Arturo Freyre-Rodriguez and Rosalina Flores-Bermudez

Facultad de Ciencias, UNAM

Circuito Exterior

Ciudad Universitaria

Coyoacan, D.F.

Mexico

C.P. 04510

525-5622-4838

arf@hp.fciencias.unam.mx

Magnetic Dipole Interaction Analyzer

Abstract (50-75 words)

This apparatus consist of two bar magnets inside a leaning tube in a repulsive position and a motion sensor in front of the upper tube aperture. The oscillatory motion of a free magnet is analyze when it falls towards the other that is fixed in the tube. The motion graphs exhibit important results as this: one can appreciate that when a magnet falls into another, it really rebounds, almost the same as a ball rebounds on the floor.

Construction of Apparatus:

The complete Magnetic Dipole Interaction Analyzer set up is shown in the next picture.

The fundamental part of the apparatus is a plexiglass tube with two magnets inside. One magnet is suspended above the other that is fixed in the tube.

This tube is supported by a plywood structure that consists of a box made from four pieces of wood:

As you can see in the picture, we used some plastic spheres to fix the magnet in an adequate position. The box has two PVC nipples that connect it to the base of the apparatus.

With four wood rods fixed to the box, we support the motion sensor, which is fixed to another plywood piece:

The following picture shows the box, rods and motion sensor arrangement with a playwood base. Observe that the apparatus has two white PVC nuts to incline the plexiglass tube. There are two facts that are important to take into account:

1. The distance between the sensor surface of the motion sensor and the aperture of the tube should be of around 6 cm.

2. The upper surface of the fixed magnet should be upper than the second plywood surface.

Now in order to test our apparatus we connect the motion sensor to the interfase and open Data Studio asking for a graph display of the data. And CARFULLY we let drop the free magnet with the appropriated orientation for repulsion:

To extract the free magnet, one has to incline the apparatus as is shown in the following picture:

The apparatus has a protractor to measure the inclined angle, but if you have software to measure distances and angles, we suggest take digital pictures of the magnets and then take measurement in a computer like this:

Use of Apparatus:

We describe some of the activities that can be developed using the Magnetic Dipole Interaction Analyzer:

1. With one of the magnets fixed at one end of the tube and the other suspended almost freely, when the tube is holding at different angles with respect to a horizontal line, the magnetic force between the magnets is determined as a function of their separation. The procedure is taking measurements of this angle and measurements of the distant between the centers of the two magnets. (See the following pictures).

The first two pictures, shows how to take two angle values which have to be averaged to obtain the inclined angle of the tube. These measurements will be taking aligning two angle measurements in the protractor with two edges of the tube as is shown. The figures also show how to measure the distant between the centers of the two magnets.

Knowing that a magnet weights 72.8 g, the force between the magnets is determined by the relation:

The following figure shows a log-log graph for a sample of data taken using the apparatus, it illustrates the behavior of the 1/r⁴ force between two dipoles.

2. In a second activity we measure the potential energy of the interaction as a function of separation. From plots of the force vs distance, and potential energy vs distance, it is possible to demonstrate that: . To do this, we first select a position vs time graph, obtained with a motion sensor, when free magnet falls toward the other and oscillates. (See following figure)

Observe that this figure shows a horizontal line corresponding to the position of the upper surface of the bottom magnet, and when the time passes, the free magnet falls from a distance of 15 cm from the motion sensor and starts to oscillate. We select this oscillation region and invert the graph as shown in figure.

Observe that in the maxima and minima points the free magnet is stopped. That means that its total energy is only potential energy, its kinetic energy is cero. Let us observe what happens during the time interval from almost 15.45 to 15.63 seconds. (See following figure)

The graph shows that the free magnet travels 16.5 cm from a point of maximal potential energy, in which it is at rest, to a point of minimal potential energy, in which it is at rest also. As the magnetic potential difference between those points should be equal to the gravitational potential energy gained by the magnet, we multiply the position values by mg to obtain the next figure, which shows a potential energy difference of 0.118 J.

From first activity, we know that the force between two dipoles behaves as 1/r⁴. So we can calculate the value of this force between the magnets, when their distance changes in the same interval that travels the magnet. We graph this force vs distance, and determine the area under the curve as shown in the next figure.

This figure demonstrates that . That is, the magnetic potential difference energy (0.116 J), is almost equal to the integral of the force in the region that is analyzed (0.113 J).

3. In the third activity we graph the oscillation motion of the freely magnet around a stable equilibrium point, and then measured the frequency of small oscillations as a function of the magnets separation, that is different inclined angle. As is the case for any stable equilibrium, expanding the potential in a Taylor series revels that small oscillation around the equilibrium point will be approximately simple harmonic with frequency:

where:

, evaluated at the equilibrium point.

As the magnet is brought into equilibrium at successively smaller separations, the force changes more rapidly with distance, and the oscillation frequency increases. Since the gravitational force is independent of separations, evaluation of the derivative yields:

and

Thus a log-log graph of oscillation frequency versus separation will show a line with slope –5/2.

The following figure is the completed graph obtained when the free magnet falls. We select some of these graphs corresponding to different equilibrium distances. That means the tube with magnets at different inclination angle.

From this graph we select small oscillations around the equilibrium, and we obtained the next figure.

Data of the oscillation frequencies for magnets at different equilibrium distances give the result shown in the following figure, with a slop value of -2.51.

4. As last activity, it is important to comment how the measure technique that this apparatus used, enable to visualize how the special type of interaction between two magnetic dipoles, produce a motion remarkable similar to the rebound of a toy ball on the floor. This is appreciated in some above figures, but it is especially appreciated when we analyzed the corresponding velocity graph: