface tension gives about 10% effect on our measurement. By taking it into account, the measured result for the NaCl solution is 1.78 sigmas off while the measurement for the Cupper sulfate pentahydrate solution (I), (II), and (III) are 1.38 sigmas, 1.64 sigmas, and 0.58 sigmas away. Therefore, the result of this experiment about 1.35 sigma off by expectation. This measurement can be improved by developing more precise way for the determination of the center of the reflected beam and for the measurement of the magnetic field flux density. Considering the effect of the surface tension will make this experiment improved.

Table 2. Measured magnetic susceptibility for the selected liquid samples

<a name="References "></a>References

• 1. Noriyuki Hirota et al., “Rise and fall of surface level of water solutions under high magnetic field,” Jpn. J. Appl. Phys. 34, L991–L993 (Aug. 1, 1995).

  2. J. Huang and D. D. Gray, “Magnetic control of convection in noncondunting diamagnetic fluids.” The American Physical Society. 58, 5164-5167 (Oct. 1998).

  3. Z. Chen and E. D. Dahlberg, “Deformation of water by a magnetic field,” Phys. Teach. 49, 144 (2011).

  4. Z. Chen and E. D. Dahlberg, “A simple technique to measure the magnetic susceptibility of liquids,” American Institute of Physics. 93 (2012).

  5. CRC Handbook of Chemistry and Physics, 92^nd ed. (CRC, 2012).