Deceleration of a Conducting Disc with Eddy Currents

TABLE OF CONTENT

1. Introduction and Theory

2. Aim

3. Experimental Procedure
3.1 The main experiment
3.2 Additional experiments
3.3 Acceleration experiment
3.4 Risk assessment

4. Results
4.1 Extracting from the raw data
4.2 Accounting for friction and other energy losses
4.3 Fitting a regression line
4.4 Main experiment
4.5 Acceleration experiment

5. Interpretation
5.1 Discussion of uncertainties
5.2 Conclusion

References and Picture Credits

1. Introduction and Theory

When an electric conductor is exposed to a changing magnetic field, there is a force on each electron in the conductor, namely F = Bev, where B is the magnetic field strength and v the velocity of the particle. This causes an overall induced electromotive force which is equal to the rate of change of flux linkage, ϕ = BAN sinθ, with respect to time (Faraday’s Law). The emf is such that it opposes the direction of change of flux (Lenz’s Law):

The induced emf causes a so-called eddy current to flow, which again has a magnetic field. The magnetic field causing the induction and the field of the induces current the interfere. In situations where either the magnet or the conductor is rotating, the interference can cause a change in velocity.

This can happen in two different ways. In the first scenario, the disc rotates and a stationary magnet is fixed next to the disc. So, relative to the disc, the magnet is moving and hence there is a change in flux and as the magnetic field is at 90° to the direction of motion of the conductor, the flux linkage is maximised. This change of flux then induces an emf across the disc, and a current flows in the direction opposing the direction of rotation. This magnetic field and that of the magnet then interfere forming a retarding effect.

Otherwise, a disc can follow a spinning magnet, or a magnet fixed to an axis allowing free rotation follows a spinning conductor disc. Here, the magnetic field induced in the conductor is attracted by the other field and hence, there is an acceleration effect.

2. Aim

The aim of this coursework is to examine the relationship between the deceleration effect of the eddy currents and the initial velocity of the disc as well as the strength of the magnetic field. Additionally, I planned to compare this to the acceleration effect that can be produced using a different set-up in which a freely rotating magnet is following a motorised conductor disc.

3. Experimental Procedure

3.1 The Main Experiment

The basic idea of the first experiment was to have an aluminium disc that rotates through the field of an electromagnet and is then decelerated. The software used for tracking the data given out by the light gates was EasySense and it didn’t allow measurements with a decent sampling rate for longer than 5 minutes. Therefore, I had to find combination of voltage and number of turns of the coil in the electromagnet that produced a detectable effect in that time interval. This was a voltage between 5 to 15 volts and a coil number of 500.

illustration not visible in this excerpt

Figure 1, 2: Arrangement of the electromagnet

In order to accelerate the disc reliably in the first place, a motor has been connected via an elastic band to the disc. The disc was then accelerated for a given amount of time until it was disconnected from the motor, in order to let it rotate freely, and at the same instant, the electromagnet was switched on to start the deceleration.The potential difference across the motor was constantly 3.00 ± 0.10 V.

illustration not visible in this excerpt

Figure 3: Connection to the motor

The angular velocity was to be measured using light gates. To get these working well, I had to try out several different arrangements. My first approach was to have two light gates and a piece of cardboard of known width stuck to the edge of the disc.

illustration not visible in this excerpt

Figure 4, 5: First approach to the measurement of the angular velocity of the disc However, this method was not satisfactory as the sampling rate of the light sensors too low to detect the cardboard. It could often pass through the gates without being captured. Low spurious frequencies (aliases) were given out which is typical for sampling with too low a sampling rate. The Sampling-Theorem by C.E. Shannon states that the sampling frequency should be at least twice as high as the frequency of the signal to be measured.¹ This is the reason why CD signals have a frequency of 44,100 Hz as the maximum frequency that can be heard by the human ear is 20,000 Hz. This problem with missing passes still occured in my final set-up, but only at the very high speeds I used in some additional experiments.

