Determining The Permittivity Of Free Space

The purpose of this experiment was to determine a value for the permittivity of free space. This was done by measuring the force between a charged sphere and an earthed metal plate. We obtained a value for the permittivity of free space of 12.0±0.8×10?12 Fm?1 which is in disagreement with the accepted value 8.854187 817. ..×10?12 Fm?1 . Likely causes for this were random errors in the force meter.

1. Introduction The permittivity of free space ?0 , also known as the electric constant, is a fundamental constant of physics. It describes how electric fields behave in a vacuum and therefore how charges interact in a vacuum. The value of ?0 is known to be exactly [1] 8.854187 817. ..×10?12 Fm?1 . This value is known to be exact because the constant can be expressed in terms of the magnetic constant ?0 and the speed of light c. Despite this, accurately measuring the electric constant is still important because it allows for the confirmation of theory by comparison of theoretical and experimental determinations of the constant. In our experiment a conducting sphere was charged and placed in a force meter, a metal plate acting as another charge was then introduced. The force was recorded, for different separations, and this was used to calculate ?0 . A similar set up could be used to calibrate a force meter as the force between sphere and plate is easily calculable. By adjusting a force meter so that it agreed exactly with the calculated value, a meter can be calibrated. Calibration of instruments is important for precision experiments and, therefore, methods for calibration are vital.

2. Methods Coulomb`s law [2] describes the force acting between two charged spheres. It is however, difficult to position two charged spheres such that the radial vector between the two is exactly vertical. A better method, therefore, is to charge one sphere and place this above an earthed metal plate. The plate acts as a mirror and so the system behaves identically to having two oppositely charged spheres. The metal plate was placed atop a vertically adjustable stand with a scale. Before charging the sphere it was first placed in the force meter. The force meter and sphere were then lowered to approximately the maximum height of the adjustable stand. The adjustable stand was then raised until the force meter reading changed significantly, indicating that the plate and sphere were just touching. From this initial height the adjustable stand could then be lowered to increase the separation. The sphere was then charged using a high voltage power supply with an exposed metal point. The charge on the sphere was then measured using an electrostatic amplifier both before and after the force reading were taken. The mean of the two charge readings was then calculated to get an associated charge value for each separation and force reading. For small separations the discharge of the sphere was too quick and so reliable force meter readings could not be obtained. The smallest separation used therefore was 6mm. Force readings were taken at a number of different separations starting from 6mm and extending to the maximum separation that could be achieved using the adjustable stand, 27mm. The force meter fluctuated dramatically so multiple repeats were taken for each separation, each time the charge was recorded before and after the sphere was brought over the earthed plate. A graph of charge Q divided by the square root of the force F, was plotted against twice the separation 2s plus the diameter of the sphere d. The gradient was used to obtain a value of ?0 .

4. Discussion As mentioned our value of 12.0±0.8×10?12 Fm?1 does not agree with the accepted value for the permittivity of free space. It follows therefore that there must have been errors in our experimental procedure beyond the uncertainties of the measurement devices. There are a number of potential sources of this error. The most likely cause of the discrepancy between our experimentally determined value and the accepted theoretical value are errors in the force meter readings. During the experiment the force meter was seen to fluctuate dramatically which made it difficult to determine what the value of force being measured was. Although efforts were made to compensate for this by taking several repeats of each reading, further investigation has revealed that many more repeats would have been necessary to fully eradicate this source of error. We used data logging software connected to the force meter and discovered fluctuations of several milli-newtons about the mean. These random fluctuations were of similar magnitude to the readings themselves and so therefore represent a significant random error in the force meter readings. We also attempted to determine the accuracy of the force meter. This was done by weighing small objects both using the force meter and a mass balance. The readings given by the mass balance were seen to be far more stable and consistently lower than the readings from the force meter. This was repeated for various weights. A Spearman`s rank test was performed on the results (see errors appendix) and there was found to be no correlation between the magnitude of the weight of the object and the error on the force meter reading. Again this suggests that the force meter introduced a large random error into our data. Possible causes for this could be vibrations in the laboratory as people moved around or the effects of other electrical equipment in the room. A possible way of reducing this error would be to damp the stand on which the force meter was held. Another potential source of error could be that the discharge of the sphere should be an exponential decay and therefore taking charge readings at the start and end of each force reading and finding the mean may not be accurate. However further investigation into this showed that over the time scales being considered the charge drop off was approximately linear so this error should be negligible. Finally there is potentially some error in measuring the separation due to uncertainty as to when the plate and sphere are just touching. However error from this is, again, likely to be negligible compared to force meter error.

5. Conclusions In conclusion, the purpose of our experiment was to determine a value for the permittivity of free space. This was done by measuring the force between a charged sphere and an earthed metal plate as a function of separation and of the charge. We obtained a value for ?0 of 12.0±0.8×10?12 Fm?1 which does not agree with the accepted value of 8.854187 817. ..×10?12 Fm?1 . Although efforts were made during the course of the experiment to minimise any errors, further investigation revealed that there were large random errors in the force meter readings. These errors potentially caused the discrepancy between our result and the expected value. In order to remove these errors the force meter could be connected to a data logger and set to record the force over an extended period of time. By recording the initial charge and then extrapolating the force meter data to find the initial force a more accurate value for the electric constant could be found. The meter could also be placed on a damped stand to reduce vibration errors.

References [1] S G Karshenboim. (2013). Progress in the accuracy of the fundamental physical constants: 2010 CODATA recommended values. Physics-Uspekhi. 56 (9), 883-909. [2] Hugh D. Young & Roger A. Freedman (2012). University Physics with Modern Physics. Harlow: Pearson Education Limited. [3] Hughes, I.G; Hase, T.P.A (2010). Measurements and their uncertainties. Oxford: Oxford University Press. 16