Lab Report Coulomb's Law

Coulomb’s Law Introduction The objective of this laboratory experiment was to observe and determine how the electrostatic force between two charged, conducting spheres depends on the distance between these two spheres and the charge placed on them. Coulomb’s Law states that when two point charges, Q1 and Q2, are separated by a distance R from each other the magnitude of electrostatic force, F, between them is given by an equation F = (k*Q1*Q2) / R2, where k is the proportionality constant, specifically known as Coulomb’s Law constant, and is equal to 8.99*109 N*m2/C2, thus giving the final units of N. This particular laboratory experiment uses a special apparatus, to confirm the validity of this law, named PASCO Coulomb Balance. The Coulomb Balance is a very sensitive torsion balance consisting of two spheres, with one being attached to a rod that is counter-balanced and suspended on a thin torsion wire, and the other being attached to a special sliding assembly that is free to slide along the axis, thus increasing or decreasing the distance between the two spheres. The spheres are made of a conductive material and can be easily given a charge by the means of a powerful and stable kilovolt power supply. The power supply has a charging probe to put a charge on a sphere and a grounding probe to ground the charge away from the sphere. The distance between the centers of the two spheres is measured by the means of the ruler, attached to the sliding assembly, as the spheres are brought closer together or farther away from each other. When an identical, positive charge is given to both spheres, from the laws of physics, we would expect the charges to repel each other. As the stationary, uniformly charged spheres is brought closer to the sphere suspended on a torsion wire, the suspended sphere rotates away from the stationary sphere through a certain angle ϴ, which displaces the suspended sphere from its equilibrium position. In order to bring the sphere back to its equilibrium position one must turn the dial, which is attached to the torsion wire, in the opposite direction of the initial angular displacement. The dial has angular gratings on it, thus allowing the experimenter to record the value of the displaced angle. As we can see from this setup, the angle ϴ is directly proportional the electrostatic force F, thus ϴ ∝ F → ϴ ∝ (k*Q1*Q2) / R2. This experiment does not require the calculation of the electrostatic forces between two charges, thus we are not given the proportionality constant that would relate force, F, to the angular displacement, ϴ, as well as we are not given the charge magnitudes in Coulomb, but rather in arbitrary values of 1, 2, 3, 4, 5, and 6. In reality these values also have the units, and the units are kV, or kilovolts, but in that case we would also want to the mathematical relationship between Coulombs and Volts that was not provided in this experiment. Even though we were not able to directly calculate the magnitude of the electrostatic forces between the two spheres, from the proportionality relationship, we can still graphically show the underlying dependency of the force F ∝ ϴ on the distance R between the spheres and the charge Q on each conducting sphere. It is very important to note that when we are looking at Coulomb’s Law, we are considering two point particles that have a specific charge on them and are positioned a certain distance apart from each other. In our experiment we are looking at two spheres that have a finite radius and define our distance of separation, R, as a distance between the centers of each sphere. Now due to the fact that charges on a sphere are uniformly dispersed on the surface of the spheres, we must account for the fact that the actual distance is not between the centers of the spheres, but it is more like an average of all the distances between each random point on the surface of the sphere, where every distance between two random points can be slightly greater, slightly less, or equal to R. To account for this uncertainty, we introduce the new value R’ = R (1 + 2.3β), with β equivalent to β = a3 / R3, where a is the radius of the sphere equivalent to a = 1.90 cm. Thus the Coulomb’s Law in our case would be given by an equation F = (k*Q1*Q2) / R’2. In this experiment we will attempt to graphically verify that the value of the exponent of R is in fact 2, and that the force is directly proportional to the product of the two charges. From the nature of proportionality, we would expect the second graph to be a straight line that goes through the origin, which we derived from the relation F = (k/R2)*Q1Q2, where k/R2 is a fixed value and is a slope of our graph. As previously mentioned the apparatus of this experiment consisted of PASCO Coulomb Balance and a Kilovolt Power Supply. Procedures and Calculations The first part of the experiment was focused on determining the later graphically showing the dependence of F ∝ ϴ on R’. The charge placed on each sphere is kept constant at 6 kV, and the only thing that is varied is the distance between the centers of the spheres. We were to obtain the angular displacement of the suspended sphere for the values of R at 20 cm, 14 cm, 10 cm, 7 cm, 6 cm, and 5 cm. Using equation R’ = R (1 + 2.3β) we calculated the value of R’ for each value of R. In order to construct the graphs, we needed to obtain the values of 1/(R’)2, ln R’, and ln ϴ. The values of R, ϴ , R’, 1/(R’)2, ln R’, and ln ϴ are tabulated in Table 1. Using the obtained values from Table 1 we plotted ln ϴ vs. ln R’ (see Graph 1). If we were to write down the linear equation of the line going through the data points we would write lnϴ = ln b + n*ln R’, where ln b is the point of intersection with the y-axis. Removing the natural logarithm from this equation we obtain ϴ = b (R’)n and