To avoid this, I attached a semi-circular piece of cardboard to the edge of the disc which covered exactly one half of the circumference of the disc. I also abandoned the second light gate, as it made the data very confusing. Hence, the light gates would always measure the time taken to do half a revolution which makes it easy to calculate the angular velocity.

illustration not visible in this excerpt

Figure 6, 7: Second approach to the measurement of the angular velocity of the disc

This layout now allowed me to do the measurements I wished to do. I decided to repeat every measurement once. The plan was as follows:

illustration not visible in this excerpt

Additionally, I did two runs without the magnet, in order to find the deceleration due to friction so that I could subtract this from the total deceleration in the other experiments in order to find the deceleration due to the induction.

3.2 Additional Experiments

Moreover, I used the EasySense magnetic field sensor to not only measure the magnetic field strength produced by the magnet at different voltages, but also to investigate the magnitude to the fluctuations due to the heat produced in the coil.

Figure 8, 9: Measuring the magnetic field strength

I measured the magnetic field strength over a period of 2 minutes and at the three different voltages used. Moreover, I repeated each measurement at three different distances from the iron core.The thickness of the aluminium layer on the disc is 0.6 cm and the spacing between the disc and the iron core in the experiments was 0.3 cm. To find the force at the top, in the middle and on the bottom of the disc, I therefore measured it at 0.3, 0.6 and 0.9 cm distance, as shown in the diagram. This was only to get an idea about how much the magnetising force varies with separation.

illustration not visible in this excerpt

Figure 10, 11: Considering the thickness of the aluminium layer in the context of changing magnetic field strength.

The field to be measured was an axial field. I used both the ±10 mT and the ±100 mT sensor, accordingly to the order of magnitude of the field strength as I started using the ±10 mT sensor and only the later measurements required a greater range.

The results on the next page show the mean of the axial field in mT and the voltage in V over the period of 2 minutes.

illustration not visible in this excerpt

The values for the voltage in the left-had column show the approximate values. The uncertainty in these values is as follows:

I. 5.00 ± 0.10 V

II. 10.00 ± 0.10 V

III. 15.00 ± 0.20 V

The power supply got less reliable at potential differences above ∼12 V. As I connected an EasySense voltmeter across the magnet, I could find possible aberrations from the anticipated value and put it into consideration.

3.3 Acceleration of a disc

In order to accelerate a disc, the same physical phenomenon is used. Here, either a conducting disc follows a spinning magnet, or a magnet fixed to an axis allowing free rotation follows a spinning conductor disc (see Introduction). The spinning magnet is the better-known example, so I decided to have a go at the second one. The Physics Departement had an aluminium disc mounted to a motor, so I only had to find a way of arranging the magnets. I chose to fix two very strong coin-sized magnets to a disc by sticking two flat pieces of metal to the back and subsequently glue the magnet to the disc. The disc was fixed firmly to a screw to which a plastic ring was loosely attached on the back-side and a screw nut prevented it from slipping down. Then, the ring could be clamped to a stand, allowing free rotation. The two magnets were placed on one diameter and equally far away from the centre of the disc (2.25 cm).

illustration not visible in this excerpt

Figure 12, 13: Arranging the freely rotating magnet.

This apparatus was then put face to face with the spinning disc:

illustration not visible in this excerpt

Figure 14, 15: Arranging the discs.

As the strength of the magnets could not be changed in this experiment, the aim was to compare the speeds of the two discs rather than speed and magnetic field strength.This time, I used the laser pistol for either disc which gave the angular velocity in rotations per minute (RPM). As they had a “hold”-button, I could press it simultaneously and hence take measurements of the velocity every 20 seconds. Reflecting tape had to be attached to the two discs. During the first measurements, the angular velocity of the magnets was about three times higher than that of the the motor-driven disc, which had to be an experimental error. I concluded, that the shiny surface of the disc to which the magnets were attached probably confused the instrument. Therefore, I fitted white paper along the circumference of the reflecting tape (see figures 15 and 18) and subsequently obtained reasonable results.

[...]

¹ From: An Introduction to the Sampling Theorem, http://www.ti.com/lit/an/snaa079c/snaa079c.pdf, page 2