A Measure of Magnetic Moment and Earth’s Magnetic Field

From the Relationship of Frequency and Current

 

 

Amanda Hickman

 

Harvey Mudd College

 

 

30 November, 2004

 

 

 

Abstract:

 

            An alternate method of determining the magnetic moment of a bar magnet and the earth’s magnetic field was studied. The frequency of oscillation of a bar magnet placed in the center of a current coil with a magnetic field parallel to earth’s magnetic field can be measured as a function of the current in the loop. The data was compared to a theoretical relationship derived from Maxwell’s equations of electromagnetism. A reduced χ2 of 0.5 suggests that although the data is in agreement with the theory the error in the measurements may have been overestimated.

Introduction

            The determination of the magnetic field of earth, Bh, and the magnetic dipole moment of a material provide insight into the nature of how the material interacts in its environment and in the presence of other magnetic fields. The magnetic dipole moment (M) is defined as two times the monopole strength of each pole times the distance between the two poles. The value of M provides insight into the strength of the magnetic material which is useful in the design of electronic generators and motors (Halliday). Understanding the magnetic field of the earth at any given point is necessary since it creates a force and torque on magnetic materials.

            An experiment that determined the magnetic dipole moment of a bar magnet and the earth’s magnetic field was previously conducted. However, in the original experiment the method used to determine the values of M and Bh also predicted the length between the poles of the magnet to be greater than the physical length of the magnet. When these values were then used to calculate the current in the coil, there was an average percent difference of 14%. These inconsistencies required a more accurate determination of M and Bh.

            In order to more accurately determine the values of M and Bh, a new experiment was conducted. A bar magnet was suspended from a small beaker and centered inside a current loop with 14 turns. The coil was oriented so that its magnetic field was parallel to the earth’s magnetic field. The frequency of oscillation of the bar magnet was then measured at various levels of current in the coil.

Theory

            Using Maxwell’s Equations the magnetic fields produced by the coil and the earth can be quantified resulting in an expression that describes the relationship between the frequency of an oscillator and the current in the coil. The bar magnet experiences a torque from the combined magnetic fields of the coil and the earth, given by

                                                                        .                                                               (1)

From rotational mechanics this expression can be simplified further when  is perpendicular to  so,

                                                               =                                            (2)

where I is the moment of inertia of the bar magnet and f is the frequency of torsional oscillation of the bar magnet. Furthermore the total magnetic field is the sum of the magnetic field from earth and the coil so,

                                                            =  =                                  (3)

Ampere’s Law tells us that the magnetic field produced by a current loop of N turns is given by,

                                                                                                                           (4)

where R is the radius of the loop,  is the permeability constant defined as , and the direction of   is arranged parallel to . By combining equations (3) and (4) an expression for frequency as a function of current is given by,

                                                                                                        (5)

Experiment

            Equation (5) predicts that the relationship between the square of the frequency and the current in the loop is linear. As such the magnetic dipole moment of the bar magnet and the magnetic field of the earth can be determined from the slope and y-intercept of the line. An experiment was designed to test the period of a bar magnet suspended by a thread of negligible torsion in the center of a current loop attached to a power supply and ammeter. The magnetic field from the loop can be adjusted using the power supply to regulate the amount of current flowing through the loop. As diagrammed in Figure 1, an inverted beaker sitting on a magnetometer rail was used to suspend the bar magnet at the center of the current loop.

            Figure 1 Schematic view of experimental setup to test the period of oscillation of a bar   magnet in the magnetic field of a coil and earth. The magnetometer is aligned so that the

            two fields are parallel. The current is adjusted at the power supply and the bar magnet is

            disturbed by another magnet.

 

            The experiment was designed so that the two magnetic fields were parallel and thus added to each other. In order to align the magnetic fields properly, the current must not deflect the magnet but instead oscillate the magnet about the same point as when the current was off. Once the apparatus was properly aligned, ten periods of the bar magnet at a displacement angle of 15° ± 5° was measured from 0 to 0.800 A of current in the loop. The period was measured with reasonable certainty by measuring the period of ten oscillations and effectively reducing the uncertainty in estimating the exact completion of one period.

 

 

Results

            The graph in Figure 2 is in agreement with equation (5). The relationship between the square of the frequency and the current in the coil is linear. The reduced chi squared of 0.5 suggests that the data is well-represented by the fit, but the uncertainty may be overestimated. It was determined that the larger source of error was in the frequency. The uncertainty was thus calculated by propagating the uncertainty in the measure of ten periods.

            Using the curve fit option in Kaleidagraph, the values of M and Bh were determined by fitting the data to equation (5). Using this fit the magnetic dipole moment M was found to equal 0.62 ± 0.01 Am2, and the magnetic field of earth in the laboratory where the experiment was conducted was found to equal. These values are in agreement with the suggested limits given in the laboratory manual.        

                        Figure 2 A plot of the square of the frequency as a function of current                                        measured in amperes. The error bars reflect error in the frequency that results by

                        propagating uncertainty in the measure of the period. The linear fit has a slope of

                        0.116 ±0.002, y-intercept equal to 0.0521 ±0.0005, and reduced chi square of 0.5.

 

            In addition to the suggested values, the newly calculated values of M and Bh can be compared to the values calculated in the original experiment. Plotting the angle a compass was deflected as a function of the distance the bar magnet was placed from the compass resulted in Bh= and M = 0.71 ± 0.02 Am2 (Physics 2-1). The percent difference between the originally calculated values and those calculated in the new experiement is 15.5% in M and 5.0%. in Bh.  It is important to note that the fit from the original experiment determined the distance between the poles of the magnet to be longer than the physical length of the magnet which would suggest that the original calculation of M and Bh are not reliable.

            Another way to compare the original values of M and Bh to the newly calculated values is to replace the value of M and Bh in the calculation of current from the original experiment with the newly derived values.  After recalculating the current, it was compared to the current read from the ammeter. The calculation in the original experiment yielded an average percent difference of 14%(Lab Book 30). Using the newly calculated values of M = 0.62 ± 0.01 Am2 and Bh = , the average percent difference is 1.9%. This value supports the conclusion that the newly calculated values of M and Bh are more accurate than those previously determined.

Conclusions

            Taking advantage of the torque exerted on a bar magnet in a magnetic field, an accurate measure of the magnetic dipole moment and the magnetic field of earth was determined. As predicted by equation (5), the relationship between the square of the frequency of the magnet centered in a coil where the two fields are parallel is linear. From the slope and y-intercept of a curve fit, the value of M = 0.62 ± 0.01 Am2 and Bh = were determined.

            In further experimentation, a more accurate determination of the error in each period measurement should be addressed. By collecting more points for each current, a standard deviation could serve as the uncertainty in the period. This measure could then be propagated to determine the error in the square of the frequency. However, further experimentation is not necessary to conclude that the method used to determine M and Bh is more accurate than the method used in the original experiment. The dramatic reduction in error when calculating the current shows that this method of determining frequency as a function of current yields accurate values for M and Bh.

Acknowledgements

            I would like to recognize my lab partner, Michael Pugh, in both the original experiment and in the further study of the magnetic dipole moment and magnetic field of earth.

References

Halliday, David, Robert Resnick, and Kenneth S. Krane.  Physics.  5th edition.  Vol. 2. New

            York: John Wiley and Sons, Inc., 2002. 2 vols.

Lab Book.  Electricity and Optics Laboratory. Harvey Mudd College. Fall, 2004.

Physics 53 Electricity and Optics Laboratory Course Manual.  Harvey Mudd College.  Fall,

            2004.