observe that this equation is very similar to Coulomb’s Law, where b = αkQ1Q2 and α being the proportionality constant to equate ϴ and F, given by ϴ = αF. Provided by the previous relationships we obtain ϴ = (αkQ1Q2) / (R’)2 = (αkQ1Q2)*(R’)-2 If our experiment is in accordance with Coulomb’s Law, we should expect n, the slope of the trend line in Graph 1 given by equation lnϴ = ln b + n*ln R’, to be equal to n = -2. From Graph 1 we obtain the slope of the trend line to be equal to n = -1.7833. The percent error difference between the expected and obtained values is 10.8%. Next, using values from Table 1, we plotted ϴ vs. 1/(R’)2 (see Graph 2). This graph provides us with another viable piece of information. We can clearly see that as R → ∞, our force F ∝ ϴ → 0. Our trend line through the points on the graph seems to be going through the origin, but in reality it will never get to zero, because there will be always some component of force between infinitely far charge particles. This graph represents the inversely proportional dependence of the force F on the squared distance R2 between the two charges. In the second part of the experiment we are told to vary the charges on the spheres, while keeping the separation distance R between them constant at 5 cm. By varying the charges on the spheres, we observe the dependence of the electrostatic force on the charge when everything else is kept constant. In our experiment both spheres receive the identical charge, thus Q1*Q2 = Q2. We are asked to graphically represent the proportional dependence of the electrostatic force F ∝ ϴ on the product of the charges of particles Q1*Q2 = Q2. Since we are not dealing with the charges that could be represented in Coulombs, we are using the arbitrary values of 1 through 6, with 6 corresponding to the value of 6 on the kilovolt power supply. The product of charges and the corresponding degrees of torsion are presented in Table 2. As we look at Graph 3 we can clearly observe that the trend line passing through our data points is linear and approximately goes through the origin of the graph. As mentioned in the introductory part of this report, we were expecting to see a straight line passing through the origin. From the following relation F = (k/R2)*Q1Q2, we would expect k/R2 to be a fixed value and represent a slope of our graph if we were to plot F vs. Q1*Q2 (in Coulombs). Since we are dealing with the angular displacement ϴ that is proportional to our F, such that ϴ = αF, the slope of our graph would be αk/R2, with a proportionality constant α accounting for the fact that we are not dealing with Coulomb's, but rather with some values. The reason we would expect this graph to be a straight line is due to the fact that our slope is always constant, as we are keeping the separation distance R constant. The reason we would expect this graph to pass through the origin is due to the fact that when product of two charges is equal to 0, the magnitude of electrostatic force F would also be 0. Conclusion In this laboratory experiment we were able to observe and graphically determine that the electrostatic force between the two charged spheres is proportionally dependent on the product of the two charges, as well as inversely proportionally dependent on the squared separation distance between the two charges. Also, we had graphically confirmed that the exponent value of R in Coulomb’s Law is in fact equal to 2. The result obtained from our graphical representation was in agreement with our hypothesis with the percent error difference of 10.8% between the expected value and the observed value. Table 1 Trial 1 2 3 R (cm) 20 14 10 ϴ (°) 42 72 123 R’ (cm) 20.04 14.08 10.16 1/(R’)2 0.00249 0.00504 0.00969 ln R’ 2.9977 2.6448 2.3185 ln ϴ 3.7377 4.2767 4.8122 4 5 6 7 6 5 202 310 425 7.32 6.44 5.63 0.01866 0.02411 0.03155 1.9906 1.8625 1.7281 5.3083 5.7366 6.0521 4 9 135 5 4 55 6 1 25 Table 2 Trial Q 1 * Q2 ϴ (°) 1 36 425 2 25 320 3 16 215 Graph 1 Graph 2 Graph 3 Questions: 1.) When we are placing two charges of the same sign on two different spheres, from the laws of physics, we would expect these spheres to repel each other. The charges of one sphere want to be as far apart from each other as they want to be from the charges of the second sphere, especially when these spheres are brought in a close proximity to each other. Following from this assumption we would expect the charge density to shift to the side of the sphere, thus the new average center of charges would be farther away from the actual center of the sphere. The closer they are brought together, the closer this point would be to the surface of the sphere. 2.) Following the derivation from Question 1, we would expect the new average center of charges to shift towards the surface of the sphere when the two spheres are brought together. When the spheres are far apart the average center of charges is slightly off the center of the sphere, but as they are brought closer together to the point where distance of separation between the spheres R ≈ 2a, the average center of charges is almost at the surface of the sphere, where R’ ≈ 4a, thus the actual distance is almost twice as large as it would be if we were to take the distance between the centers of the spheres. 3.) The possible errors that could have arose throughout this laboratory experiment could be related to a possible charge leakage to the negatively charged surroundings or unequally positively charged surrounding objects. This issue could be corrected by providing a constant source of charge to the spheres. The torsion wire was constantly varying due to slight shaking and wind as well as air from the air conditioner, such that it constantly affected our equilibrium point. This issue could was corrected by changing the position of the equipment to face away from the source of the air.